[PDF] Geometrization of the local Langlands correspondence

Abstract

Following the idea of [Far16], we develop the foundations of the geometric Langlands program on the Fargues--Fontaine curve. In particular, we define a category of \ell-adic sheaves on the stack \mathrm{Bun}_G of G-bundles on the Fargues--Fontaine curve, prove a geometric Satake equivalence over the Fargues--Fontaine curve, and study the stack of L-parameters. As applications, we prove finiteness results for the cohomology of local Shimura varieties and general moduli spaces of local shtukas, and define L-parameters associated with irreducible smooth representations of G(E), a map from the spectral Bernstein center to the Bernstein center, and the spectral action of the category of perfect complexes on the stack of L-parameters on the category of \ell-adic sheaves on \mathrm{Bun}_G.

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GGeometrization of the local Langlands correspondence

Laurent Fargues and Peter Scholze a r X i v : . [ m a t h . R T ] F e b bstract. Following the idea of [

Far16 ], we develop the foundations of the geometric Langlandsprogram on the Fargues–Fontaine curve. In particular, we deﬁne a category of (cid:96) -adic sheaves onthe stack Bun G of G -bundles on the Fargues–Fontaine curve, prove a geometric Satake equivalenceover the Fargues–Fontaine curve, and study the stack of L -parameters. As applications, we proveﬁniteness results for the cohomology of local Shimura varieties and general moduli spaces of localshtukas, and deﬁne L -parameters associated with irreducible smooth representations of G ( E ), amap from the spectral Bernstein center to the Bernstein center, and the spectral action of thecategory of perfect complexes on the stack of L -parameters on the category of (cid:96) -adic sheaves onBun G . ontents Chapter I. Introduction 7I.1. The local Langlands correspondence 7I.2. The big picture 8I.3. The Fargues–Fontaine curve 18I.4. The geometry of Bun G (cid:96) -adic sheaves on Bun G L -parameters 32I.9. Construction of L -parameters 34I.10. The spectral action 37I.11. The origin of the ideas 39I.12. Acknowledgments 43I.13. Notation 44Chapter II. The Fargues–Fontaine curve and vector bundles 45II.1. The Fargues–Fontaine curve 47II.2. Vector bundles on the Fargues–Fontaine curve 56II.3. Further results on Banach–Colmez spaces 74Chapter III. Bun G | Bun G | D ´et (Bun G ) 165 V.1. Classifying stacks 166V.2. ´Etale sheaves on strata 169V.3. Local charts 171V.4. Compact generation 175V.5. Bernstein–Zelevinsky duality 177V.6. Verdier duality 178V.7. ULA sheaves 180Chapter VI. Geometric Satake 183VI.1. The Beilinson–Drinfeld Grassmannian 186VI.2. Schubert varieties 192VI.3. Semi-inﬁnite orbits 197VI.4. Equivariant sheaves 202VI.5. Aﬃne ﬂag variety 204VI.6. ULA sheaves 207VI.7. Perverse Sheaves 211VI.8. Convolution 219VI.9. Fusion 221VI.10. Tannakian reconstruction 226VI.11. Identiﬁcation of the dual group 230VI.12. Cartan involution 234Chapter VII. D (cid:4) ( X ) 237VII.1. Solid sheaves 239VII.2. Four functors 246VII.3. Relative homology 251VII.4. Relation to D ´et D lis (Bun G ) 265Chapter VIII. L -parameter 271VIII.1. The stack of L -parameters 273VIII.2. The singularities of the moduli space 276VIII.3. The coarse moduli space 280VIII.4. Excursion operators 285VIII.5. Modular representation theory 287Chapter IX. The Hecke action 307IX.1. Condensed ∞ -categories 310IX.2. Hecke operators 311IX.3. Cohomology of local Shimura varieties 314IX.4. L -parameter 317IX.5. The Bernstein center 317IX.6. Properties of the correspondence 320IX.7. Applications to representations of G ( E ) 324 ONTENTS 5

Chapter X. The spectral action 331X.1. Rational coeﬃcients 332X.2. Elliptic parameters 337X.3. Integral coeﬃcients 339Bibliography 343HAPTER I

Introduction

I.1. The local Langlands correspondence

The local Langlands correspondence aims at a description of the irreducible smooth representa-tions π of G ( E ), for a reductive group G over a local ﬁeld E . Until further notice, we will simplifyour life by assuming that G is split; the main text treats the case of general reductive G , requiringonly minor changes.The case where E is archimedean, i.e. E = R or E = C , is the subject of Langlands’ classicalwork [ Lan89 ]. Based on the work of Harish-Chandra, cf. e.g. [

HC66 ], Langlands associates toeach π an L -parameter, that is a continuous homomorphism ϕ π : W E → (cid:98) G ( C )where W E is the Weil group of E = R , C (given by W C = C × resp. a nonsplit extension 1 → W C → W R → Gal( C / R ) → (cid:98) G is the Langlands dual group. This is the split reductive group over Z whose root datum is dual to the root datum of G . The map π (cid:55)→ ϕ π has ﬁnite ﬁbres, and a lot ofwork has been done on making the ﬁbres, the so-called L -packets, explicit. If G = GL n , the map π (cid:55)→ ϕ π is essentially a bijection.Throughout this paper, we assume that E is nonarchimedean, of residue characteristic p > F q . Langlands has conjectured that one can still naturally associate an L -parameter ϕ π : W E → (cid:98) G ( C )to any irreducible smooth representation π of G ( E ). In the nonarchimedean case, W E is thedense subgroup of the absolute Galois group Gal( E | E ), given by the preimage of Z ⊂ Gal( F q | F q )generated by the Frobenius x (cid:55)→ x q . This begs the question where such a parameter should comefrom. In particular,(1) How does the Weil group W E relate to the representation theory of G ( E )?(2) How does the Langlands dual group (cid:98) G arise?The goal of this paper is to give a natural construction of a parameter ϕ π (only depending ona choice of isomorphism C ∼ = Q (cid:96) ), and in the process answer questions (1) and (2). I.2. The big picture

In algebraic geometry, to any ring A corresponds a space Spec A . The starting point of ourinvestigations is a careful reﬂection on the space Spec E associated with E . Note that the group G ( E ) is the automorphism group of the trivial G -torsor over Spec E , while the Weil group of E isessentially the absolute Galois group of E , that is the (´etale) fundamental group of Spec E . Thus, G ( E ) relates to coherent information (especially G -torsors) on Spec E , while W E relates to ´etaleinformation on Spec E . Moreover, the perspective of G -torsors is a good one: Namely, for generalgroups G there can be nontrivial G -torsors E on Spec E , whose automorphism groups are thenthe so-called pure inner forms of Vogan [ Vog93 ]. Vogan realized that from the perspective of thelocal Langlands correspondence, and in particular the parametrization of the ﬁbres of π (cid:55)→ ϕ π , itis proﬁtable to consider all pure inner forms together; in particular, he was able to formulate aprecise form of the local Langlands conjecture (taking into account the ﬁbres of π (cid:55)→ ϕ π ) for pureinner forms of (quasi)split groups. All pure inner forms together arise by looking at the groupoidof all G -bundles on Spec E : This is given by[ ∗ /G ](Spec E ) = (cid:71) [ α ] ∈ H (Spec E,G ) [ ∗ /G α ( E )] , where H (Spec E, G ) is the set of G -torsors on Spec E up to isomorphism, and G α the correspondingpure inner form of G . Also, we already note that representations of G ( E ) are equivalent to sheaveson [ ∗ /G ( E )] (this is a tautology if G ( E ) were a discrete group; in the present context of smoothrepresentations, it is also true for the correct notion of “sheaf”), and hence sheaves on[ ∗ /G ](Spec E ) = (cid:71) [ α ] ∈ H (Spec E,G ) [ ∗ /G α ( E )] , are equivalent to tuples ( π α ) [ α ] of representations of G α ( E ). Looking at the ´etale side of the correspondence, we observe that the local Langlands correspon-dence makes the Weil group W E of E appear, not its absolute Galois group Gal( E | E ). Recall that W E ⊂ Gal( E | E ) is the dense subgroup given as the preimage of the inclusion Z ⊂ Gal( F q | F q ) ∼ = (cid:98) Z ,where Gal( F q | F q ) is generated by its Frobenius morphism x (cid:55)→ x q . On the level of geometry, thischange corresponds to replacing a scheme X over F q with the (formal) quotient X F q / Frob.In the function ﬁeld case E = F q (( t )), we are thus led to replace Spec E by Spec ˘ E/ϕ Z where˘ E = F q (( t )). We can actually proceed similarly in general, taking ˘ E to be the completion of themaximal unramiﬁed extension of E . For a natural deﬁnition of π , one then has π (Spec( ˘ E ) /ϕ Z ) = W E — for example, Spec ˘ E → Spec( ˘ E ) /ϕ Z is a W E -torsor, where ˘ E is a separable closure.Let us analyze what this replacement entails on the other side of the correspondence: Lookingat the coherent theory of Spec ˘ E/ϕ Z , one is led to study ˘ E -vector spaces V equipped with ϕ -linearautomorphisms σ . This is known as the category of isocrystals Isoc E . The category of isocrystalsis much richer than the category of E -vector spaces, which it contains fully faithfully. Namely, by Needless to say, the following presentation bears no relation to the historical developments of the ideas, whichas usual followed a far more circuitous route. We will discuss some of our original motivation in Section I.11 below. The point of replacing [ ∗ /G ( E )] by [ ∗ /G ](Spec E ) was also stressed by Bernstein. .2. THE BIG PICTURE 9 the Dieudonn´e–Manin classiﬁcation, the category Isoc E is semisimple, with one simple object V λ for each rational number λ ∈ Q . The endomorphism algebra of V λ is given by the central simple E -algebra D λ of Brauer invariant λ ∈ Q / Z . Thus, there is an equivalence of categoriesIsoc E ∼ = (cid:77) λ ∈ Q Vect D λ ⊗ V λ . Here, if one writes λ = sr with coprime integers s, r , r >

0, then V λ ∼ = ˘ E r is of rank r with σ givenby the matrix  . . . π s  where π ∈ E is a uniformizer.Above, we were considering G -torsors on Spec E , thus we should now look at G -torsors in Isoc E .These are known as G -isocrystals and have been extensively studied by Kottwitz [ Kot85 ], [

Kot97 ].Their study has originally been motivated by the relation of isocrystals to p -divisible groups and ac-cordingly a relation of G -isocrystals to the special ﬁbre of Shimura varieties (parametrizing abelianvarieties with G -structure, and thus p -divisible groups with G -structure). Traditionally, the setof G -isocrystals is denoted B ( E, G ), and for b ∈ B ( E, G ) we write E b for the corresponding G -isocrystal. In particular, Kottwitz has isolated the class of basic G -isocrystals; for G = GL n , a G -isocrystal is just a rank n isocrystal, and it is basic precisely when it has only one slope λ . Thereis an injection H (Spec E, G ) (cid:44) → B ( E, G ) as any G -torsor on Spec E “pulls back” to a G -torsor inIsoc E ; the image lands in B ( E, G ) basic . For any b ∈ B ( E, G ) basic , the automorphism group of E b isan inner form G b of G ; the set of such inner forms of G is known as the extended pure inner formsof G . Note that for G = GL n , there are no nontrivial pure inner forms of G , but all inner formsof G are extended pure inner forms, precisely by the occurence of all central simple E -algebrasas H λ for some slope λ . More generally, if the center of G is connected, then all inner forms of G can be realized as extended pure inner forms. Kaletha, [ Kal14 ], has extended Vogan’s resultson pure inner forms to extended pure inner forms, giving a precise form of the local Langlandscorrespondence (describing the ﬁbres of π (cid:55)→ ϕ π ) for all extended pure inner forms and therebyshowing that G -isocrystals are proﬁtable from a purely representation-theoretic point of view. Wewill actually argue below that it is best to include G b for all b ∈ B ( E, G ), not only the basic b ; theresulting automorphism groups G b are then inner forms of Levi subgroups of G . Thus, we are ledto consider the groupoid of G -torsors in Isoc E , G -Isoc ∼ = (cid:71) [ b ] ∈ B ( E,G ) [ ∗ /G b ( E )] . Sheaves on this are then tuples of representations ( π b ) [ b ] ∈ B ( E,G ) of G b ( E ). The local Langlandsconjecture, including its expected functorial behaviour with respect to passage to inner forms andLevi subgroups, then still predicts that for any irreducible sheaf F — necessarily given by anirreducible representation π b of G b ( E ) for some b ∈ B ( E, G ) — one can associate an L -parameter ϕ F : W E → (cid:98) G ( C ). To go further, we need to bring geometry into the picture: Indeed, it will be via geometry that(sheaves on the groupoid of) G -torsors on Spec ˘ E/ϕ Z will be related to the fundamental group W E of Spec ˘ E/ϕ Z . The key idea is to study a moduli stack of G -torsors on Spec ˘ E/ϕ Z .There are several ways to try to deﬁne such a moduli stack. The most naive may be thefollowing. The category Isoc E is an E -linear category. We may thus, for any E -algebra A , consider G -torsors in Isoc E ⊗ E A . The resulting moduli stack will then actually be (cid:71) b ∈ B ( E,G ) [ ∗ /G b ] , an Artin stack over E , given by a disjoint union of classifying stacks for the algebraic groups G b . This perspective is actually instrumental in deﬁning the G b as algebraic groups. However, itis not helpful for the goal of further geometrizing the situation. Namely, sheaves on [ ∗ /G b ] arerepresentations of the algebraic group G b , while we are interested in representations of the locallyproﬁnite group G b ( E ).A better perspective is to treat the choice of F q as auxiliary, and replace it by a general F q -algebra R . In the equal characteristic case, we can then replace ˘ E = F q (( t )) with R (( t )). This carriesa Frobenius ϕ = ϕ R acting on R . To pass to the quotient Spec R (( t )) /ϕ Z , we need to assume thatthe Frobenius of R is an automorphism, i.e. that R is perfect. (The restriction to perfect R willbecome even more critical in the mixed characteristic case. For the purpose of considering (cid:96) -adicsheaves, the passage to perfect rings is inconsequential, as ´etale sheaves on a scheme X and on itsperfection are naturally equivalent.) We are thus led to the moduli stack on perfect F q -algebras G - I soc : { perfect F q -algebras } → { groupoids } : R (cid:55)→ { G -torsors on Spec R (( t )) /ϕ Z } . These are also known as families of G -isocrystals over the perfect scheme Spec R . (Note the curly I in G - I soc, to distinguish it from the groupoid G -Isoc.)This deﬁnition can be extended to the case of mixed characteristic. Indeed, if R is a perfect F q -algebra, the analogue of R [[ t ]] is the unique π -adically complete ﬂat O E -algebra (cid:101) R with (cid:101) R/π = R ;explicitly, (cid:101) R = W O E ( R ) = W ( R ) ⊗ W ( F q ) O E in terms of the p -typical Witt vectors W ( R ) or theramiﬁed Witt vectors W O E ( R ). Thus, if E is of mixed characteristic, we deﬁne G - I soc : { perfect F q -algebras } → { groupoids } : R (cid:55)→ { G -torsors on Spec( W O E ( R )[ π ]) /ϕ Z } . We will not use the stack G - I soc in this paper. However, it has been highlighted recently amongothers implicitly by Genestier–V. Laﬀorgue, [ GL17 ], and explicitly by Gaitsgory, [

Gai16 , Section4.2], and Zhu, [

Zhu20 ], and one can hope that the results of this paper have a parallel expression interms of G - I soc, so let us analyze it further in this introduction. It is often deﬁned in the followingslightly diﬀerent form. Namely, v-locally on R , any G -torsor over R (( t )) resp. W O E ( R )[ π ] is trivialby a recent result of Ansch¨utz [ Ans18 ]. Choosing such a trivialization, a family of G -isocrystals isgiven by some element of LG ( R ), where we deﬁne the loop group LG ( R ) = G ( R (( t ))) (resp . LG ( R ) = G ( W O E ( R )[ π ])) . Changing the trivialization of the G -torsor amounts to σ -conjugation on LG , so as v-stacks G - I soc = LG/ Ad ,σ LG is the quotient of LG under σ -conjugation by LG . .2. THE BIG PICTURE 11 The stack G - I soc can be analyzed. More precisely, we have the following result. Theorem

I.2.1 . The prestack G - I soc is a stack for the v-topology on perfect F q -algebras. Itadmits a stratiﬁcation into locally closed substacks G - I soc b ⊂ G - I soc for b ∈ B ( E, G ) , consisting of the locus where at each geometric point, the G -isocrystal is isomorphicto E b . Moreover, each stratum G - I soc b ∼ = [ ∗ /G b ( E )] is a classifying stack for the locally proﬁnite group G b ( E ) . The loop group LG is an ind-(inﬁnite dimensional perfect scheme), so the presentation G - I soc = LG/ Ad ,σ LG is of extremely inﬁnite nature. We expect that this is not an issue with the presentation, butthat the stack itself has no good ﬁniteness properties; in particular note that all strata appearto be of the same dimension 0, while admitting nontrivial specialization relations. Xiao–Zhu (see[ XZ17 ], [

Zhu20 ]) have nonetheless been able to deﬁne a category D ( G - I soc , Q (cid:96) ) of (cid:96) -adic sheaveson G - I soc, admitting a semi-orthogonal decomposition into the various D ( G - I soc b , Q (cid:96) ). Each D ( G - I soc b , Q (cid:96) ) ∼ = D ([ ∗ /G b ( E )] , Q (cid:96) ) is equivalent to the derived category of the category of smoothrepresentations of G b ( E ) (on Q (cid:96) -vector spaces). Here, as usual, we have to ﬁx an auxiliary prime (cid:96) (cid:54) = p and an isomorphism C ∼ = Q (cid:96) .At this point we have deﬁned a stack G - I soc, with a closed immersion i : [ ∗ /G ( E )] ∼ = G - I soc ⊂ G - I soc , thus realizing a fully faithful embedding i ∗ : D ( G ( E ) , Q (cid:96) ) (cid:44) → D ( G - I soc , Q (cid:96) )of the derived category of smooth representations of G ( E ) into the derived category of Q (cid:96) -sheaveson G - I soc. It is in this way that we “geometrize the representation theory of G ( E )”.The key additional structure that we need are the Hecke operators: These will simultaneouslymake the Weil group W E (i.e. π (Spec ˘ E/ϕ Z )) and, by a careful study, also the Langlands dualgroup (cid:98) G appear. Recall that Hecke operators are related to modiﬁcations of G -torsors, and areparametrized by a point x of the curve where the modiﬁcation happens, and the type of themodiﬁcation at x (which can be combinatorially encoded in terms of a cocharacter of G — thiseventually leads to the appearance of (cid:98) G ). Often, the eﬀect of Hecke operators is locally constantfor varying x . In that case, letting x vary amounts to an action of π ( X ), where X is the relevantcurve; thus, the curve should now be Spec ˘ E/ϕ Z .Thus, if we want to deﬁne Hecke operators, we need to be able to consider modiﬁcations of G -isocrystals. These modiﬁcations ought to happen at a section of Spec R (( t )) → Spec R (resp. a This result seems to be well-known to experts, but we are not aware of a full reference. For the v-descent (evenarc-descent), see [

Iva20 , Lemma 5.9]. The stratiﬁcation is essentially constructed in [

RR96 ]; the local constancy ofthe Kottwitz map is proved in general in Corollary III.2.8. The identiﬁcation of the strata in some cases is in [

CS17 ,Proposition 4.3.13], and the argument works in general, up to translating all occurences of p -divisible groups intofamilies of isocrystals. non-existent map Spec( W ( R ) ⊗ W ( F q ) E ) → Spec R ). Unfortunately, the map R → R (( t )) doesnot admit any sections. In fact, we would certainly want to consider continuous sections; suchcontinuous sections would then be in bijection with maps F q (( t )) = ˘ E → R . In other words, inagreement with the motivation from the previous paragraph, the relevant curve should be Spec ˘ E ,or really Spec ˘ E modulo Frobenius — so we can naturally hope to get actions of π (Spec ˘ E/ϕ Z ) bythe above recipe.However, in order for this picture to be realized we need to be in a situation where we havecontinuous maps F q (( t )) → R . In other words, we can only hope for sections if we put ourselvesinto a setting where R is itself some kind of Banach ring.This ﬁnally brings us to the setting considered in this paper. Namely, we replace the categoryof perfect F q -schemes with the category of perfectoid spaces Perf = Perf F q over F q . Locally any S ∈ Perf is of the form S = Spa( R, R + ) where R is a perfectoid Tate F q -algebra: This meansthat R is a perfect topological algebra that admits a topologically nilpotent unit (cid:36) ∈ R (called apseudouniformizer) making it a Banach algebra over F q (( (cid:36) )). Moreover, R + ⊂ R is an open andintegrally closed subring of powerbounded elements. Often R + = R ◦ is the subring of powerboundedelements, and we consequently use the abbreviation Spa R = Spa( R, R ◦ ). The geometric (rank 1)points of S are given by Spa C for complete algebraically closed nonarchimedean ﬁelds C , and asusual understanding geometric points is a key ﬁrst step. We refer to [ SW20 ] for an introductionto adic and perfectoid spaces.For any S = Spa( R, R + ), we need to deﬁne the analogue of Spec R (( t )) /ϕ Z , taking the topologyof R into account. Note that for discrete R (cid:48) , we haveSpa R (cid:48) (( t )) = Spa R (cid:48) × Spa F q Spa F q (( t )) , and we are always free to replace Spec R (cid:48) (( t )) /ϕ Z by Spa R (cid:48) (( t )) /ϕ Z as they have the same categoryof vector bundles. This suggests that the analogue of Spec R (cid:48) (( t )) isSpa( R, R + ) × Spa F q Spa F q (( t )) = D ∗ Spa(

R,R + ) , a punctured open unit disc over Spa( R, R + ), with coordinate t . Note thatSpa( R, R + ) × Spa F q Spa F q (( t )) ⊂ Spa R + × Spa F q Spa F q [[ t ]] = Spa R + [[ t ]]is the locus where t and (cid:36) ∈ R + are invertible, where (cid:36) is a topologically nilpotent unit of R . Thelatter deﬁnition can be extended to mixed characteristic: We letSpa( R, R + ) ˙ × Spa F q Spa E ⊂ Spa R + ˙ × Spa F q Spa O E := Spa W O E ( R + )be the open subset where π and [ (cid:36) ] ∈ W O E ( R + ) are invertible. This space is independent of thechoice of (cid:36) as for any other such (cid:36) (cid:48) , one has (cid:36) | (cid:36) (cid:48) n and (cid:36) (cid:48) | (cid:36) n for some n ≥

1, and then thesame happens for their Teichm¨uller representatives. We note that the symbol ˙ × is purely symbolic:There is of course no map of adic spaces Spa E → Spa F q along which a ﬁbre product could betaken. Definition

I.2.2 . The Fargues–Fontaine curve (for the local ﬁeld E , over S = Spa( R, R + ) ∈ Perf ) is the adic space over E deﬁned by X S = X S,E = (cid:0) Spa(

R, R + ) × Spa F q Spa F q (( t )) (cid:1) /ϕ Z , .2. THE BIG PICTURE 13 respectively X S = X S,E = (cid:0) Spa(

R, R + ) ˙ × Spa F q Spa E (cid:1) /ϕ Z , where the Frobenius ϕ acts on ( R, R + ) . A novel feature, compared to the discussion of G -isocrystals, is that the action of ϕ is free andtotally discontinuous, so the quotient by ϕ is well-deﬁned in the category of adic spaces. In fact,on Y S = Spa( R, R + ) ˙ × Spa F q Spa E ⊂ Spa W O E ( R + ) one can compare the absolute values of π and[ (cid:36) ]. As both are topologically nilpotent units, the ratiorad = log( | [ (cid:36) ] | ) / log( | π | ) : | Y S | → (0 , ∞ )gives a well-deﬁned continuous map. The Frobenius on | Y S | multiplies rad by q , proving that theaction is free and totally discontinuous.We note that in the function ﬁeld case E = F q (( t )), the space Y S = S × Spa F q Spa F q (( t )) = D ∗ S is precisely a punctured open unit disc over S . In this picture, the radius function measures thedistance to the origin: Close to the origin, the radius map is close to 0, while close to the boundaryof the open unit disc it is close to ∞ . The quotient by ϕ is however not an adic space over S anymore, as ϕ acts on S . Thus, X S = Y S /ϕ Z = D ∗ S /ϕ Z is locally an adic space of ﬁnite type over S , but not globally so. This space, for S = Spa C ageometric point, has been ﬁrst studied by Hartl–Pink [ HP04 ].If S = Spa C is a geometric point but E is general, this curve (or rather a closely relatedschematic version) has been extensively studied by Fargues–Fontaine [ FF18 ], where it was shownthat it plays a central role in p -adic Hodge theory. From the perspective of adic spaces, it hasbeen studied by Kedlaya–Liu [ KL15 ]. In particular, in this case where S is a point, X S is indeeda curve: It is a strongly noetherian adic space whose connected aﬃnoid subsets are spectra ofprincipal ideal domains. In particular, in this situation there is a well-behaved notion of “classicalpoints”, referring to those points that locally correspond to maximal ideals. These can be classiﬁed.In the equal characteristic case, the description of Y S = D ∗ S = S × Spa F q Spa F q (( t ))shows that the closed points are in bijection with maps S → Spa F q (( t )) up to Frobenius; where nowone has to take the quotient under t (cid:55)→ t q . In mixed characteristic, the situation is more subtle,and brings us to the tilting construction for perfectoid spaces. Proposition

I.2.3 . If E is of mixed characteristic and S = Spa C is a geometric point, theclassical points of X C are in bijection with untilts C (cid:93) | E of C , up to the action of Frobenius. Here, we recall that for any complete algebraically closed ﬁeld C (cid:48) | E , or more generally anyperfectoid Tate ring R , one can form the tilt R (cid:91) = lim ←− x (cid:55)→ x p R, where the addition is deﬁned on the ring of integral elements in terms of the bijection R (cid:91) + =lim ←− x (cid:55)→ x p R + ∼ = lim ←− x (cid:55)→ x p R + /π , where now x (cid:55)→ x p is compatible with addition on R + /π . Then R (cid:91) is a perfectoid Tate algebra of characteristic p . Geometrically, sending Spa( R, R + ) to Spa( R (cid:91) , R (cid:91) + )deﬁnes a tilting functor on perfectoid spaces T (cid:55)→ T (cid:91) , preserving the underlying topological spaceand the ´etale site, cf. [ SW20 ].One sees that the classical points of X S , for S = Spa C a geometric point, are in bijection withuntilts S (cid:93) of S together with a map S → Spa E , modulo the action of Frobenius. Recall from[ SW20 ] that for any adic space Z over W ( F q ), one deﬁnes a functor Z ♦ : Perf → Sets : S (cid:55)→ { S (cid:93) , f : S (cid:93) → Z } sending a perfectoid space S over F q to pairs S (cid:93) of an untilt of S , and a map S (cid:93) → Z . If Z is ananalytic adic space, then Z ♦ is a diamond, that is a quotient of a perfectoid space by a pro-´etaleequivalence relation. Then the classical points of X S are in bijection with the S -valued points ofthe diamond (Spa ˘ E ) ♦ /ϕ Z . More generally, for any S ∈ Perf, maps S → (Spa ˘ E ) ♦ /ϕ Z are in bijection with degree 1 Cartierdivisors D S ⊂ X S , so we deﬁne Div = (Spa ˘ E ) ♦ /ϕ Z . We warn the reader the action of the Frobenius here is a geometric Frobenius. In particular, itonly exists on (Spa ˘ E ) ♦ , not on Spa ˘ E , in case E is of mixed characteristic. However, one still has π (Div ) = W E .This ends our long stream of thoughts on the geometry of Spec E : We have arrived at theFargues–Fontaine curve, in its various incarnations. To orient the reader, we recall them here:(i) For any complete algebraically closed nonarchimedean ﬁeld C | F q , the curve X C = X C,E , astrongly noetherian adic space over E , locally the adic spectrum of a principal ideal domain. Onecan also construct a schematic version X alg C , with the same classical points and the same categoryof vector bundles. The classical points are in bijection with untilts C (cid:93) | E of C , up to Frobenius.(ii) More generally, for any perfectoid space S ∈ Perf, the “family of curves” X S , again an adicspace over E , but no longer strongly noetherian. If S is aﬃnoid, there is a schematic version X alg S ,with the same category of vector bundles.(iii) The “mirror curve” Div = (Spa ˘ E ) ♦ /ϕ Z , which is only a diamond. For any S ∈ Perf, thisparametrizes “degree 1 Cartier divisors on X S ”.A peculiar phenomenon here is that there is no “absolute curve” of which all the others arethe base change. Another peculiar feature is that the space of degree 1 Cartier divisors is not thecurve itself.Again, it is time to study G -torsors. This leads to the following deﬁnition. Definition

I.2.4 . Let

Bun G : Perf → { groupoids } : S (cid:55)→ { G - torsors on X S } be the moduli stack of G -torsors on the Fargues–Fontaine curve. Remark

I.2.5 . Let us stress here that while “the Fargues–Fontaine curve” is not really a well-deﬁned notion, “the moduli stack of G -torsors on the Fargues–Fontaine curve” is. .2. THE BIG PICTURE 15 As X S maps towards Spa ˘ E/ϕ Z , there is a natural pullback functor G -Isoc → Bun G ( S ). Thefollowing result is in most cases due to Fargues [ Far18b ], completed by Ansch¨utz, [

Ans19 ]. Theorem

I.2.6 . If S = Spa C is a geometric point, the map B ( G ) → Bun G ( S ) / ∼ = is a bijection. In particular, any vector bundle on X S is a direct sum of vector bundles O X S ( λ ) associated to D − λ , λ ∈ Q .Under this bijection, b ∈ B ( G ) is basic if and only if the corresponding G -torsor E b on X S issemistable in the sense of Atiyah–Bott [ AB83 ] . However, it is no longer true that the automorphism groups are the same. On the level of thestack, we have the following result.

Theorem

I.2.7 . The prestack

Bun G is a v-stack. It admits a stratiﬁcation into locally closedsubstacks i b : Bun bG ⊂ Bun G for b ∈ B ( G ) consisting of the locus where at each geometric point, the G -torsor is isomorphic to E b . Moreover, each stratum Bun bG ∼ = [ ∗ / (cid:101) G b ] is a classifying space for a group (cid:101) G b that is an extension of the locally proﬁnite group G b ( E ) by a“unipotent group diamond”.The semistable locus Bun ss G ⊂ Bun G is an open substack, and Bun ss G = (cid:71) b ∈ B ( G ) basic [ ∗ /G b ( E )] . Remark

I.2.8 . The theorem looks formally extremely similar to Theorem I.2.1. However,there is a critical diﬀerence, namely the closure relations are reversed: For Bun G , the inclusion ofBun bG for b ∈ B ( G ) basic is an open immersion while it was a closed immersion in Theorem I.2.1.Note that basic b ∈ B ( G ) correspond to semistable G -bundles, and one would indeed expect thesemistable locus to be an open substack. Generally, Bun G behaves much like the stack of G -bundleson the projective line. Remark

I.2.9 . We deﬁne a notion of Artin stacks in this perfectoid setting, and indeed Bun G is an Artin stack; we refer to Section I.4 for a more detailed description of our geometric resultson Bun G . This shows that Bun G has much better ﬁniteness properties than G - I soc, even if it isdeﬁned on more exotic test objects.We can deﬁne a derived category of (cid:96) -adic sheaves D (Bun G , Q (cid:96) )on Bun G . This admits a semi-orthogonal decomposition into all D (Bun bG , Q (cid:96) ), and D (Bun bG , Q (cid:96) ) ∼ = D ([ ∗ /G b ( E )] , Q (cid:96) ) ∼ = D ( G b ( E ) , Q (cid:96) )is equivalent to the derived category of smooth G b ( E )-representations. Remark

I.2.10 . It is reasonable to expect that this category is equivalent to the category D ( G - I soc , Q (cid:96) ) deﬁned by Xiao–Zhu. However, we do not pursue this comparison here.Finally, we can deﬁne the Hecke stack that will bring all key players together. Consider theglobal Hecke stack Hck G parametrizing pairs ( E , E (cid:48) ) of G -bundles on X S , together with a map S → Div giving rise to a degree 1 Cartier divisor D S ⊂ X S , and an isomorphism f : E| X S \ D S ∼ = E (cid:48) | X S \ D S that is meromorphic along D S . This gives a correspondenceBun G h ←− Hck

G h −→ Bun G × Div . To deﬁne the Hecke operators, we need to bound the modiﬁcation, i.e. bound the poles of f along D S . This is described by the local Hecke stack H ck G , parametrizing pairs of G -torsors on thecompletion of X S along D S , together with an isomorphism away from D S ; thus, there is a naturalmap Hck G → H ck G from the global to the local Hecke stack. Geometrically, H ck G admits aSchubert stratiﬁcation in terms of the conjugacy classes of cocharacters of G ; in particular, thereare closed Schubert cells H ck G, ≤ µ for each conjugacy class µ : G m → G . By pullback, this deﬁnesa correspondence Bun G h , ≤ µ ←−−− Hck G, ≤ µ h , ≤ µ −−−→ Bun G × Div where now h , ≤ µ and h , ≤ µ are proper. One can then consider Hecke operators Rh , ≤ µ, ∗ h ∗ , ≤ µ : D (Bun G , Λ) → D (Bun G × Div , Λ) . The following theorem ensures that Hecke operators are necessarily locally constant as onevaries the point of Div , and hence give rise to representations of π (Div ) = W E . In the following,we are somewhat cavalier about the precise deﬁnition of D ( − , Q (cid:96) ) employed, and the notion of W E -equivariant objects: The ﬁne print is addressed in the main text. Theorem

I.2.11 . Pullback along the map

Div → [ ∗ /W E ] induces an equivalence D (Bun G × Div , Q (cid:96) ) ∼ = D (Bun G × [ ∗ /W E ] , Q (cid:96) ) ∼ = D (Bun G , Q (cid:96) ) BW E . Thus, Hecke operators produce W E -equivariant objects in D (Bun G , Q (cid:96) ), making the Weil groupappear naturally.One also wants to understand how Hecke operators compose. This naturally leads to the studyof D ( H ck G , Q (cid:96) ) as a monoidal category, under convolution. Here, we have the geometric Satakeequivalence. In the setting of usual smooth projective curves (over C ), this was established inthe papers of Lusztig [ Lus83 ], Ginzburg [

Gin90 ] and Mirkovi´c–Vilonen [

MV07 ]. The theorembelow is a ﬁrst approximation; we will actually prove a more precise version with Z (cid:96) -coeﬃcients,describing all perverse sheaves on H ck G , and applying to the Beilinson–Drinfeld Grassmannians inthe spirit of Gaitsgory’s paper [ Gai07 ]. Theorem

I.2.12 . There is a natural monoidal functor from

Rep (cid:98) G to D ( H ck G , Q (cid:96) ) . Remark

I.2.13 . Our proof of Theorem I.2.12 follows the strategy of Mirkovi´c–Vilonen’s proof,and in particular deﬁnes a natural symmetric monoidal structure on the category of perverse sheavesby using the fusion product. This requires one to work over several copies of the base curve, and .2. THE BIG PICTURE 17 let the points collide. It is a priori very surprising that this can be done in mixed characteristic,as it requires a space like Spa Q p ˙ × Spa F p Spa Q p . Spaces of this type do however exist as diamonds,and this was one of the main innovations of [ SW20 ]. Remark

I.2.14 . Using a degeneration of the local Hecke stack, which is essentially the B +dR -aﬃne Grassmannian of [ SW20 ], to the Witt vector aﬃne Grassmannian, Theorem I.2.12 gives anew proof of Zhu’s geometric Satake equivalence for the Witt vector aﬃne Grassmannian [

Zhu17 ].In fact, we even prove a version with Z (cid:96) -coeﬃcients, thus also recovering the result of Yu [ Yu19 ]. Remark

I.2.15 . Regarding the formalism of (cid:96) -adic sheaves, we warn the reader that we arecheating slightly in the formulation of Theorem I.2.12; the deﬁnition of D (Bun G , Q (cid:96) ) implicit aboveis not the same as the one implicit in Theorem I.2.12. With torsion coeﬃcients, the problem woulddisappear, and in any case the problems are essentially of technical nature.Thus, this also makes the Langlands dual group (cid:98) G appear naturally. For any representation V of (cid:98) G , we get a Hecke operator T V : D (Bun G , Q (cid:96) ) → D (Bun G , Q (cid:96) ) BW E . Moreover, the Hecke operators commute and T V ⊗ W ∼ = T V ◦ T W | ∆( W E ) where we note that T V ◦ T W naturally takes values in W E × W E -equivariant objects; the restrictionon the right means the restriction to the action of the diagonal copy ∆( W E ) ⊂ W E × W E .At this point, the representation theory of G ( E ) (which sits fully faithfully in D (Bun G , Q (cid:96) )), theWeil group W E , and the dual group (cid:98) G , all interact with each other naturally. It turns out that thiscategorical structure is precisely what is needed to construct L -parameters for (Schur-)irreducibleobjects A ∈ D (Bun G , Q (cid:96) ), and in particular for irreducible smooth representations of G ( E ). Wewill discuss the construction of L -parameters below in Section I.9.We note that the whole situation is exactly parallel to the Betti geometric Langlands situa-tion considered by Nadler–Yun [ NY19 ], and indeed the whole strategy can be described as “thegeometric Langlands program on the Fargues–Fontaine curve”. It is curious that our quest wasto understand the local Langlands correspondence in an arithmetic setting, for potentially veryramiﬁed representations, and eventually we solved it by relating it to the global Langlands corre-spondence in a geometric setting, in the everywhere unramiﬁed setting.In the rest of this introduction, we give a more detailed overview of various aspects of thispicture:(i) The Fargues–Fontaine curve (Section I.3);(ii) The geometry of the stack Bun G (Section I.4);(iii) The derived category of (cid:96) -adic sheaves on Bun G (Section I.5);(iv) The geometric Satake equivalence (Section I.6);(v) Finiteness of the cohomology of Rapoport–Zink spaces, local Shimura varieties, and more gen-eral moduli spaces of shtukas (Section I.7);(vi) The stack of L -parameters (Section I.8);(vii) The construction of L -parameters (Section I.9); (viii) The spectral action (Section I.10);(ix) The origin of the ideas ﬂeshed out in this paper (Section I.11).These items largely mirror the chapters of this paper, and each chapter begins with a repriseof these introductions. I.3. The Fargues–Fontaine curve

The Fargues–Fontaine curve has been studied extensively in the book of Fargues–Fontaine[

FF18 ] and further results, especially in the relative situation, have been obtained by Kedlaya–Liu[

KL15 ]. In the ﬁrst chapter, we reprove these foundational results, thereby also collecting andunifying certain results (proved often only for E = Q p ).The ﬁrst results concern the Fargues–Fontaine curve X C = X S when S = Spa C for somecomplete algebraically closed nonarchimedean ﬁeld C | F q . We deﬁne a notion of classical points of X C in that case; they form a subset of | X C | . The basic ﬁniteness properties of X C are summarizedin the following result. Theorem

I.3.1 . The adic space X C is locally the adic spectrum Spa(

B, B + ) where B is aprincipal ideal domain; the classical points of Spa(

B, B + ) ⊂ X C are in bijection with the maximalideals of B . For each classical point x ∈ X C , the residue ﬁeld of x is an untilt C (cid:93) of C over E ,and this induces a bijection of the classical points of X C with untilts C (cid:93) of C over E , taken up tothe action of Frobenius. In the equal characteristic case, Theorem I.3.1 is an immediate consequence of the presentation X C = D ∗ C /ϕ Z and classical results in rigid-analytic geometry. In the p -adic case, we use tiltingto reduce to the equal characteristic case. At one key turn, in order to understand Zariski closedsubsets of X C , we use the result that Zariski closed implies strongly Zariski closed [ BS19 ]. Usingthese ideas, we are able to give an essentially computation-free proof.A key result is the classiﬁcation of vector bundles.

Theorem

I.3.2 . The functor from

Isoc E to vector bundles on X C induces a bijection on iso-morphism classes. In particular, there is a unique stable vector bundle O X C ( λ ) of any slope λ ∈ Q ,and any vector bundle E can be written as a direct sum of stable bundles. We give a new self-contained proof of Theorem I.3.2, making critical use of the v-descent resultsfor vector bundles obtained in [

Sch17a ] and [

SW20 ], and basic results on the geometry of Banach–Colmez spaces established here. The proof in the equal characteristic case by Hartl–Pink [

HP04 ]and the proof of Kedlaya in the p -adic case [ Ked04 ] relied on heavy computations, while the proofof Fargues–Fontaine [

FF18 ] relied on the description of the Lubin–Tate and Drinfeld moduli spacesof π -divisible O -modules. Our proof is related to the arguments of Colmez in [ Col02 ].Allowing general S ∈ Perf F q , we deﬁne the moduli space of degree 1 Cartier divisors as Div =Spd ˘ E/ϕ Z . Given a map S → Div , one can deﬁne an associated closed Cartier divisor D S ⊂ X S ;locally, this is given by an untilt D S = S (cid:93) ⊂ X S of S over E , and this embeds Div into the spaceof closed Cartier divisors on X S (justifying the name). Another important result is the followingampleness result, cf. [ KL15 , Proposition 6.2.4], which implies that one can deﬁne an algebraicversion of the curve, admitting the same theory of vector bundles. .3. THE FARGUES–FONTAINE CURVE 19

Theorem

I.3.3 . Assume that S ∈ Perf is aﬃnoid. For any vector bundle E on X S , the twist E ( n ) is globally generated and has no higher cohomology for all n (cid:29) . Deﬁning the graded ring P = (cid:77) n ≥ H ( X S , O X S ( n )) and the scheme X alg S = Proj P , there is a natural map of locally ringed spaces X S → X alg S , pullbackalong which deﬁnes an equivalence of categories of vector bundles, preserving cohomology.If S = Spa C for some complete algebraically closed nonarchimedean ﬁeld C , then X alg C is aregular noetherian scheme of Krull dimension , locally the spectrum of a principal ideal domain,and its closed points are in bijection with the classical points of X C . We also need to understand families of vector bundles, i.e. vector bundles E on X S for general S . Here, the main result is the following. Theorem

I.3.4 . Let S ∈ Perf and let E be a vector bundle on X S . Then the function taking apoint s ∈ S to the Harder–Narasimhan polygon of E| X s deﬁnes a semicontinuous function on S . Ifit is constant, then E admits a global Harder–Narasimhan stratiﬁcation, and pro-´etale locally on S one can ﬁnd an isomorphism with a direct sum of O X S ( λ ) ’s.In particular, if E is everywhere semistable of slope , then E is pro-´etale locally trivial, and thecategory of such E is equivalent to the category of pro-´etale E -local systems on S . The key to proving Theorem I.3.4 is the construction of certain global sections of E . To achievethis, we use v-descent techniques, and an analysis of the spaces of global sections of E ; these areknown as Banach–Colmez spaces, and were ﬁrst introduced (in slightly diﬀerent terms) in [ Col02 ]. Definition

I.3.5 . Let E be a vector bundle on X S . The Banach–Colmez space BC ( E ) associatedwith E is the locally spatial diamond over S whose T -valued points, for T ∈ Perf S , are given by BC ( E )( T ) = H ( X T , E| X T ) . Similarly, if E is everywhere of only negative Harder–Narasimhan slopes, the negative Banach–Colmez space BC ( E [1]) is the locally spatial diamond over S whose T -valued points are BC ( E [1])( T ) = H ( X T , E| X T ) . Implicit here is that this functor actually deﬁnes a locally spatial diamond. For this, we calculatesome key examples of Banach–Colmez spaces. For example, if E = O X S ( λ ) with 0 < λ ≤ [ E : Q p ](resp. all positive λ if E is of equal characteristic), then BC ( E ) is representable by a perfectoid openunit disc (of dimension given by the numerator of λ ). A special case of this is the identiﬁcationof BC ( O X S (1)) with the universal cover of a Lubin–Tate formal group law, yielding a very closerelation between Lubin–Tate theory, and thus local class ﬁeld theory, and the Fargues–Fontainecurve; see also [ Far18a ]. On the other hand, for larger λ , or negative λ , Banach–Colmez spacesare more exotic objects; for example, the negative Banach–Colmez space BC ( O X C ( − ∼ = ( A C (cid:93) ) ♦ /E is the quotient of the aﬃne line by the translation action of E ⊂ A C (cid:93) . We remark that ourproof of the classiﬁcation theorem, Theorem I.3.2, ultimately relies on the negative result that BC ( O X C ( − For the proof of Theorem I.3.4, a key result is that projectivized Banach–Colmez spaces( BC ( E ) \ { } ) /E × are proper — they are the relevant analogues of “families of projective spaces over S ”. In particular,their image in S is a closed subset, and if the image is all of S , then we can ﬁnd a nowhere vanishingsection of E after a v-cover, as then the projectivized Banach–Colmez space is a v-cover of S . Fromhere, Theorem I.3.4 follows easily. I.4. The geometry of

Bun G Let us discuss the geometry of Bun G . Here, G can be any reductive group over a nonarchimedeanlocal ﬁeld E , with residue ﬁeld F q of characteristic p . Recall that Kottwitz’ set B ( G ) = B ( E, G ) of G -isocrystals can be described combinatorially, by two discrete invariants. The ﬁrst is the Newtonpoint ν : B ( G ) → ( X ∗ ( T ) + Q ) Γ , where T is the universal Cartan of G and Γ = Gal( E | E ). More precisely, any G -isocrystal E deﬁnesa slope morphism D → G ˘ E where D is the diagonalizable group with cocharacter group Q ; itsdeﬁnition reduces to the case of GL n , where it amounts to the slope decomposition of isocrystals.Isomorphisms of G -isocrystals lead to conjugate slope morphisms, and this deﬁnes the map ν .The other map is the Kottwitz invariant κ : B ( G ) → π ( G E ) Γ . Its deﬁnition is indirect, starting from tori, passing to the case of G with simply connected derivedgroup, and ﬁnally to the general case by z-extensions. Then Kottwitz shows that( ν, κ ) : B ( G ) → ( X ∗ ( T ) + Q ) Γ × π ( G E ) Γ is injective. Moreover, κ induces a bijection between B ( G ) basic and π ( G E ) Γ . The non-basicelements can be described in terms of Levi subgroups.Using ν and κ , one can deﬁne a partial order on B ( G ) by declaring b ≤ b (cid:48) if κ ( b ) = κ ( b (cid:48) ) and ν b ≤ ν b (cid:48) with respect to the dominance order.Up to sign, one can think of ν , resp. κ , as the Harder–Narasimhan polygon, resp. ﬁrst Chernclass, of a G -bundle. Theorem

I.4.1 . The prestack

Bun G satisﬁes the following properties. (i) The prestack

Bun G is a stack for the v-topology. (ii) The points | Bun G | are naturally in bijection with Kottwitz’ set B ( G ) of G -isocrystals. Actually, we only know this for sure if E is p -adic; in the function ﬁeld case, we supply a small extra argumentcircumventing the issue. .4. THE GEOMETRY OF Bun G (iii) The map ν : | Bun G | → B ( G ) → ( X ∗ ( T ) + Q ) Γ is semicontinuous, and κ : | Bun G | → B ( G ) → π ( G E ) Γ is locally constant. Equivalently, the map | Bun G | → B ( G ) is continuous when B ( G ) is equippedwith the order topology. (iv) For any b ∈ B ( G ) , the corresponding subfunctor i b : Bun bG = Bun G × | Bun G | { b } ⊂ Bun G is locally closed, and isomorphic to [ ∗ / (cid:101) G b ] , where (cid:101) G b is a v-sheaf of groups such that (cid:101) G b → ∗ isrepresentable in locally spatial diamonds with π (cid:101) G b = G b ( E ) . The connected component (cid:101) G ◦ b ⊂ (cid:101) G b of the identity is cohomologically smooth of dimension (cid:104) ρ, ν b (cid:105) . (v) In particular, the semistable locus

Bun ss G ⊂ Bun G is open, and given by Bun ss G ∼ = (cid:71) b ∈ B ( G ) basic [ ∗ /G b ( E )] . (vi) For any b ∈ B ( G ) , there is a map π b : M b → Bun G that is representable in locally spatial diamonds, partially proper and cohomologically smooth, where M b parametrizes G -bundles E together with an increasing Q -ﬁltration whose associated graded is,at all geometric points, isomorphic to E b with its slope grading. The v-stack M b is representable inlocally spatial diamonds, partially proper and cohomologically smooth over [ ∗ /G b ( E )] . (vii) The v-stack

Bun G is a cohomologically smooth Artin stack of dimension . As examples, let us analyze the case of GL and GL . For GL , and general tori, everything issemistable, so Pic := Bun GL ∼ = (cid:71) Z [ ∗ /E × ] . For GL , the Kottwitz invariant gives a decompositionBun GL = (cid:71) α ∈ Z Bun α GL . Each connected component has a unique semistable point, given by the basic element b ∈ B (GL ) basic with κ ( b ) = α . For b ∈ B (GL ) basic ∼ = Z , the corresponding group G b ( E ) is given by GL ( E )when b ∈ Z , and by D × when b ∈ Z \ Z , where D | E is the quaternion algebra.The non-semistable points of Bun GL are given by extensions of line bundles, which are of theform O ( i ) ⊕ O ( j ) for some i, j ∈ Z , with 2 α = i + j . Let us understand the simplest degenerationinside Bun GL , which is from O ( ) to O ⊕ O (1). The individual strata here are[ ∗ /D × ] , [ ∗ / Aut(

O ⊕ O (1))] . Here Aut(

O ⊕ O (1)) = (cid:18) E × BC ( O (1))0 E × (cid:19) . Here BC ( O (1)) is representable by a perfectoid open unit disc Spd F q [[ t /p ∞ ]].In this case, the local chart M b for Bun GL parametrizes rank 2 bundles E written as anextension 0 → L → E → L (cid:48) → L ∼ = O and L (cid:48) ∼ = O (1). Fixing such isomorphisms deﬁnes a E × × E × -torsor (cid:102) M b → M b with (cid:102) M b = BC ( O ( − G is closely related to the structure of negative Banach–Colmez spaces. Italso shows that while the geometry of Bun G is quite nonstandard, it is still fundamentally a ﬁnite-dimensional and “smooth” situation.For general G , we still get a decomposition into connected componentsBun G = (cid:71) α ∈ π ( G ) Γ Bun αG and each connected component Bun αG admits a unique semistable point.By a recent result of Viehmann [ Vie21 ], the map | Bun G | → B ( G ) is a homeomorphism. Thishad previously been proved for G = GL n by Hansen [ Han17 ] based on [

BFH + ]; that argumentwas extended to some classical groups in unpublished work of Hamann.Let us say some words about the proof of Theorem I.4.1. Part (i) has essentially been provedin [ SW20 ], and part (ii) follows from the result of Fargues and Ansch¨utz, Theorem I.2.6. In part(iii), the statement about ν reduces to GL n by an argument of Rapoport–Richartz [ RR96 ], whereit is Theorem I.3.4. The statement about κ requires more work, at least in the general case: Ifthe derived group of G is simply connected, one can reduce to tori, which are not hard to handle.In general, one approach is to use z-extensions (cid:101) G → G to reduce to the case of simply connectedderived group. For this, one needs that Bun (cid:101) G → Bun G is a surjective map of v-stacks; we provethis using Beauville–Laszlo uniformization. Alternatively, one can use the abelianized Kottwitz setof Borovoi [ Bor98 ], which we prove to behave well relatively over a perfectoid space S . Part (iv)is a also consequence of Theorem I.3.4. Part (v) is a consequence of parts (iii) and (iv). The keypoint is then part (vi), which will imply (vii) formally. The properties of M b itself are easy toestablish — the analysis for GL above generalizes easily to show that (cid:102) M b is a successive extensionof negative Banach–Colmez spaces. The key diﬃculty is to prove that π b : M b → Bun G is cohomologically smooth. Note that as we are working with perfectoid spaces, there are no tangentspaces, and we cannot hope to prove smoothness via deformation theory. To attack this problem,we nonetheless prove a general “Jacobian criterion of cohomological smoothness”. The setup hereis the following.Let S be a perfectoid space, and let Z → X S be a smooth map of (sousperfectoid) adic spaces;this means that Z is an adic space that is locally ´etale over a ﬁnite-dimensional ball over X S . In thissituation, we can deﬁne a v-sheaf M Z → S parametrizing sections of Z → X S , i.e. the S (cid:48) -valuedpoints, for S (cid:48) /S a perfectoid space, are given by the maps s : X S (cid:48) → Z lifting X S (cid:48) → X S . Foreach such section, we get the vector bundle s ∗ T Z/X S on S (cid:48) , where T Z/X S is the tangent bundle. .5. (cid:96) -ADIC SHEAVES ON Bun G Naively, deformations of S (cid:48) → M Z , i.e. of X S (cid:48) → Z over X S (cid:48) → X S , should correspond to globalsections H ( X S (cid:48) , s ∗ T Z/X S ), and obstructions to H ( X S (cid:48) , s ∗ T Z/X S ). If s ∗ T Z/X S has everywhere onlypositive Harder–Narasimhan slopes, then this vanishes locally on S (cid:48) . By analogy with the classicalsituation, we would thus expect the open subspace M sm Z ⊂ M Z , where s ∗ T Z/X S has positive Harder–Narasimhan slopes, to be (cohomologically) smooth over S .Our key geometric result conﬁrms this, at least if Z → X S is quasiprojective. Theorem

I.4.2 . Assume that Z → X S can, locally on S , be embedded as a Zariski closedsubset of an open subset of (the adic space) P nX S . Then M Z → S is representable in locally spatialdiamonds, compactiﬁable, and of locally ﬁnite dim . trg . Moreover, the open subset M sm Z ⊂ M Z iscohomologically smooth over S . In the application, the space Z → X S will be the ﬂag variety parametrizing Q -ﬁltrations on agiven G -torsor E on X S . Then M b will be an open subset of M sm Z .The proof of Theorem I.4.2 requires several innovations. The ﬁrst is a notion of formal smooth-ness, in which inﬁnitesimal thickenings (that are not available in this perfectoid setting) are replacedby small ´etale neighborhoods. This leads to a notion with a close relation to the notion of absoluteneighborhood retracts [ Bor67 ] in classical topology. We prove that virtually all examples of coho-mologically smooth maps are also formally smooth, including Banach–Colmez spaces and Bun G . Wealso prove that M sm Z → S is formally smooth, which amounts to some delicate estimates, spreadingsections X T → Z into small neighborhoods of T ⊂ T , for any Zariski closed immersion T ⊂ T of aﬃnoid perfectoid spaces — here we crucially use the assumption that all Harder–Narasimhanslopes are positive. Coupled with the theorem that Zariski closed implies strongly Zariski closed[ BS19 ] this makes it possible to write M sm Z , up to (cohomologically and formally) smooth maps,as a retract of a space that is ´etale over a ball over S . Certainly in classical topology, this is notenough to ensure cohomological smoothness — a coordinate cross is a retract of R — but it doesimply that the constant sheaf F (cid:96) is universally locally acyclic over S . For this reason, and otherapplications to sheaves on Bun G as well as geometric Satake, we thus also develop a general theoryof universally locally acyclic sheaves in our setting. To ﬁnish the proof, we use a deformation tothe normal cone argument to show that the dualizing complex is “the same” as the one for theBanach–Colmez space BC ( s ∗ T Z/X S ). I.5. (cid:96) -adic sheaves on

Bun G For our results, we need to deﬁne the category of (cid:96) -adic sheaves on Bun G . More precisely, wewill deﬁne for each Z (cid:96) -algebra Λ a category D (Bun G , Λ)of sheaves of Λ-modules on Bun G . If Λ is killed by some power of (cid:96) , such a deﬁnition is the mainachievement of [ Sch17a ]. Our main interest is however the case Λ = Q (cid:96) . In the case of schemes (ofﬁnite type over an algebraically closed ﬁeld), the passage from torsion coeﬃcients to Q (cid:96) -coeﬃcientsis largely formal: Roughly, D ( X, Q (cid:96) ) = Ind(lim ←− n D bc ( X, Z /(cid:96) n Z ) ⊗ Z (cid:96) Q (cid:96) ) . Behind this deﬁnition are however strong ﬁniteness results for constructible sheaves; in particular,the morphism spaces between constructible sheaves are ﬁnite. For Bun G , or for the category ofsmooth representations, there are still compact objects (given by compactly induced representationsin the case of smooth representations), but their endomorphism algebras are Hecke algebras, whichare inﬁnite-dimensional. A deﬁnition along the same lines would then replace all Hecke algebrasby their (cid:96) -adic completions, which would drastically change the category of representations.Our deﬁnition of D (Bun G , Λ) in general involves some new ideas, employing the idea of solidmodules developed by Clausen–Scholze [ CS ] in the context of the pro-´etale (or v-)site; in the end, D (Bun G , Λ) is deﬁned as a certain full subcategory D lis (Bun G , Λ) ⊂ D (cid:4) (Bun G , Λ)of the category D (cid:4) (Bun G , Λ) of solid complexes of Λ-modules on the v-site of Bun G . The formalismof solid sheaves, whose idea is due to Clausen and the second author, is developed in Chapter VII.It presents some interesting surprises; in particular, there is always a left adjoint f (cid:92) to pullback f ∗ ,satisfying base change and a projection formula. (In return, Rf ! fails to exist in general.) Theorem

I.5.1 . Let Λ be any Z (cid:96) -algebra. (i) Via excision triangles, there is an inﬁnite semiorthogonal decomposition of D (Bun G , Λ) into thevarious D (Bun bG , Λ) for b ∈ B ( G ) . (ii) For each b ∈ B ( G ) , pullback along Bun bG ∼ = [ ∗ / (cid:101) G b ] → [ ∗ /G b ( E )] gives an equivalence D ([ ∗ /G b ( E )] , Λ) ∼ = D (Bun bG , Λ) , and D ([ ∗ /G b ( E )] , Λ) ∼ = D ( G b ( E ) , Λ) is equivalent to the derived category of the category of smoothrepresentations of G b ( E ) on Λ -modules. (iii) The category D (Bun G , Λ) is compactly generated, and a complex A ∈ D (Bun G , Λ) is compactif and only if for all b ∈ B ( G ) , the restriction i b ∗ A ∈ D (Bun bG , Λ) ∼ = D ( G b ( E ) , Λ) is compact, and zero for almost all b . Here, compactness in D ( G b ( E ) , Λ) is equivalent to lying inthe thick triangulated subcategory generated by c - Ind G b ( E ) K Λ as K runs over open pro- p -subgroupsof G b ( E ) . (iv) On the subcategory D (Bun G , Λ) ω ⊂ D (Bun G , Λ) of compact objects, there is a Bernstein–Zelevinsky duality functor D BZ : ( D (Bun G , Λ) ω ) op → D (Bun G , Λ) ω with a functorial identiﬁcation R Hom(

A, B ) ∼ = π (cid:92) ( D BZ ( A ) ⊗ L Λ B ) for B ∈ D (Bun G , Λ) , where π : Bun G → ∗ is the projection. The functor D BZ is an equivalence,and D BZ is naturally equivalent to the identity. It is compatible with usual Bernstein–Zelevinskyduality on D ( G b ( E ) , Λ) for basic b ∈ B ( G ) . .5. (cid:96) -ADIC SHEAVES ON Bun G (v) An object A ∈ D (Bun G , Λ) is universally locally acyclic (with respect to Bun G → ∗ ) if and onlyif for all b ∈ B ( G ) , the restriction i b ∗ A ∈ D (Bun bG , Λ) ∼ = D ( G b ( E ) , Λ) is admissible, i.e. for all pro- p open subgroups K ⊂ G b ( E ) , the complex ( i b ∗ A ) K is perfect. Univer-sally locally acyclic complexes are preserved by Verdier duality, and satisfy Verdier biduality. This theorem extends many basic notions from representation theory — ﬁnitely presentedobjects, admissible representations, Bernstein–Zelevinsky duality, smooth duality — to the settingof D (Bun G , Λ).Parts (i) and (ii) are easy when Λ is (cid:96) -power torsion. In general, their proofs invoke theprecise deﬁnition of D (Bun G , Λ) = D lis (Bun G , Λ) and are somewhat subtle. Part (iii) uses that ii b ∗ admits a left adjoint, which will then automatically preserve compact objects (inducing compactgenerators). Using the diagram [ ∗ /G b ( E )] q b ←− M b π b −→ Bun G , this left adjoint is deﬁned as π b(cid:92) q ∗ b . The veriﬁcation that this is indeed a left adjoint amounts insome sense to the assertion that M b is “strictly local” along the closed subspace [ ∗ /G b ( E )] ⊂ M b in the sense that for all A ∈ D ( M b , Λ), the restriction R Γ( M b , A ) → R Γ([ ∗ /G b ( E )] , A )is an isomorphism. This builds on a detailed analysis of the topological nature of M b , in particularthat (cid:102) M b \ ∗ is a spatial diamond, and Theorem I.5.2 below. For part (iv), the constructions in (iii)imply the existence of D BZ ( A ) on a class of generators, thus in general, and similar arguments tothe ones in (iii) prove the biduality. Finally, part (v) is essentially a formal consequence.The key cohomological result for the proof is the following result, applied to (cid:102) M b \∗ (or quotientsof it). It plays on the subtle point that the point ∗ is not quasiseparated. Theorem

I.5.2 . Let X be a spatial diamond such that f : X → ∗ is partially proper, and ofﬁnite dim . trg . Then for any aﬃnoid perfectoid space S , the base change X S = X × S naturallyadmits two ends. Taking compactly supported cohomology with respect to one end (but no supportcondition at the other end), one has R Γ ∂ - c ( X S , A ) = 0 for all A ∈ D + (cid:4) ( X, Z (cid:96) ) (resp. all A ∈ D (cid:4) ( X, Z (cid:96) ) if f is (cid:96) -cohomologically smooth). As an example, if X = Spa F q (( t )), then X S = D ∗ S is an open unit disc over S , whose two endsare the origin and the boundary, and one has R Γ ∂ - c ( D ∗ S , Z (cid:96) ) = 0 . In particular, the cohomology of Spa F q [[ t ]] agrees with sections on the closed point, showing thatSpa F q [[ t ]] is “strictly local”. The same phenomenon is at work for M b . I.6. The geometric Satake equivalence

In order to deﬁne the Hecke operators, we need to prove the geometric Satake equivalence,taking representations of the dual group (cid:98) G to sheaves on the local Hecke stack. In order to analyzecompositions of Hecke operators, it will in fact be necessary to analyze modiﬁcations at severalpoints.Thus, for any ﬁnite set I , we consider the moduli space (Div ) I parametrizing degree 1 Cartierdivisors D i ⊂ X S , i ∈ I . Locally on S , each D i deﬁnes an untilt S (cid:93)i of S over E , and one can formthe completion B + of O X S along the union of the D i . Inverting the D i deﬁnes a localization B of B + . One can then deﬁne a positive loop group L +(Div ) I G and loop group L (Div ) I G , with valuesgiven by G ( B + ) resp. G ( B ); for brevity, we will simply write L + G and LG here. One can thendeﬁne the local Hecke stack H ck IG = [ L + G \ LG/L + G ] → (Div ) I . For d = | I | , this is in fact already deﬁned over the moduli space Div d = (Div ) d / Σ d of degree d Cartier divisors. We will often break symmetry, and ﬁrst take the quotient on the right to deﬁnethe Beilinson–Drinfeld GrassmannianGr IG = LG/L + G → (Div ) I so that H ck IG = L + G \ Gr IG . The Beilinson–Drinfeld Grassmannian Gr IG → (Div ) I is a small v-sheaf that can be written asan increasing union of closed subsheaves that are proper and representable in spatial diamonds, bybounding the relative position; this is one main result of [ SW20 ]. On the other hand, L + G canbe written as an inverse limit of truncated positive loop groups, which are representable in locallyspatial diamonds and cohomologically smooth; moreover, on each bounded subset, it acts throughsuch a ﬁnite-dimensional quotient. This essentially reduces the study of all bounded subsets of H ck IG to Artin stacks.In particular, one can write the local Hecke stack as an increasing union of closed substacksthat are quasicompact over (Div ) I , by bounding the relative position. In the following, we assumethat the coeﬃcients Λ are killed by some power of (cid:96) , so that we can use the theory from [ Sch17a ].Let D ´et ( H ck IG , Λ) bd ⊂ D ´et ( H ck IG , Λ)be the full subcategory of all objects with quasicompact support over (Div ) I . This is a monoidalcategory under convolution (cid:63) . Here, we use the convolution diagram H ck IG × (Div ) I H ck IG ( p ,p ) ←−−−− L + G \ LG × L + G LG/L + G m −→ H ck IG and deﬁne A (cid:63) B = Rm ∗ ( p ∗ A ⊗ L Λ p ∗ B ) . The map m is ind-proper (its ﬁbres are Gr IG ), and in particular proper on any bounded subset;thus, proper base change ensures that this deﬁnes an associative monoidal structure.On D ´et ( H ck IG , Λ) bd , one can deﬁne a relative perverse t -structure (where an object is perverse ifand only if it is perverse over any geometric ﬁbre of (Div ) I ). For this t -structure, the convolution .6. THE GEOMETRIC SATAKE EQUIVALENCE 27 (cid:63) is left t -exact (and t -exactness only fails for issues related to non-ﬂatness over Λ). To prove thatthere is a well-deﬁned t -structure, and the preservation of perversity under convolution, we adaptBraden’s theorems [ Bra03 ] on hyperbolic localization, and a degeneration to the Witt vector aﬃneGrassmannian [

Zhu17 ], [

BS17 ]. We will discuss hyperbolic localization further below.We remark that there is no general theory of perverse sheaves in p -adic geometry, the issue be-ing that it is diﬃcult to unambiguously assign a dimension to a point of an adic space (cf. [ Tem21 ]for what is known about topological transcendence degrees of points, and the subtleties especiallyin characteristic p ). In particular, we would not know how to deﬁne a notion of perverse sheaf on(Div ) I in general, which is the reason we revert to asking perversity only in the ﬁbres. Here, weuse that all geometric ﬁbres of the stack H ck IG → (Div ) I have only countably many points enu-merated explicitly in terms of dominant cocharacters µ i , and one can assign by hand the dimension (cid:80) i (cid:104) ρ, µ i (cid:105) of the corresponding open Schubert cells. Remark

I.6.1 . It seems to have been overlooked in the literature that for any map f : X → S locally of ﬁnite type between schemes, one can deﬁne a relative perverse t -structure, with relativeperversity equivalent to perversity on all geometric ﬁbres. It is a nontrivial result that there indeedis such a t -structure; this will appear in forthcoming work of Hansen and the second author. Thus,there is a good notion of “families of perverse sheaves”.Moreover, one can restrict to the complexes A ∈ D ´et ( H ck IG , Λ) bd that are universally locallyacyclic over (Div ) I . This condition is also preserved under convolution. Definition

I.6.2 . The Satake category

Sat IG (Λ) ⊂ D ´et ( H ck IG , Λ) bd is the category of all A ∈ D ´et ( H ck IG , Λ) bd that are perverse, ﬂat over Λ (i.e., for all Λ -modules M ,also A ⊗ L Λ M is perverse), and universally locally acyclic over (Div ) I . Intuitively, Sat IG (Λ) are the “ﬂat families of perverse sheaves on H ck IG → (Div ) I ”, whereﬂatness refers both to the geometric aspect of ﬂatness over (Div ) I (encoded in universal localacyclicity) and the algebraic aspect of ﬂatness in the coeﬃcients Λ. The Satake category Sat IG (Λ)is a monoidal category under convolution. Moreover, it is covariantly functorial in I .In fact, the monoidal structure naturally upgrades to a symmetric monoidal structure. Thisrelies on the fusion product, for which it is critical to allow general ﬁnite sets I . Namely, givenﬁnite sets I , . . . , I k , letting I = I (cid:116) . . . (cid:116) I k , one has an isomorphism H ck IG × (Div ) I (Div ) I ; I ,...,I k ∼ = k (cid:89) j =1 H ck I j G × (Div ) I (Div ) I ; I ,...,I k where (Div ) I ; I ,...,I k ⊂ (Div ) I is the open subset where x i (cid:54) = x i (cid:48) whenever i, i (cid:48) ∈ I lie in diﬀerent I j ’s. The exterior tensor product then deﬁnes a functor (cid:2) kj =1 : k (cid:89) j =1 Sat I j G (Λ) → Sat I ; I ,...,I k G (Λ) where Sat I ; I ,...,I k G (Λ) is the variant of Sat IG (Λ) for H ck IG × (Div ) I (Div ) I ; I ,...,I k . However, the re-striction functor Sat IG (Λ) → Sat I ; I ,...,I k G (Λ)is fully faithful, and the essential image of the exterior product lands in its essential image. Thus,we get a natural functor ∗ kj =1 : k (cid:89) j =1 Sat I j G (Λ) → Sat IG (Λ) , independent of the ordering of the I j . In particular, for any I , we get a functorSat IG (Λ) × Sat IG (Λ) → Sat I (cid:116) IG (Λ) → Sat IG (Λ) , using functoriality of Sat JG (Λ) in J , which deﬁnes a symmetric monoidal structure ∗ on Sat IG (Λ),commuting with (cid:63) . This is called the fusion product. In general, for any symmetric monoidalcategory ( C , ∗ ) with a commuting monoidal structure (cid:63) , the monoidal structure (cid:63) necessarily agreeswith ∗ ; thus, the fusion product reﬁnes the convolution product. (As usual in geometric Satake, weactually need to change ∗ slightly by introducing certain signs into the commutativity constraint,depending on the parity of the support of the perverse sheaves.)Moreover, restricting A ∈ Sat IG (Λ) to Gr IG and taking the pushforward to (Div ) I , all cohomol-ogy sheaves are local systems of Λ-modules on (Div ) I . By a version of Drinfeld’s lemma, theseare equivalent to representations of W IE on Λ-modules. This deﬁnes a symmetric monoidal ﬁbrefunctor F I : Sat IG (Λ) → Rep W IE (Λ) , where Rep W IE (Λ) is the category of continuous representations of W IE on ﬁnite projective Λ-modules.Using a version of Tannaka duality, one can then build a Hopf algebra in the Ind-category ofRep W IE (Λ) so that Sat IG (Λ) is given by its category of representations (internal in Rep W IE (Λ)). Forany ﬁnite set I , this is given by the tensor product of I copies of the corresponding Hopf algebrafor I = {∗} , which in turn is given by some aﬃne group scheme G (cid:86) over Λ with W E -action. Theorem

I.6.3 . There is a canonical isomorphism G (cid:86) ∼ = (cid:98) G with the Langlands dual group, underwhich the action of W E on G (cid:86) agrees with the usual action of W E on (cid:98) G up to an explicit cyclotomictwist. If √ q ∈ Λ , the cyclotomic twist can be trivialized, and Sat IG (Λ) is naturally equivalent to thecategory of ( (cid:98) G (cid:111) W E ) I -representations on ﬁnite projective Λ -modules. This theorem is thus a version of the theorem of Mirkovi´c–Vilonen [

MV07 ], coupled with thereﬁnements of Gaitsgory [

Gai07 ] for general I . (We remark that we formulate a theorem validfor any Λ, not necessarily regular; such a formulation does not seem to be in the literature. Also,we give a purely local proof: Most proofs require a globalization on a (usual) curve.) Contrary toMirkovi´c–Vilonen, we actually construct an explicit pinning of G (cid:86) . For the proof, one can restrictto Λ = Z /(cid:96) n Z ; passing to a limit over n , one can actually build a group scheme over Z (cid:96) . Its genericﬁbre is reductive, as the Satake category with Q (cid:96) -coeﬃcients is (geometrically) semisimple: Forthis, we again use the degeneration to the Witt vector aﬃne Grassmannian and the decompositiontheorem for schemes. To identify the reductive group, we argue ﬁrst for tori, and then for rank1 groups, where everything reduces to G = PGL which is easy to analyze by using the minus-cule Schubert cell. Here, the pinning includes a cyclotomic twist as of course the cohomology of .6. THE GEOMETRIC SATAKE EQUIVALENCE 29 the minuscule Schubert variety P of Gr PGL contains a cyclotomic twist. Afterwards, we applyhyperbolic localization in order to construct symmetric monoidal functors Sat G → Sat M for anyLevi M of G , inducing dually maps M (cid:86) → G (cid:86) . This produces many Levi subgroups of G (cid:86) Q (cid:96) fromwhich it is easy to get the isomorphism with (cid:98) G Q (cid:96) , including a pinning. As these maps M (cid:86) → G (cid:86) areeven deﬁned integrally, and (cid:98) G ( Z (cid:96) ) ⊂ (cid:98) G ( Q (cid:96) ) is a maximal compact open subgroup by Bruhat–Titstheory, generated by the rank 1 Levi subgroups, one can then deduce that G (cid:86) ∼ = (cid:98) G integrally, againwith an explicit (cyclotomic) pinning.We will also need the following addendum regarding a natural involution. Namely, the localHecke stack H ck G has a natural involution sw given by reversing the roles of the two G -torsors; inthe presentation in terms of LG , this is induced by the inversion on LG . Then sw ∗ induces naturallyan involution of Sat G (Λ), and this involution can be upgraded to a symmetric monoidal functorcommuting with the ﬁbre functor, thus realizing a W E -equivariant automorphism of G (cid:86) ∼ = (cid:98) G . Proposition

I.6.4 . The action of sw ∗ on Sat G induces the automorphism of (cid:98) G that is theCartan involution of the split group (cid:98) G , conjugated by (cid:98) ρ ( − . Critical to all of our arguments is the hyperbolic localization functor. In the setting of theBeilinson–Drinfeld Grassmannian, assume that P + , P − ⊂ G are two opposite parabolics, withcommon Levi M . We get a diagram Gr IP + q + (cid:124) (cid:124) p + (cid:35) (cid:35) Gr IG Gr IM . Gr IP − q − (cid:98) (cid:98) p − (cid:59) (cid:59) We get two “constant term” functorsCT + = R ( p + ) ! ( q + ) ∗ , CT − = R ( p − ) ∗ R ( q − ) ! : D ´et (Gr IG , Λ) bd → D ´et (Gr IM , Λ) bd , and one can construct a natural transformation CT − → CT + . The functor CT + correspondsclassically to the Satake transform, of integrating along orbits under the unipotent radical of U + .Hyperbolic localization claims that the transformation CT − → CT + is an equivalence when re-stricted to L + G -equivariant objects. This has many consequences; note that CT + is built from leftadjoint functors while CT − is built from right adjoint functors, so if they are isomorphic, hyperboliclocalization has the best of both worlds. In particular, hyperbolic localization commutes with allcolimits and all limits, preserves (relative) perversity, universal local acyclicity, commutes with anybase change, etc. .This is in fact a special case of the following more general assertion. Let S be any small v-stack,and f : X → S be a proper map that is representable in spatial diamonds with dim . trg f < ∞ .Assume that there is an action of G m on X/S , where G m ( R, R + ) = R × . The ﬁxed points X ⊂ X of the G m -action form a closed substack. We assume that one can deﬁne an attractor locus X + ⊂ X and a repeller locus X − ⊂ X , given by disjoint unions of locally closed subspaces, on which the t ∈ G m -action admits a limit as t → t → ∞ ). We get a diagram X + q + (cid:126) (cid:126) p + (cid:33) (cid:33) X X ,X − q − (cid:96) (cid:96) p − (cid:61) (cid:61) generalizing (bounded parts of) the above diagram if one chooses a cocharacter µ : G m → G whosedynamic parabolics are P + , P − . One can deﬁne L + = R ( p + ) ! ( q + ) ∗ , L − = R ( p − ) ∗ R ( q − ) ! : D ´et ( X, Λ) → D ´et ( X , Λ)and a natural transformation L − → L + . The following is our version of Braden’s theorem [ Bra03 ],cf. also [

Ric19 ]. Theorem

I.6.5 . The transformation L − → L + is an equivalence when restricted to the essentialimage of D ´et ( X/ G m , Λ) → D ´et ( X, Λ) . The proof makes use of the following principle: If Y → S is partially proper with a G m -action such that the quotient stack Y / G m is qcqs over S , then again Y admits two ends, and thepartially compactly supported cohomology of Y with coeﬃcients in any A ∈ D ´et ( Y / G m , Λ) vanishesidentically.

I.7. Cohomology of moduli spaces of shtuka

At this point, we have deﬁned D (Bun G , Λ) , and using the geometric Satake equivalence and the diagramHck IG q (cid:47) (cid:47) h (cid:123) (cid:123) h (cid:38) (cid:38) H ck IG Bun G Bun G × (Div ) I one can deﬁne the Hecke operator T V = Rh ∗ ( h ∗ ⊗ L Λ q ∗ S V ) : D (Bun G , Λ) → D (Bun G × (Div ) I , Λ)for any V ∈ Sat IG (Λ), where S V is the corresponding sheaf on H ck IG . This works at least if Λ iskilled by some power of (cid:96) . We can in fact extend this functor to all Z (cid:96) -algebras Λ. Moreover,its image lies in the full subcategory of those objects that are locally constant in the direction of(Div ) I , thereby giving a functor T V : D (Bun G , Λ) → D (Bun G , Λ) BW IE .7. COHOMOLOGY OF MODULI SPACES OF SHTUKA 31 to the category of W IE -equivariant objects in D (Bun G , Λ). The proof is surprisingly formal: Onereduces to I = {∗} by an inductive argument, and then uses that Div = Spd ˘ E/ϕ Z is still just apoint. More precisely, one uses that D (Bun G , Λ) → D (Bun G × Spd (cid:98) E, Λ)is an equivalence.

Remark

I.7.1 . To deﬁne D (Bun G , Λ) BW IE , we need to upgrade D (Bun G , Λ) to a condensed ∞ -category; then it is the notion of W IE -equivariant objects for the condensed group W IE .A ﬁrst consequence of our results is that T V , forgetting the W IE -equivariance, preserves ﬁnitenessproperties. Note that T V ◦ T W ∼ = T V ⊗ W as the geometric Satake equivalence is monoidal. Thisformally implies that T V is left and right adjoint to T V ∗ . From here, it is not hard to prove thefollowing result. Theorem

I.7.2 . The functor T V : D (Bun G , Λ) → D (Bun G , Λ) preserves compact objects anduniversally locally acyclic objects. Moreover, it commutes with Bernstein–Zelevinsky and Verdierduality in the sense that there are natural isomorphisms D BZ ( T V ( A )) ∼ = T sw ∗ V ∨ ( D BZ ( A )) and R H om( T V ( A ) , Λ) ∼ = T (sw ∗ V ) ∨ R H om( A, Λ) . Here sw ∗ is the involution of Sat IG which by Proposition I.6.4 is induced by the Cartan involutionof (cid:98) G , conjugated by (cid:98) ρ ( − (cid:96) n for some n ; for thegeneral formulation, we would need to discuss more precisely the foundational issues surroundingthe derived categories. In [ SW20 , Lecture XXIII], for any collection { µ i } i of conjugacy classes ofcocharacters with ﬁelds of deﬁnition E i /E and b ∈ B ( G ), there is deﬁned a tower of moduli spacesof local shtukas f K : (Sht ( G,b,µ • ) ,K ) K ⊂ G ( E ) → (cid:89) i ∈ I Spd ˘ E i as K ranges over compact open subgroups of G ( E ), equipped with compatible ´etale period maps π K : Sht ( G,b,µ • ) ,K → Gr tw G, (cid:81) i ∈ I Spd ˘ E i , ≤ µ • . Here, Gr tw G, (cid:81) i ∈ I Spd ˘ E i → (cid:81) i ∈ I Spd ˘ E is a certain twisted form of the convolution aﬃne Grassman-nian, cf. [ SW20 , Section 23.5]. Let W be the exterior tensor product (cid:2) i ∈ I V µ i of highest weightrepresentations, and S W the corresponding sheaf on Gr tw G, (cid:81) i ∈ I Spd ˘ E i . We continue to write S W forits pullback to Sht ( G,b,µ • ) ,K . Corollary

I.7.3 . The sheaf Rf K ! S W ∈ D ([ ∗ /G b ( E )] × (cid:89) i ∈ I Spd ˘ E i , Λ) is equipped with partial Frobenii, thus descends to an object of D ([ ∗ /G b ( E )] × (cid:89) i ∈ I Spd ˘ E i /ϕ Z i , Λ) . This object lives in the full subcategory D ( G b ( E ) , Λ) B (cid:81) i ∈ I W Ei ⊂ D ([ ∗ /G b ( E ))] × (cid:89) i ∈ I Spd ˘ E i /ϕ Z i , Λ) , and its restriction to D ( G b ( E ) , Λ) is compact. In particular, for any admissible representation ρ of G b ( E ) , the object R Hom G b ( E ) ( Rf K ! S W , ρ ) ∈ D (Λ) B (cid:81) i ∈ I W Ei is a representation of (cid:81) i ∈ I W E i on a perfect complex of Λ -modules. Taking the colimit over K , thisgives rise to a complex of admissible G ( E ) -representations lim −→ K R Hom G b ( E ) ( Rf K ! S W , ρ ) equipped with a (cid:81) i ∈ I W E i -action.If ρ is compact, then so is lim −→ K R Hom G b ( E ) ( Rf K ! S W , ρ ) as a complex of G ( E ) -representations. Specializing to I = {∗} and µ minuscule, we get local Shimura varieties, and this proves theﬁniteness properties of [ RV14 , Proposition 6.1] unconditionally, as well as [

RV14 , Remark 6.2 (iii)].We note that those properties seem inaccessible using only the deﬁnition of the moduli spaces ofshtukas, i.e. without the use of Bun G . I.8. The stack of L -parameters Let us discuss the other side of the Langlands correspondence, namely (the stack of) L -parameters. This has been previously done by Dat–Helm–Kurinczuk–Moss [ DHKM20 ] and Zhu[

Zhu20 ]. One wants to deﬁne a scheme whose Λ-valued points, for a Z (cid:96) -algebra Λ, are the contin-uous 1-cocycles ϕ : W E → (cid:98) G (Λ) . (Here, we endow (cid:98) G with its usual W E -action, that factors over a ﬁnite quotient Q of W E . Asdiscussed above, the diﬀerence between the two actions disappears over Z (cid:96) [ √ q ], and we ﬁnd itmuch more convenient to use the standard normalization here, so that we can sometimes make useof the algebraic group (cid:98) G (cid:111) Q .)There seems to be a mismatch here, in asking for an algebraic stack, but continuous cocycles.Interestingly, there is a way to phrase the continuity condition that produces a scheme. Namely,we consider Λ as a condensed Z (cid:96) -algebra that is “relatively discrete over Z (cid:96) ”. Abstract Z (cid:96) -modules M embed fully faithfully into condensed Z (cid:96) -modules, via sending M to M disc ⊗ Z (cid:96), disc Z (cid:96) . Theorem

I.8.1 . There is a scheme Z ( W E , (cid:98) G ) over Z (cid:96) whose Λ -valued points, for a Z (cid:96) -algebra Λ , are the condensed -cocycles ϕ : W E → (cid:98) G (Λ) , where we regard Λ as a relatively discrete condensed Z (cid:96) -algebra. The scheme Z ( W E , (cid:98) G ) is a unionof open and closed aﬃne subschemes Z ( W E /P, (cid:98) G ) as P runs through open subgroups of the wild .8. THE STACK OF L -PARAMETERS 33 inertia subgroup of W E , and each Z ( W E /P, (cid:98) G ) is a ﬂat local complete intersection over Z (cid:96) ofdimension dim G . The point here is that the inertia subgroup of W E has a Z (cid:96) -factor, and this can map in interestingways to Λ when making this deﬁnition. To prove the theorem, following [ DHKM20 ] and [

Zhu20 ]we deﬁne discrete dense subgroups W ⊂ W E /P by discretizing the tame inertia, and the restriction Z ( W E /P, (cid:98) G ) → Z ( W, (cid:98) G ) is an isomorphism, where the latter is clearly an aﬃne scheme.We can also prove further results about the (cid:98) G -action on Z ( W E , (cid:98) G ), or more precisely each Z ( W E /P, (cid:98) G ). For this result, we need to exclude some small primes, but if G = GL n , all primes (cid:96) are allowed; for classical groups, all (cid:96) (cid:54) = 2 are allowed. More precisely, we say that (cid:96) is very goodfor (cid:98) G if the following conditions are satisﬁed.(i) The (algebraic) action of W E on (cid:98) G factors over a ﬁnite quotient Q of order prime to (cid:96) .(ii) The order of the fundamental group of the derived group of (cid:98) G is prime to (cid:96) (equivalently, π Z ( G )is of order prime to (cid:96) ).(iii) If G has factors of type B , C , or D , then (cid:96) (cid:54) = 2; if it has factors of type E , F , or G , then (cid:96) (cid:54) = 2 ,

3; and if it has factors of type E , then (cid:96) (cid:54) = 2 , , Theorem

I.8.2 . Assume that (cid:96) is a very good prime for (cid:98) G . Then H i ( (cid:98) G, O ( Z ( W E /P, (cid:98) G ))) = 0 for i > and the formation of the invariants O ( Z ( W E /P, (cid:98) G )) (cid:98) G commutes with any base change.The algebra O ( Z ( W E /P, (cid:98) G )) (cid:98) G admits an explicit presentation in terms of excursion operators, O ( Z ( W E /P, (cid:98) G )) (cid:98) G = colim ( n,F n → W ) O ( Z ( F n , (cid:98) G )) (cid:98) G where the colimit runs over all maps from a free group F n to W ⊂ W E /P , and Z ( F n , (cid:98) G ) ∼ = (cid:98) G n with the simultaneous twisted (cid:98) G -conjugation.Moreover, the ∞ -category Perf( Z ( W E /P, (cid:98) G ) / (cid:98) G ) is generated under cones and retracts by theimage of Rep( (cid:98) G ) → Perf( Z ( W E /P, (cid:98) G ) / (cid:98) G ) , and Ind Perf( Z ( W E /P, (cid:98) G )) is equivalent to the ∞ -category of modules over O ( Z ( W E /P, (cid:98) G )) in Ind Perf( B (cid:98) G ) .All of these results also hold with Q (cid:96) -coeﬃcients, without any assumption on (cid:96) . With Q (cid:96) -coeﬃcients, these results are simple, as the representation theory of (cid:98) G is semisimple.However, with Z (cid:96) -coeﬃcients, these results are quite subtle, and we need to dive into modularrepresentation theory of reductive groups. Very roughly, the proof of the theorem proceeds byanalyzing the closed (cid:98) G -orbits in the stack of L -parameters ﬁrst, and then use a deformation to thenormal cone to understand the behaviour near any (cid:98) G -orbit. We make critical use of some resultsof Touz´e–van der Kallen [ TvdK10 ]. Let us make some further remarks about the closed (cid:98) G -orbits.First, the closed (cid:98) G -orbits in Z ( W E /P, (cid:98) G ) L , for any algebraically closed ﬁeld L over Z (cid:96) , corre-spond to the semisimple L -parameter ϕ : W E → (cid:98) G ( L ), and also biject to the geometric points ofSpec O ( Z ( W E /P, (cid:98) G )) (cid:98) G . Here, an L -parameter is semisimple if, whenever it factors over a para-bolic (cid:98) P ⊂ (cid:98) G , it also factors over a Levi (cid:99) M ⊂ (cid:98) P . Any semisimple parameter is in fact continuousfor the discrete topology on L , i.e. trivial on an open subgroup of I E . If L = F (cid:96) and Q is of orderprime to (cid:96) , then ϕ is semisimple if and only if it factors over a ﬁnite quotient of order prime to (cid:96) . Its orbit in Z ( W E /P, (cid:98) G ) L is then given by (cid:98) G/S ϕ where S ϕ ⊂ (cid:98) G is the centralizer of ϕ , which isin that case the ﬁxed points under the action of a ﬁnite solvable group F of automorphisms of (cid:98) G ,where the order of F is prime to (cid:96) . We need the following result. Theorem

I.8.3 . Let H be a reductive group over an algebraically closed ﬁeld L of characteristic (cid:96) . Let F be a ﬁnite group of order prime to (cid:96) acting on H . Then H F is a smooth linear algebraicgroup whose connected component is reductive, and with π H F of order prime to (cid:96) . If F is solvable,the image of Perf( BH ) → Perf( BH F ) generates under cones and retracts. The last part of this theorem is proved by a very explicit (and exhausting) analysis of allpossible cases. It would be desirable to have a more enlightening argument, possibly also removingthe assumption that F is solvable. In fact, we would expect similar results to hold true in thecase where W is replaced by the fundamental group of a Riemann surface. Our arguments are notgeneral enough to handle that case. Remark

I.8.4 . While the hypotheses imposed on (cid:96) are surely not optimal, we are quite surethat some hypothesis on (cid:96) is required in Theorem I.8.2. For example, if (cid:98) G = PGL and (cid:96) = 2,we expect problems to arise. For example, one can show that for X = (cid:98) G with the conjugationaction by (cid:98) G , the ∞ -category Perf( X/ (cid:98) G ) is not generated under cones and retracts by the imageof Rep( (cid:98) G ). Our guess would be that the condition that (cid:96) does not divide the order of π ( (cid:98) G der ) isessential.On the other hand, we expect that, for example by the use of z-embeddings [ Kal14 , Section 5],one can reduce all relevant questions (like the general construction of maps on Bernstein centersdiscussed below, or the construction of the spectral action) to the case where (cid:96) does not divide theorder of π ( (cid:98) G der ). We are not taking this up here. I.9. Construction of L -parameters Finally, we can discuss the construction of L -parameters. Assume ﬁrst for simplicity that Λ = Q (cid:96) with ﬁxed √ q ∈ Q (cid:96) , and let A ∈ D (Bun G , Q (cid:96) ) be any Schur-irreducible object, i.e. End( A ) = Q (cid:96) .For example, A could correspond to an irreducible smooth representation of G ( E ), taking theextension by zero along [ ∗ /G ( E )] (cid:44) → Bun G . Then, following V. Laﬀorgue [ Laf18 ], we can deﬁneexcursion operators as follows. For any representation V of ( (cid:98) G (cid:111) W E ) I over Q (cid:96) , together with maps α : Q (cid:96) → V | (cid:98) G , β : V | (cid:98) G → Q (cid:96) when restricted to the action of the diagonal copy (cid:98) G ⊂ ( (cid:98) G (cid:111) W E ) I , and elements γ i ∈ W E for i ∈ I ,we can deﬁne the endomorphism A T α −→ T V ( A ) ( γ i ) i ∈ I −−−−→ T V ( A ) T β −→ A of A , deﬁning an element of Q (cid:96) . With all the formalism in place, the following result is essentiallydue to V. Laﬀorgue [ Laf18 , Proposition 11.7].

Proposition

I.9.1 . There is a unique continuous semisimple L -parameter ϕ A : W E → (cid:98) G ( Q (cid:96) ) .9. CONSTRUCTION OF L -PARAMETERS 35 such that for all ( I, V, α, β, ( γ i ) i ∈ I ) as above, the excursion operator A T α −→ T V ( A ) ( γ i ) i ∈ I −−−−→ T V ( A ) T β −→ A is given by multiplication with the scalar Q (cid:96) α −→ V ( ϕ A ( γ i ) i ∈ I ) −−−−−−−→ V β −→ Q (cid:96) . Note that in fact, the excursion operators deﬁne elements in the Bernstein center of G ( E ), asthey deﬁne endomorphisms of the identity functor. From this perspective, let us make the followingdeﬁnition. Definition

I.9.2 . (i) The Bernstein center of G ( E ) is Z ( G ( E ) , Λ) = π End(id D ( G ( E ) , Λ) ) = lim ←− K ⊂ G ( E ) Z (Λ[ K \ G ( E ) /K ]) where K runs over open pro- p subgroups of G ( E ) , and Λ[ K \ G ( E ) /K ] = End G ( E ) ( c - Ind G ( E ) K Λ) isthe Hecke algebra of level K . (ii) The geometric Bernstein center of G is Z geom ( G, Λ) = π End(id D lis (Bun G , Λ) ) . Inside Z geom ( G, Λ) , we let Z geomHecke ( G, Λ) be the subring of all endomorphisms f : id → id commutingwith Hecke operators, in the sense that for all V ∈ Rep( (cid:98) G I ) and A ∈ D lis (Bun G , Λ) , one has T V ( f ( A )) = f ( T V ( A )) ∈ End( T V ( A )) . (iii) The spectral Bernstein center of G is Z spec ( G, Λ) = O ( Z ( W E , (cid:98) G ) Λ ) (cid:98) G , the ring of global functions on the quotient stack Z ( W E , (cid:98) G ) Λ / (cid:98) G . The inclusion D ( G ( E ) , Λ) (cid:44) → D lis (Bun G , Λ) induces a map of algebra Z geom ( G, Λ) → Z ( G ( E ) , Λ).Now the construction of excursion operators, together with Theorem I.8.2 imply the following.Here Λ is a Z (cid:96) [ √ q ]-algebra. Proposition

I.9.3 . Assume that (cid:96) is invertible in Λ , or (cid:96) is a very good prime for (cid:98) G . Thenthere is a canonical map Z spec ( G, Λ) → Z geomHecke ( G, Λ) ⊂ Z geom ( G, Λ) , and in particular a map Ψ G : Z spec ( G, Λ) → Z ( G ( E ) , Λ) . In fact, even for general (cid:96) , one gets similar maps, up to replacing Z ( W E , (cid:98) G ) (cid:12) (cid:98) G by a universallyhomeomorphic scheme. The construction of L -parameters above is then a consequence of this mapon Bernstein centers. The existence of such an integral map is due to Helm–Moss [ HM18 ] in thecase G = GL n . Remark

I.9.4 . In the function-ﬁeld case, a similar construction has been given by Genestier–Laﬀorgue [

GL17 ]. We expect that the two constructions agree, and that proving this is withinreach (as both are based on the cohomology of moduli spaces of shtukas), but have not thoughtmuch about it.We make the following conjecture regarding independence of (cid:96) . For its formulation, we notethat there is a natural Q -algebra Z spec ( G, Q ) whose base change to Q (cid:96) is Z spec ( G, Q (cid:96) ) for any (cid:96) (cid:54) = p ;in fact, one can take the global functions on the stack of L -parameters that are continuous for thediscrete topology (i.e. trivial on an open subgroup of W E ); see also [ DHKM20 ]. Conjecture

I.9.5 . There is a (necessarily unique) map Z spec ( G, Q ( √ q )) → Z ( G ( E ) , Q ( √ q )) that after base extension to any Q (cid:96) for (cid:96) (cid:54) = p recovers the composite Z spec ( G, Q (cid:96) ( √ q )) → Z geom ( G, Q (cid:96) ( √ q )) → Z ( G ( E ) , Q (cid:96) ( √ q )) . This would ensure that the L -parameters we construct are independent of (cid:96) in the relevantsense. Further conjectures about this map and its relation to the stable Bernstein center havebeen formulated by Haines [ Hai14 ] (see also [

BKV15 ], [

SS13 , Section 6]). In particular, it isconjectured that for G quasisplit, the map Ψ G is injective, and its image can be characterized asthose elements of the Bernstein center of G ( E ) whose corresponding distribution is invariant understable conjugation.One can also construct the map to the Bernstein center in terms of moduli spaces of localshtukas, as follows. For simplicity, we discuss this again only if Λ is a Z /(cid:96) n -algebra for some n . Given I and V as above, we can consider a variant Sht ( G, ,V ) ,K of the spaces Sht ( G,b, ≤ µ • ) ,K considered above, where the bound is given by the support of V and we ﬁx the element b = 1.They come with an ´etale period map π K : Sht ( G, ,V ) ,K → Gr tw G, (cid:81) ni =1 Spd ˘ E and a perverse sheaf S V . When restricted to the geometric diagonal x : Spd (cid:98) E → n (cid:89) i =1 Spd ˘ E, they become a corresponding moduli space of shtukas with one leg f ∆ K : Sht ( G, ,V | (cid:98) G ) → Spd (cid:98) E with the sheaf S V | (cid:98) G . The sheaf S V | (cid:98) G admits maps α (resp. β ) from (resp. to) the sheaf i ∗ Λ, where i : G ( E ) /K = Sht ( G, , Q (cid:96) ) ,K (cid:44) → Sht ( G, ,V | (cid:98) G ) ,K is the subspace of shtukas with no legs. This producesan endomorphism c -Ind G ( E ) K Λ α −→ Rf ∆ K ! S V | (cid:98) G = ( Rf K ! S V ) x ( γ i ) i ∈ I −−−−→ ( Rf K ! S V ) x = Rf ∆ K ! S V | (cid:98) G β −→ c -Ind G ( E ) K Λ . Here, the action of ( γ i ) i ∈ I is deﬁned by Corollary I.7.3. It follows from the deﬁnitions that thisis precisely the previous construction applied to the representation c -Ind G ( E ) K Λ. Note that theseendomorphisms are G ( E )-equivariant, so deﬁne elements in the Hecke algebraΛ[ K \ G ( E ) /K ] = End G ( E ) ( c -Ind G ( E ) K Λ); .10. THE SPECTRAL ACTION 37 in fact, these elements are central (as follows by comparison to the previous construction). Takingthe inverse limit over K , one gets the elements in the Bernstein center of G ( E ). Concerning the L -parameters we construct, we can prove the following basic results. Theorem

I.9.6 . (i) If G = T is a torus, then π (cid:55)→ ϕ π is the usual Langlands correspondence. (ii) The correspondence π (cid:55)→ ϕ π is compatible with twisting. (iii) The correspondence π (cid:55)→ ϕ π is compatible with central characters. (iv) The correspondence π (cid:55)→ ϕ π is compatible with passage to congradients. (v) If G (cid:48) → G is a map of reductive groups inducing an isomorphism of adjoint groups, π is anirreducible smooth representation of G ( E ) and π (cid:48) is an irreducible constitutent of π | G (cid:48) ( E ) , then ϕ π (cid:48) is the image of ϕ π under the induced map (cid:98) G → (cid:99) G (cid:48) . (vi) If G = G × G is a product of two groups and π is an irreducible smooth representation of G ( E ) , then π = π (cid:2) π for irreducible smooth representations π i of G i ( E ) , and ϕ π = ϕ π × ϕ π under (cid:98) G = (cid:98) G × (cid:98) G . (vii) If G = Res E (cid:48) | E G (cid:48) is the Weil restriction of scalars of a reductive group G (cid:48) over some ﬁnite sep-arable extension E (cid:48) | E , so that G ( E ) = G (cid:48) ( E (cid:48) ) , then L -parameters for G | E agree with L -parametersfor G (cid:48) | E (cid:48) . (viii) The correspondence π (cid:55)→ ϕ π is compatible with parabolic induction. (ix) For G = GL n and supercuspidal π , the correspondence π (cid:55)→ ϕ π agrees with the usual localLanglands correspondence [ LRS93 ] , [ HT01 ] , [ Hen00 ] . Note that parts (viii) and (ix) together say that for GL n and general π , the L -parameter ϕ π iswhat is usually called the semisimple L -parameter. I.10. The spectral action

The categorical structure we have constructed actually produces something better. Let Λbe the ring of integers in a ﬁnite extension of Q (cid:96) ( √ q ). We have the stable ∞ -category C = D lis (Bun G , Λ) ω of compact objects, which is linear over Λ, and functorially in the ﬁnite set I anexact monoidal functor Rep Λ ( (cid:98) G (cid:111) Q ) I → End Λ ( C ) BW IE that is linear over Rep Λ ( Q I ); here, End Λ ( C )denotes the stable ∞ -category of Λ-linear endofunctors of C , and we regard it as being enriched incondensed Λ-modules via regarding C as enriched in relatively discrete condensed Λ-modules. Aﬁrst version of the following theorem is due to Nadler–Yun [ NY19 ] in the context of Betti geometricLanglands, and a more general version appeared in the work of Gaitsgory–Kazhdan–Rozenblyum–Varshavsky [

SW20 ], this was the construction of excursion operatorsthat we envisaged. Note that a key step here is that the cohomology of moduli spaces of local shtukas deﬁnes a localsystem on (Div ) I . It is however not clear how to prove this purely in terms of moduli space of shtukas. In the globalfunction ﬁeld case, this result has been obtained by Xue [ Xue20 ]. Note that Z ( W E , (cid:98) G ) is not quasicompact, as it has inﬁnitely many connected components; itcan be written as the increasing union of open and closed quasicompact subschemes Z ( W E /P, (cid:98) G ).We say that an action of Perf( Z ( W E , (cid:98) G ) / (cid:98) G ) on a stable ∞ -category C is compactly supported iffor all X ∈ C the functor Perf( Z ( W E , (cid:98) G ) / (cid:98) G ) → C (induced by acting on X ) factors over somePerf( Z ( W E /P, (cid:98) G ) / (cid:98) G ).We stress again that for G = GL n , all (cid:96) are very good, and for classical groups, all (cid:96) (cid:54) = 2 arevery good. Theorem

I.10.1 . Assume that (cid:96) is a very good prime for (cid:98) G . Let C be a small Λ -linear stable ∞ -category. Then giving, functorially in the ﬁnite set I , an exact Rep Λ ( Q I ) -linear monoidal functor Rep( (cid:98) G (cid:111) Q ) I → End Λ ( C ) BW IE is equivalent to giving a compactly supported Λ -linear action of Perf( Z ( W E , (cid:98) G ) Λ / (cid:98) G ) . Here, given a compactly supported Λ -linear action of Perf( Z ( W E , (cid:98) G ) Λ / (cid:98) G ) , one can produce suchan exact Rep Λ ( Q I ) -linear monoidal functor Rep Λ ( (cid:98) G (cid:111) Q ) I → End Λ ( C ) BW IE functorially in I by composing the exact Rep Λ ( Q I ) -linear symmetric monoidal functor Rep( (cid:98) G (cid:111) Q ) I → Perf( Z ( W E , (cid:98) G ) Λ / (cid:98) G ) BW IE with the action of Perf( Z ( W E , (cid:98) G ) Λ / (cid:98) G ) .The same result holds true with Λ a ﬁeld over Q (cid:96) , for any prime (cid:96) . Here, the exact Rep Λ ( Q I )-linear symmetric monoidal functorRep Λ ( (cid:98) G (cid:111) Q ) I → Perf( Z ( W E , (cid:98) G ) Λ / (cid:98) G ) BW IE is induced by tensor products and the exact Rep Λ ( Q )-linear symmetric monoidal functorRep Λ ( (cid:98) G (cid:111) Q ) → Perf( Z ( W E , (cid:98) G ) Λ / (cid:98) G ) BW E corresponding to the universal (cid:98) G (cid:111) Q -torsor, with the universal W E -equivariance as parametrizedby Z ( W E , (cid:98) G ) / (cid:98) G .The key part of the proof is actually the ﬁnal part of Theorem I.8.2 above, which eﬀectivelydescribes Perf( Z ( W E /P, (cid:98) G ) / (cid:98) G ) in terms of generators and relations, as does the present theorem.In particular, we get an action of Perf( Z ( W E , (cid:98) G ) Λ / (cid:98) G ) on D lis (Bun G , Λ), suitably compatiblewith the Hecke action.With everything in place, it is now obvious that the main conjecture is the following, cf. [

AG15 ],[

Zhu20 ], [

Hel20 ]: The formulation of the conjecture presumes a spectral action, which we have only constructed under a smallassumption on (cid:96) ; implicit is thus that a spectral action, reﬁning the Hecke action, can be deﬁned for all (cid:96) . .11. THE ORIGIN OF THE IDEAS 39 Conjecture

I.10.2 . Assume that G is quasisplit and choose Whittaker data consisting of aBorel B ⊂ G and generic character ψ : U ( E ) → O × L of the unipotent radical U ⊂ B , where L/ Q (cid:96) is some algebraic extension; also ﬁx √ q ∈ O L . Then there is an equivalence D (Bun G , O L ) ω ∼ = D b, qccoh , Nilp ( Z ( W E , (cid:98) G ) O L / (cid:98) G ) of stable ∞ -categories equipped with actions of Perf( Z ( W E , (cid:98) G ) O L / (cid:98) G ) . Under this correspondence,the structure sheaf of Z ( W E , (cid:98) G ) O L / (cid:98) G maps to the Whittaker sheaf, which is the sheaf concentratedon Bun G corresponding to the Whittaker representations c - Ind G ( E ) U ( E ) ψ . Here, we use the notion of complexes of coherent sheaves with nilpotent singular support, see[

AG15 ]. More precisely, D b, qccoh , Nilp is the ∞ -category of bounded complexes with quasicompactsupport, coherent cohomology, and nilpotent singular support. With characteristic 0 coeﬃcients,or at banal primes (cid:96) , the condition of nilpotent singular support is actually automatic.If W ψ is the Whittaker sheaf and we note ∗ the spectral action, the conjecture thus says thatPerf qc ( Z ( W E , (cid:98) G ) O L / (cid:98) G ) −→ D (Bun G , O L ) M (cid:55)−→ M ∗ W ψ is fully faithful and extends to an equivalence of stable ∞ -categories D b, qccoh , Nilp ( Z ( W E , (cid:98) G ) O L / (cid:98) G ) ∼ = D (Bun G , O L ) ω . Remark

I.10.3 . Consider the conjecture with coeﬃcients in Q (cid:96) . Ideally, the conjecture shouldalso include a comparison of t -structures. Unfortunately, we did not immediately see a good can-didate for matching t -structures. Ideally, this would compare the perverse t -structure on the left(which is well-deﬁned, for abstract reasons, and appears at least implicitly in [ CS17 ], [

CS19a ];it seems to be the “correct” t -structure for questions of local-global compatibility) with some“perverse-coherent” t -structure on the right. If so, the equivalence would also yield a bijectionbetween irreducible objects in the abelian hearts. On the left-hand side, these irreducible objectswould then be enumerated by pairs ( b, π b ) of an element b ∈ B ( G ) and an irreducible smoothrepresentation π b of G b ( E ), by using intermediate extensions. On the right-hand side, they wouldlikely correspond to a Frobenius-semisimple L -parameter ϕ : W E → (cid:98) G ( Q (cid:96) ) together with an irre-ducible representation of the centralizer S ϕ of ϕ . Independently of the categorical conjecture, onecan wonder whether these two sets are in fact canonically in bijection. I.11. The origin of the ideas

Finally, let us give some account of the historical developments of these ideas. Let us ﬁrstrecall some of our early work in the direction of local Langlands correspondences. Fargues [

Far04 ]has proved that in the cohomology of basic Rapoport–Zink spaces for GL n (and U (3)) and generalminuscule cocharacters, an appropriate version of the local Langlands correspondence is realized.Moreover, Fargues [ Far08 ] has proved the duality isomorphism between the Lubin–Tate and Drin-feld tower. Already at this point Fargues thought of this as an attempt to geometrize the Jacquet-Langlands correspondence, see [

Far08 , Theorem 2 of the Pr´eambule]. On the other hand, Scholze[

Sch13 ] has given a new proof of the local Langlands correspondence for GL n . His results pointed to the idea that there ought to exist certain sheaves on the moduli stack of p -divisible groups (which,when restricted to perfect schemes, can be regarded as a “part” of the stack GL n - I soc consideredabove), giving a certain geometrization of the local Langlands correspondence, then formulated asa certain character sheaf property (inspired by the character formulas in [ Sch13 ]). Related ob-servations were also made by Boyer (cf. e.g. [

Boy09 ]) and in unpublished work of Dat. However,Scholze was always uneasy with the very bad geometric properties of the stack of p -divisible groups.At this point, both of us had essentially left behind local Langlands to study other questions.Fargues found the fundamental curve of p -adic Hodge theory in his work with Fontaine [ FF18 ]; aninitial critical motivation for Fargues was a development of “ p -adic Hodge theory without Galoisactions”, i.e. for ﬁelds like C p . Indeed, this was required in some of his work on Rapoport–Zinkspaces. On the other hand, Scholze developed perfectoid spaces [ Sch12 ], motivated by the weight-monodromy conjecture. After his talk at a conference in Princeton in March 2011, Weinstein gavea talk about his results on the Lubin–Tate tower at inﬁnite level, which made it clear that it isin fact a perfectoid space. Scholze at the time was already eager to understand the isomorphismbetween Lubin–Tate and Drinfeld tower, and it now became clear that it should really be an iso-morphism of perfectoid spaces. This was worked out in [

SW13 ]. At the time of writing of [

SW13 ],the perspective of the Fargues–Fontaine curve had already become central, and we realized thatthe isomorphism of the towers simply amounts to two dual descriptions of the space of minusculemodiﬁcations O nX → O X ( n ) on the Fargues–Fontaine curve, depending on which bundle is ﬁxedand which one is the modiﬁcation. This was the ﬁrst clear connection between local Langlands (asencoded in the cohomology of Lubin–Tate and Drinfeld space) and the theory of vector bundleson the Fargues–Fontaine curve, which Scholze had however not taken seriously enough. Moreover,Fargues had noted in [ FF18 ], in the proof of “weakly admissible implies admissible”, that modiﬁca-tions of vector bundles were playing an important role: the Hodge ﬁltration of a ﬁltered ϕ -moduleallows one to deﬁne a new vector bundle by modifying the vector bundle associated to an isocrystali.e. by “applying a Hecke correspondence” as he said in the talk [ Far10 ] at the conference in honorof Jean-Marc Fontaine.This duality perspective also put the two dual period morphisms into the center of attention:The Hodge–de Rham period mapping, and the Hodge–Tate period mapping (which are swappedunder the duality isomorphism). Thinking about the Lubin–Tate tower as part of the moduli spaceof elliptic curves, Scholze then realized that the Hodge–Tate period map even exists globally on themoduli space of elliptic curves with inﬁnite level (on the level of Berkovich topological spaces, thishad also been observed by Fargues before). Moreover, Scholze realized that the Hodge–Tate periodmap gives a substitute for the map from the moduli space of elliptic curves to the moduli space of p -divisible groups, and that the sheaves he sought for a geometric interpretation of [ Sch13 ] have abetter chance of existing on the target of the Hodge-Tate period map, which is simply a projectivespace over C p ; he sketched these ideas in an MSRI talk [ Sch14 ]. (Again, Dat has had similarideas.) Eventually, this perspective was used in his work with Caraiani [

CS17 ], [

CS19a ] to studytorsion in the cohomology of Shimura varieties. The work with Caraiani required the classiﬁcationof G -torsors on the Fargues–Fontaine curve, which was proved by Fargues [ Far18b ].Increasingly taking the perspective of studying all geometric objects by mapping only perfectoidspaces in, the idea of diamonds emerged quickly, including the possibility of getting several copiesof Spec Q p (the earliest published incarnation of this idea is [ Wei17 ]), and of deﬁning generalmoduli spaces of p -adic shtukas. These ideas were laid out in Scholze’s Berkeley course [ SW20 ] .11. THE ORIGIN OF THE IDEAS 41 during the MSRI trimester in Fall 2014. The eventual goal was always to adapt V. Laﬀorgue’swork [ Laf18 ] to the case of p -adic ﬁelds; the original strategy was to deﬁne the desired excursionoperators via the cohomology of moduli spaces of local shtukas. At the beginning of the trimester,Scholze was still very wary about the geometric Langlands program, as it did not seem to be ableto incorporate the subtle arithmetic properties of supercuspidal representations of p -adic groups.It was thus a completely unexpected conceptual leap that in fact the best perspective for the wholesubject is to view the local Langlands correspondence as a geometric Langlands correspondenceon the Fargues–Fontaine curve, which Fargues suggested over a coﬀee break at MSRI (partlyinspired by having thought intensely about the space of G -bundles on the curve in relation to[ Far18b ]). Fargues was taking the perspective of Hecke eigensheaves then, seeking to construct forany (discrete) L -parameter ϕ an associated Hecke eigensheaf A ϕ on Bun G with eigenvalue ϕ . Thisshould deﬁne a functor ϕ (cid:55)→ A ϕ , and thus carry an action of the centralizer group S ϕ ⊂ (cid:98) G of ϕ ,and the corresponding S ϕ -isotypic decomposition of A ϕ should realize the internal structures of the L -packets. Moreover, the Hecke eigensheaf property should imply the Kottwitz conjecture [ RV14 ,Conjecture 7.3] on the cohomology of local Shimura varieties. This made everything come together.In particular, it gave a compelling geometric origin for the internal structure of L -packets, and alsomatched the recent work of Kaletha [ Kal14 ] who used basic G -isocrystals for the ﬁne study of L -packets.Unfortunately, the conjecture was formulated on extremely shaky grounds: It presumed thatone could work with the moduli stack Bun G as if it were an object of usual algebraic geometry.Of course, it also presumed that there is a version of geometric Satake, etc.pp. On the otherhand, we realized that once we could merely formulate Fargues’ conjecture, enough machinery isavailable to apply Laﬀorgue’s ideas [

Laf18 ] to get the “automorphic-to-Galois” direction and deﬁne(semisimple) L -parameters (as Genestier–Laﬀorgue [ GL17 ] did in equal characteristic).Since then, it has been a long and very painful process. The ﬁrst step was to give a gooddeﬁnition of the category of geometric objects relevant to this picture, i.e. diamonds. In particular,one had to prove that the relevant aﬃne Grassmannians have this property. This was the mainresult of the Berkeley course [

SW20 ]. For the proof, the concept of v-sheaves was introduced,which has since taken on a life of its own also in algebraic geometry (cf. [

BM18 ]). (Generally,v-descent turned out to be an extremely powerful proof technique. We use it here to reprove thebasic theorems about the Fargues–Fontaine curve, recovering the main theorems of [

FF18 ] and[

KL15 ] with little eﬀort.) Next, one had to develop a 6-functor formalism for the ´etale cohomologyof diamonds, which was achieved in [

Sch17a ], at least with torsion coeﬃcients. The passageto Q (cid:96) -coeﬃcients requires more eﬀort than for schemes, and we will comment on it below. Acentral technique of [ Sch17a ] is pro-´etale descent, and more generally v-descent. In fact, virtuallyall theorems of [

Sch17a ] are proved using such descent techniques, essentially reducing them toproﬁnite collections of geometric points. It came as a surprise to Scholze that this process ofdisassembling smooth spaces into proﬁnite sets has any power in proving geometric results, andthis realization gave a big impetus to the development of condensed mathematics (which in turnfueled back into the present project).At this point, it became possible to contemplate Fargues’ conjecture. In this respect, the ﬁrstresult that had to be established is that D ´et (Bun G , Z /(cid:96) n Z ) is well-behaved, for example satisﬁesVerdier biduality for “admissible” sheaves. We found a proof, contingent on the cohomological smoothness of a certain “chart” π b : M b → Bun G for Bun G near any b ∈ B ( G ); this was explainedin Scholze’s IH´ES course [ Sch17b ]. While for G = GL n , the cohomological smoothness of π b couldbe proved by a direct attack, in general we could only formulate it as a special case of a general“Jacobian criterion of smoothness” for spaces parametrizing sections of Z → X S for some smoothadic space Z over the Fargues–Fontaine curve. Proving this Jacobian criterion required three furtherkey ideas. The ﬁrst is the notion of “formal smoothness”, where liftings to inﬁnitesimal thickenings(that do not exist in perfectoid geometry) are replaced by liftings to actual small open (or ´etale)neighborhoods. The resulting notion is closely related to the notion of absolute neighborhoodretracts in classical topology [ Bor67 ]. Through some actual “analysis”, it is not hard to provethat the space of sections is formally smooth. Unfortunately, this does not seem to be enough toguarantee cohomological smoothness. The ﬁrst issue is that formal smoothness does not imply anyﬁnite-dimensionality. Here, the second key idea comes in, which is Bhatt’s realization [

BS19 ] thatZariski closed immersions are strongly Zariski closed in the sense of [

Sch15 , Section II.2] (contraryto a claim made by Scholze there). At this point, it would be enough to show that spaces thatare formally smooth and Zariski closed in a ﬁnite-dimensional perfectoid ball are cohomologicallysmooth. Unfortunately, despite many tries, we are still unable to prove that even the diﬀerentnotions of dimension of [

Sch17a ] (Krull dimension, dim . trg, cohomological dimension) agree forsuch spaces. This may well be the most important foundational open problem in the theory: Problem

I.11.1 . Let X ⊂ (cid:101) B nC be Zariski closed, where (cid:101) B n is a perfectoid ball. Show that X hasa well-behaved dimension. In fact, we ﬁnd it crazy that we are able to prove all sorts of nontrivial geometric results withoutever being able to unambiguously talk about dimensions!Our attacks on this failing, a third key idea comes in: Namely, the notion of universally locallyacyclic sheaves, that we also developed independently in order to prove geometric Satake. It iseasy to see that formal smoothness plus ﬁnite-dimensionality implies that the constant sheaf isuniversally locally acyclic; it remains to see that the dualizing sheaf is invertible. This can beproved by a deformation to the normal cone (using universal local acyclicity to spread the result onthe normal cone to a neighborhood). We found this argument at a conference in Luminy in July2018; an inspiration to use a deformation to the normal cone may have been Clausen’s use in theproof of the “linearization hypothesis”.These results are enough to show that D ´et (Bun G , Z /(cid:96) n Z ) is well-behaved, and are alreadyenough to prove new ﬁniteness results on the cohomology of Rapoport–Zink spaces (with torsioncoeﬃcients). Our next emphasis was on geometric Satake. This essentially required the theoryof universally locally acyclic sheaves, and a version of Braden’s hyperbolic localization theorem[ Bra03 ]. We were able to ﬁnd substitutes for both. Regarding universally locally acyclic sheaves,we were able to prove analogues of most basic theorems, however we failed to prove that in generalthey are preserved under relative Verdier duality (even while we could check it by hand in allrelevant cases). Lu–Zheng [

LZ20 ] then found a new characterization of universally locally acyclicsheaves, making stability under relative Verdier duality immediate. Their arguments immediatelytransport to our setting. Eventually we used a slightly diﬀerent characterization, but in spirit theargument is still the same as theirs. Regarding hyperbolic localization, we could not follow Braden’sarguments that rely on nice coordinate choices. Instead, we reduce all arguments to the following(simple to prove) principle: If X is a (partially proper) space with a G m -action such that [ X/ G m ] .12. ACKNOWLEDGMENTS 43 is qcqs, and A ∈ D ´et ([ X/ G m ] , Λ), then the partially compactly supported cohomology of X withcoeﬃcients in A vanishes. The idea here is that the G m -action contracts X towards one of the ends.Afterwards, the proof of geometric Satake largely follows the lines of [ MV07 ], although there arecertain improvements in the argument; in particular, we give a simple reduction to groups of rank1, and pin the isomorphism with the dual group.Using these results, one has all ingredients in place, but only working with torsion coeﬃcients.One can formally pass to (cid:96) -adically complete sheaves, but this leads to studying representations onBanach Q (cid:96) -vector spaces, which is very unnatural. During this time, Clausen came to Bonn, andClausen and Scholze started to develop condensed mathematics, and the theory of solid modules[ CS ]. They realized that one could also deﬁne solid Z (cid:96) -sheaves on schemes or diamonds, and thatthis makes it possible to study representations on discrete Q (cid:96) - or Q (cid:96) -vector spaces, as desired.We take this up here, and ﬁrst deﬁne solid Z (cid:96) -sheaves on any small v-stack, together with some5-functor formalism (involving relative homology in place of compactly supported cohomology;its right adjoint is then pullback, so there are only 5 functors), and afterwards pass to a certainsubcategory of “lisse-´etale” sheaves to deﬁne the desired category D lis (Bun G , Q (cid:96) ), with exactly thedesired properties.In the meantime, Nadler–Yun [ NY19 ] and Gaitsgory–Kazhdan–Rozenblyum–Varshavsky [

GKRV19 ]showed that the categorical structures we have now constructed — D (Bun G , Q (cid:96) ) together with theaction of Hecke operators — formally induce an action of the category of perfect complexes onthe stack of L -parameters on D (Bun G , Q (cid:96) ), giving a categorical upgrade to the construction of L -parameters based on excursion operators. (We were aware of some weak form of this, whenrestricted to elliptic parameters; this was discussed in the last lecture of [ Sch17b ], based on someunpublished results of Ansch¨utz.) Here, we make the eﬀort of proving a result with Z (cid:96) -coeﬃcients,at least under a minor assumption on (cid:96) . I.12. Acknowledgments

We apologize for the long delay in the preparation of the manuscript. Already in April 2016,an Oberwolfach Arbeitsgemeinschaft gave an introduction to these ideas. The required foundationson ´etale cohomology of diamonds (with torsion coeﬃcients) were essentially worked out in thewinter term 2016/17, where also the ARGOS seminar in Bonn studied the evolving manuscript[

Sch17a ], which was an enormous help. Next, we gave the CAGA course at the IH´ES in Spring2017, [

Sch11 ]. At the time we had essentially obtained enough results to achieve the construction of L -parameters; the missing ingredient was the Jacobian criterion. In particular, we understood thatone has to use the local charts M b to study Bun G . In more detail, these results were discussed inthe ARGOS seminar in the summer term of 2017, and the proof of geometric Satake in the ARGOSseminar in the summer term of 2018. We heartily thank all the participants of these seminars forworking through these manuscripts, and their valuable feedback. In the meantime, in the springof 2018, we revised the Berkeley notes [ SW20 ] and used [

Sch17a ] to give a full construction ofmoduli spaces of p -adic shtukas. Then we eventually found a full proof of the Jacobian criterionin July 2018 at a conference in Luminy. Our early attempts at writing up the results ran into theissue that we wanted to ﬁrst prove basic sanity results on dimensions. Later, we got sidetrackedby the development of condensed mathematics with Dustin Clausen, which eventually made itpossible to use Q (cid:96) -coeﬃcients in this paper, ﬁnally giving the most natural results. In fact, we always wanted to develop the foundations so that papers such as the paper of Kaletha–Weinstein[ KW17 ] on the Kottwitz conjecture can build on a proper foundation, and prove results aboutgeneral smooth Q (cid:96) -representations; we believe we have ﬁnally achieved this. In the winter term2020/21, Scholze gave an (online) course in Bonn about these results, and wants to thank all theparticipants for their valuable feedback. We thank the many mathematicians that we discussedwith about these results, including Johannes Ansch¨utz, David Ben-Zvi, Joseph Bernstein, BhargavBhatt, Arthur-C´esar le Bras, Ana Caraiani, Dustin Clausen, Pierre Colmez, Jean-Fran¸cois Dat,Vladimir Drinfeld, Jessica Fintzen, Dennis Gaitsgory, Toby Gee, David Hansen, Eugen Hellmann,Alex Ivanov, Tasho Kaletha, Kiran Kedlaya, Dmitry Kubrak, Vincent Laﬀorgue, Judith Ludwig,Andreas Mihatsch, Sophie Morel, Jan Nekov´aˇr, Wies(cid:32)lawa Nizio(cid:32)l, Vincent Pilloni, Michael Rapoport,Timo Richarz, Will Sawin, Tony Scholl, Antoine Touz´e, Roman Travkin, Wilberd van der Kallen,Yakov Varshavsky, Jared Weinstein, Zhiyou Wu, Xinwen Zhu, Konrad Zou, and undoubtedly manyothers. Special thanks go to David Hansen for many comments on a preliminary draft.During the preparation of this manuscript, Fargues was supported by the ANR grant ANR-14-CE25 PerCoLaTor and the ERC Advanced Grant 742608 GeoLocLang, and Scholze was supportedby a DFG Leibniz Prize, and by the DFG under the Excellence Strategy – EXC-2047/1 – 390685813. I.13. Notation

Throughout most of this paper, E denotes a nonarchimedean local ﬁeld with residue ﬁeld F q of characteristic p >

0, and we ﬁx an algebraic closure k = F q of F q . Then ˘ E is the completedunramiﬁed extension of E with residue ﬁeld k . We also ﬁx a separable closure E of E , with absoluteGalois group Γ = Gal( E | E ), containing the Weil group W E , inertia subgroup I E , and wild inertia P E . The letter P usually denotes open subgroups of P E .The group G is usually a reductive group over E ; reductive groups are always assumed to beconnected.For any topological space X , we denote by X the sheaf taking any S (in the relevant testcategory, usually a perfectoid space) to the continuous maps from | S | to X . This is in the spiritof the passage from topological spaces to condensed sets, see [ CS ]. We make occasional use of thecondensed language, but do not make use of any nontrivial results from [ CS ]. In particular, ourdiscussion of solid (cid:96) -adic sheaves is self-contained.We will occasionally use the “animated” terminology, see [ CS ], [ ˇCS19b ]. In particular, we usethe term anima for what is variously called spaces in [ Lur09 ], ∞ -groupoids, or homotopy types, andfor any ring A , the ∞ -category of animated A -algebras is the ∞ -category obtained from simplicial A -algebras by inverting weak equivalences. Thus, animated A -algebras are freely generated undersifted colimits by polynomial algebras A [ X , . . . , X n ].If C is an ( ∞ -)category equipped with an action of a group G , we write C BG for the ( ∞ -)categoryof G -equivariant objects in C . Note that the data here is really a functor BG → Cat ∞ , and C BG isby deﬁnition the limit of this diagram. (It would be more customary to write C G , but this leads toinconsistent notation.)HAPTER II The Fargues–Fontaine curve and vector bundles

The goal of this chapter is to deﬁne the Fargues–Fontaine curve, in its various incarnations,and the category of vector bundles on the Fargues–Fontaine curve. Throughout this chapter, weﬁx a nonarchimedean local ﬁeld E with residue ﬁeld F q of characteristic p . We let O E ⊂ E be thering of integers, and π a uniformizing element in E .For any perfectoid space S over F q , we introduce a curve Y S , to be thought of as the product S × Spa F q Spa O E , together with an open subset Y S ⊂ Y S given by the locus where π (cid:54) = 0. Thiscarries a Frobenius ϕ induced from the Frobenius on S , and X S is the quotient Y S /ϕ Z .The ﬁrst results concern the Fargues–Fontaine curve X C = X S when S = Spa C for somecomplete algebraically closed nonarchimedean ﬁeld C | F q . We deﬁne a notion of classical points of X C in that case; they form a subset of | X C | . The basic ﬁniteness properties of X C are summarizedin the following result. Theorem

II.0.1 (Proposition II.1.11, Corollary II.1.12, Deﬁnition/Proposition II.1.22) . Theadic space Y C is locally the adic spectrum Spa(

B, B + ) where B is a principal ideal domain; theclassical points of Spa(

B, B + ) ⊂ Y C are in bijection with the maximal ideals of B . For each classicalpoint x ∈ Y C , the residue ﬁeld of x is an untilt C (cid:93) of C over O E , and this induces a bijection ofthe classical points of Y C with untilts C (cid:93) of C over O E . A similar result holds true for Y C ⊂ Y C ,and the quotient X C = Y C /ϕ Z . In the equal characteristic case, this is an immediate consequence of Y C = D C and classi-cal results in rigid-analytic geometry. In the p -adic case, we use tilting to reduce to the equalcharacteristic case. More precisely, if E is p -adic and E ∞ is the completion of E ( π /p ∞ ), then Y C × Spa O E Spa O E ∞ is perfectoid, with tilt given by a perfectoid open unit disc (cid:101) D C . The corre-sponding map | (cid:101) D C | → |Y C | induces a surjective map on classical points, see Proposition II.1.8. Atone key turn, in order to understand Zariski closed subsets of Y C , we use the result that Zariskiclosed subspaces are invariant under tilting, to reduce to (cid:101) D C . More precisely, we recall the followingresult. Proposition

II.0.2 ([

Sch15 , Section II.2], [

BS19 ], [

Sch17a , Deﬁnition 5.7, Theorem 5.8]) . Let S = Spa( R, R + ) be an aﬃnoid perfectoid space with tilt S (cid:91) = Spa( R (cid:91) , R (cid:91) + ) . Then a closedsubspace Z ⊂ | S | is the vanishing locus of an ideal I ⊂ R if and only if Z ⊂ | S | ∼ = | S (cid:91) | is thevanishing locus of an ideal J ⊂ R (cid:91) . In that case, there is a universal perfectoid space S Z → S such that | S Z | → | S | factors over Z , and S Z = Spa( T, T + ) is aﬃnoid perfectoid with | S Z | → Z ahomeomorphism, R → T surjective, R + → T + almost surjective, and T + is the integral closure of R + in T .

456 II. THE FARGUES–FONTAINE CURVE AND VECTOR BUNDLES

A key result is the classiﬁcation of vector bundles.

Theorem

II.0.3 (Theorem II.2.14) . The functor from

Isoc E to vector bundles on X C inducesa bijection on isomorphism classes. In particular, there is a unique stable vector bundle O X C ( λ ) ofany slope λ ∈ Q , and any vector bundle E can be written as a direct sum of stable bundles. We give a new self-contained proof of the classiﬁcation theorem, making critical use of thev-descent results for vector bundles obtained in [

Sch17a ] and [

SW20 ], and basic results on thegeometry of Banach–Colmez spaces established here.Allowing general S ∈ Perf F q , we deﬁne the moduli space of degree 1 Cartier divisors as Div =Spd ˘ E/ϕ Z . Given a map S → Div , one can deﬁne an associated closed Cartier divisor D S ⊂ X S ;locally, this is given by an untilt D S = S (cid:93) ⊂ X S of S over E , and this embeds Div into thespace of closed Cartier divisors on X S . Another important result is the following ampleness result,cf. [ KL15 , Proposition 6.2.4], which implies that one can deﬁne an algebraic version of the curve,admitting the same theory of vector bundles.

Theorem

II.0.4 (Theorem II.2.6, Proposition II.2.7, Proposition II.2.9) . Assume that S ∈ Perf is aﬃnoid. For any vector bundle E on X S , the twist E ( n ) is globally generated and has no highercohomology for all n (cid:29) . Deﬁning the graded ring P = (cid:77) n ≥ H ( X S , O X S ( n )) and the scheme X alg S = Proj P , there is a natural map of locally ringed spaces X S → X alg S , pullbackalong which deﬁnes an equivalence of categories of vector bundles, preserving cohomology.If S = Spa C for some complete algebraically closed nonarchimedean ﬁeld C , then X alg C is aregular noetherian scheme of Krull dimension , locally the spectrum of a principal ideal domain,and its closed points are in bijection with the classical points of X C . We also need to understand families of vector bundles, i.e. vector bundles E on X S for general S . Here, the main result is the following, which is originally due to Kedlaya–Liu [ KL15 ]. Theorem

II.0.5 (Theorem II.2.19, Corollary II.2.20) . Let S ∈ Perf and let E be a vectorbundle on X S . Then the function taking a point s ∈ S to the Harder–Narasimhan polygon of E| X s deﬁnes a semicontinuous function on S . If it is constant, then E admits a global Harder–Narasimhan stratiﬁcation, and pro-´etale locally on S one can ﬁnd an isomorphism with a directsum of O X S ( λ ) ’s.In particular, if E is everywhere semistable of slope , then E is pro-´etale locally trivial, and thecategory of such E is equivalent to the category of pro-´etale E -local systems on S . The key to proving this theorem is the construction of certain global sections of E . To achievethis, we use v-descent techniques, and an analysis of the spaces of global sections of E ; these areknown as Banach–Colmez spaces, and were ﬁrst introduced (in slightly diﬀerent terms) in [ Col02 ]. Definition

II.0.6 . Let E be a vector bundle on X S . The Banach–Colmez space BC ( E ) associatedwith E is the locally spatial diamond over S whose T -valued points, for T ∈ Perf S , are given by BC ( E )( T ) = H ( X T , E| X T ) . I.1. THE FARGUES–FONTAINE CURVE 47

Similarly, if E is everywhere of only negative Harder–Narasimhan slopes, the negative Banach–Colmez space BC ( E [1]) is the locally spatial diamond over S whose T -valued points are BC ( E [1])( T ) = H ( X T , E| X T ) . Implicit here is that this functor actually deﬁnes a locally spatial diamond. For this, we calculatesome key examples of Banach–Colmez spaces. For example, if E = O X S ( λ ) with 0 < λ ≤ [ E : Q p ](resp. all positive λ if E is of equal characteristic), then BC ( E ) is representable by a perfectoid openunit disc (of dimension given by the numerator of λ ). A special case of this is the identiﬁcationof BC ( O X S (1)) with the universal cover of a Lubin–Tate formal group law, yielding a very closerelation between Lubin–Tate theory, and thus local class ﬁeld theory, and the Fargues–Fontainecurve. This case actually plays a special role in getting some of the theory started, and we recall itexplicitly in Section II.2.1. On the other hand, for larger λ , or negative λ , Banach–Colmez spacesare more exotic objects; for example, the negative Banach–Colmez space BC ( O X C ( − ∼ = ( A C (cid:93) ) ♦ /E is the quotient of the aﬃne line by the translation action of E ⊂ A C (cid:93) .A key result is Proposition II.2.16, stating in particular that projectivized Banach–Colmezspaces ( BC ( E ) \ { } ) /E × are proper — they are the relevant analogues of “families of projective spaces over S ”. In particular,their image in S is a closed subset, and if the image is all of S , then we can ﬁnd a nowhere vanishingsection of E after a v-cover, as then the projectivized Banach–Colmez space is a v-cover of S . II.1. The Fargues–Fontaine curveII.1.1. The curve Y C . Recall that for any perfect F q -algebra R , there is a unique π -adicallycomplete ﬂat O E -algebra (cid:101) R such that (cid:101) R = R/π . There is a unique multiplicative lift [ · ] : R → (cid:101) R of the identity R → R , called the Teichm¨uller lift. Explicitly, one can take (cid:101) R = W O E ( R )in terms of the ramiﬁed Witt vectors.The construction of the Fargues–Fontaine curve is based on this construction on the level ofperfectoid spaces S over F q . Its construction is done in three steps. First, one constructs a curve Y S ,an adic space over O E , which carries a Frobenius action ϕ . Passing to the locus Y S = Y S \ { π = 0 } ,i.e. the base change to E , the action of ϕ is free and totally discontinuous, so that one can pass tothe quotient X S = Y S /ϕ Z , which will be the Fargues–Fontaine curve.We start by constructing Y S in the aﬃnoid case. More precisely, if S = Spa( R, R + ) is anaﬃnoid perfectoid space over F q , and (cid:36) ∈ R + is a pseudouniformizer (i.e. a topologically nilpotentunit of R ), we let Y S = Spa W O E ( R + ) \ V ([ (cid:36) ]) . This does not depend on the choice of (cid:36) , as for any choice of (cid:36), (cid:36) (cid:48) ∈ R , one has (cid:36) | (cid:36) (cid:48) n , (cid:36) (cid:48) | (cid:36) n forsome n >

0. The q -th power Frobenius of R + induces an automorphism ϕ of Y S . To construct theFargues–Fontaine curve, we will eventually remove V ( π ) from Y S and quotient by ϕ , but for nowwe recall some properties of Y S . Proposition

II.1.1 . The above deﬁnes an analytic adic space Y S over O E . Letting E ∞ be thecompletion of E ( π /p ∞ ) , the base change Y S × Spa O E Spa O E ∞ is a perfectoid space, with tilt given by S × F q Spa F q [[ t /p ∞ ]] = D S, perf , a perfectoid open unit disc over S . Proof.

One can cover Y S by the subsets Y S, [0 ,n ] := {| π | n ≤ | [ (cid:36) ] | (cid:54) = 0 } ⊂ Y S , which arerational subsets of Spa W O E ( R + ), where n > p for simplicity. Then Y S, [0 ,n ] = Spa( B S, [0 ,n ] , B + S, [0 ,n ] )where B S, [0 ,n ] = W O E ( R + ) (cid:104) π n [ (cid:36) ] (cid:105) [ (cid:36) ] ]and B + S, [0 ,n ] ⊂ B S, [0 ,n ] is the integral closure of W O E ( R + ) (cid:104) π n [ (cid:36) ] (cid:105) . To see that Y S is an adic space(i.e. the structure presheaf is a sheaf) and Y S × Spa O E Spa O E ∞ is perfectoid, it is enough toprove that B S, [0 ,n ] (cid:98) ⊗ O E O E ∞ is a perfectoid Tate algebra. Indeed, the algebra B S, [0 ,n ] splits oﬀ B S, [0 ,n ] (cid:98) ⊗ O E O E ∞ as a direct factor as topological B S, [0 ,n ] -module, and hence the sheaf property forperfectoid spaces gives the result for Y S, [0 ,n ] and thus all of Y S (cf. the sousperfectoid property of[ HK20 ], [

SW20 , Section 6.3]). Using the Frobenius automorphism of (

R, R + ), one can in factassume that n = 1.Let us abbreviate A = B S, [0 , (cid:98) ⊗ O E O E ∞ and A + ⊂ A the integral closure of B + S, [0 , (cid:98) ⊗ O E O E ∞ . In particular A +0 = ( W O E ( R + ) (cid:98) ⊗ O E O E ∞ )[( π [ (cid:36) ] ) /p ∞ ] ∧ [ (cid:36) ] ⊂ A + , and A = A +0 [ (cid:36) ] ]. Note that A +0 / [ (cid:36) ] = ( R + /(cid:36) ⊗ F q O E ∞ /π )[ t /p ∞ ] / ( π /p m − [ (cid:36) ] /p m t /p m ) ∼ = R + /(cid:36) [ t /p ∞ ] . This implies already that A +0 is integral perfectoid by [ BMS18 , Lemma 3.10 (ii)], and thus nec-essarily (cf. [

BMS18 , Lemma 3.21]) A +0 → A + is an almost isomorphism and A +0 [ (cid:36) ] ] ∼ = A isperfectoid. Moreover, one can see that the tilt of A is given by R (cid:104) t /p ∞ (cid:105) , where t (cid:93) = π [ (cid:36) ] , whichcorresponds to the subset {| t | ≤ | (cid:36) | (cid:54) = 0 } ⊂ S × F q Spa F q [[ t /p ∞ ]] = D S, perf . (cid:3) Proposition

II.1.2 . For any perfectoid space T over F q , giving an untilt T (cid:93) of T together witha map T (cid:93) → Y S of analytic adic spaces is equivalent to giving an untilt T (cid:93) together with a map T (cid:93) → Spa O E , and a map T → S . In other words, there is a natural isomorphism Y ♦ S ∼ = Spd O E × S. I.1. THE FARGUES–FONTAINE CURVE 49

Proof.

Changing notation, we need to see that for any perfectoid space T over O E , givinga map T → Y S is equivalent to giving a map T (cid:91) → S . Let T = Spa( A, A + ). Giving a map T → Y S is equivalent to giving a map W O E ( R + ) → A + such that the image of [ (cid:36) ] in A isinvertible. By the universal property of W O E ( R + ) in case R + is perfect, this is equivalent to givinga map R + → ( A + ) (cid:91) such that the image of (cid:36) in A (cid:91) is invertible. But this is precisely a map T = Spa( A, A + ) → S = Spa( R, R + ). (cid:3) In particular, there is a natural map |Y S | ∼ = |Y ♦ S | ∼ = | (Spa O E ) ♦ × S | → | S | . The following proposition ensures that we may glue Y S for general S , i.e. for any perfectoidspace S there is an analytic adic space Y S equipped with an isomorphism Y ♦ S ∼ = Spd O E × S (and in particular a map |Y S | → | S | ) such that for U = Spa( R, R + ) ⊂ S an aﬃnoid subset, thecorresponding pullback of Y S is given by Y U . Proposition

II.1.3 . If S (cid:48) ⊂ S is an aﬃnoid subset, then Y S (cid:48) → Y S is an open immersion,with |Y S (cid:48) | (cid:47) (cid:47) (cid:15) (cid:15) |Y S | (cid:15) (cid:15) S (cid:48) (cid:47) (cid:47) S cartesian. Proof.

Let Z ⊂ Y S be the open subset corresponding to |Y S | × | S | | S (cid:48) | ⊂ |Y S | . Then byfunctoriality of the constructions, we get a natural map Y S (cid:48) → Z . To see that it is an isomorphism,we can check after base change to O E ∞ (as the maps on structure sheaves are naturally splitinjective). The base change of Y S (cid:48) and Z become perfectoid, and hence it suﬃces to see that onegets an isomorphism after passing to diamonds, where it follows from Proposition II.1.2. (cid:3) Next, we recall the “sections of Y S → S ”. Proposition

II.1.4 ([

SW20 , Proposition 11.3.1]) . Let S be a perfectoid space over F q . Thefollowing objects are in natural bijection. (i) Sections of Y ♦ S → S ; (ii) Morphisms S → Spd O E ; (iii) Untilts S (cid:93) over O E of S .Moreover, given an untilt S (cid:93) over O E of S , there is a natural closed immersion of adic spaces S (cid:93) (cid:44) → Y S that presents S (cid:93) as a closed Cartier divisor in Y S . Proof.

The equivalence of (i), (ii) and (iii) is a direct consequence of Proposition II.1.2. Thus,let S (cid:93) be an untilt of S over O E . We may work locally, so assume S = Spa( R, R + ) is aﬃnoid. Then S (cid:93) = Spa( R (cid:93) , R (cid:93) + ) is aﬃnoid perfectoid as well, and R (cid:93) + = W O E ( R + ) /ξ for some nonzerodivisor ξ ∈ W O E ( R + ) that can be chosen to be of the form π − a [ (cid:36) ] for some a ∈ W O E ( R + ) and suitable topologically nilpotent (cid:36) ∈ R (choose (cid:36) ∈ R + a pseudouniformizingelement such that (cid:36) (cid:93) | π , and write π = (cid:36) (cid:93) θ ( a ) for some a ). To see that S (cid:93) deﬁnes a closed Cartierdivisor in Y S , that is to say the sequence0 → O Y S ξ −→ O Y S → i ∗ O S (cid:93) → i : S (cid:93) (cid:44) → Y S , we need to see that for any open aﬃnoid U = Spa( A, A + ) ⊂ Y S withaﬃnoid perfectoid pullback V = Spa( B, B + ) ⊂ S (cid:93) , the sequence0 → A ξ −→ A → B → S (cid:93) = V ( ξ ) ⊂ Y S . In particular, replacing S by V (cid:91) , we can assume that V = S (cid:93) . In that case, any neighborhood of S (cid:93) = V ( ξ ) in Y S contains {| ξ | ≤ | [ (cid:36) ] | n } for some n >

0, so we can assume that U is of this form.Endow A with the spectral norm, where we normalize the norm on each completed residue ﬁeldof Y S by | [ (cid:36) ] | = q . We claim that with this choice of norm, one has | ξa | ≥ q − n | a | for all a ∈ A . In particular, this implies that ξ : A → A is injective, and has closed image (as thepreimage of any Cauchy sequence in the image is a Cauchy image). On the other hand, R (cid:93) is theseparated completion of A/ξ , so B = A/ξ .To verify the claimed inequality, it is enough to see that the norm of | a | is equal to the supremumnorm over {| ξ | = | [ (cid:36) | n } . In fact, it is enough to consider the points in the Shilov boundary, i.e. thosepoints Spa( C, O C ) → U that admit a specialization Spa( C, C + ) → Y S whose image is not containedin U ; any such is necessarily contained in {| ξ = [ (cid:36) ] n } . This will in fact hold for all functions on U × Spa O E Spa O E ∞ , for which the claim reduces to the tilt, which is an aﬃnoid subset of D S, perf .By approximation, it then reduces to the case of aﬃnoid subsets of D S , where it is well-known thatthe maximum is taken on the Shilov boundary. (Note that this question immediately reduces tothe case that S is a geometric point.) (cid:3) Remark

II.1.5 . The preceding Cartier divisor satisﬁes the stronger property of being a “relativeCartier divisor” in the sense that for all s ∈ S its pullback to Y Spa( K ( s ) ,K ( s ) + ) is a Cartier divisor.Now let us analyze the case S = Spa C for some complete algebraically closed nonarchimedeanﬁeld over F q . Example

II.1.6 . Assume that E = F q (( t )) is of equal characteristic. Then Y C = D C is an openunit disc over C , with coordinate t . In particular, inside |Y C | , we have the subset of classical points |Y C | cl ⊂ |Y C | , which can be identiﬁed as |Y C | cl = { x ∈ C | | x | < } . I.1. THE FARGUES–FONTAINE CURVE 51

Note that these classical points are in bijection with maps O E → C (over F q ), i.e. with “untilts of C over O E ”.With suitable modiﬁcations, the same picture exists also when E is of mixed characteristic. Definition/Proposition

II.1.7 . Any untilt C (cid:93) of C over O E deﬁnes a closed Cartier divisor Spa C (cid:93) (cid:44) → Y S , and in particular a closed point of |Y C | . This induces an injection from the set ofsuch untilts to |Y C | .The set of classical points |Y C | cl ⊂ |Y C | is deﬁned to be the set of such points. Proof.

We have seen that any untilt C (cid:93) deﬁnes such a map Spa C (cid:93) (cid:44) → Y S . As it is a closedCartier divisor, the corresponding point is closed in |Y C | . One can recover C (cid:93) as the completedresidue ﬁeld at the point, together with the map W O E ( O C ) → O C (cid:93) , which induces the isomorphism O C ∼ = O (cid:91)C (cid:93) and thus C ∼ = ( C (cid:93) ) (cid:91) , giving the untilt structure on C (cid:93) ; this shows that the map isinjective. (cid:3) Recall that Y C is preperfectoid. In fact, if one picks a uniformizer π ∈ E and lets E ∞ be thecompletion of E ( π /p ∞ ), then Y C × O E O E ∞ is perfectoid, and its tilt is given bySpa C × Spa O (cid:91)E ∞ ∼ = Spa C × Spa F q [[ t /p ∞ ]] . Thus, we get a map | D C | = | Spa C × Spa F q [[ t ]] | ∼ = | Spa C × Spa F q [[ t /p ∞ ]] | ∼ = |Y C × O E O E ∞ | → |Y C | . Proposition

II.1.8 . Under this map, the classical points | D C | cl = { x ∈ C | | x | < } ⊂ | D C | are exactly the preimage of the classical points |Y C | cl ⊂ |Y C | . Unraveling the deﬁnitions, one sees that the map { x ∈ C | | x | < } = | D C | cl → |Y C | cl sends any x ∈ C with | x | < π − [ x ]). In particular, theproposition shows that any classical point of Y C can be written in this form. Proof.

This is clear as classical points are deﬁned in terms of maps of diamonds, which arecompatible with this tilting construction on topological spaces. (cid:3)

The formation of classical points is also compatible with changing C in the following sense. Proposition

II.1.9 . Let C (cid:48) | C be an extension of complete algebraically closed nonarchimedeanﬁelds over F q , inducing the map Y C (cid:48) → Y C . A point x ∈ |Y C | is classical if and only if its preimagein |Y C (cid:48) | is a classical point. Moreover, if x ∈ |Y C | is a rank- -point that is not classical, then thereis some C (cid:48) | C such that the preimage of x contains a nonempty open subset of |Y C (cid:48) | . In other words, one can recognize classical points as those points that actually stay pointsafter any base change; all other rank 1 points actually contain whole open subsets after some basechange.

Proof.

It is clear that if x is classical, then its preimage is a classical point. Conversely,if x ∈ |Y C | is a rank 1 point, and S = Spd K ( x ), the point x is given by a morphism S → Spa C × Spd O E . If the preimage of x is a classical point, the induced morphism S → Spa C becomes an isomorphism after pullback via Spa C (cid:48) → Spa C . Since S is a v-sheaf ([ Sch17a ,Proposition 11.9]) and Spa C (cid:48) → Spa C a v-cover, the morphism S → Spa C is an isomorphism,and thus x is a classical point.Now assume that x is nonclassical rank-1-point; we want to ﬁnd C (cid:48) | C such that the preimageof x contains an open subset of |Y C (cid:48) | . By Proposition II.1.8, it is enough to prove the similar resultfor D C , using that | D C (cid:48) | → |Y C (cid:48) | is open. Thus, assume x ∈ | D C | is a non-classical point. Let C (cid:48) be a completed algebraic closure of the corresponding residue ﬁeld. Then the preimage of x in | D C (cid:48) | has a tautological section (cid:101) x ∈ D C (cid:48) ( C (cid:48) ) which is a classical point, and the preimage of x containsa small disc B ( (cid:101) x, r ) ⊂ D C (cid:48) for some r >

0. Indeed, this follows from the description of the rank 1points of D C as being either the Gauss norm for some disc B ( x, r ) ⊂ D C of radius r >

0, or theinﬁmum of such over a decreasing sequence of balls (but with radii not converging to zero). SeeLemma II.1.10. (cid:3)

Lemma

II.1.10 . Let x ∈ D C ( C ) , ρ ∈ (0 , , and x ρ ∈ | D C | be the Gauss norm with radius ρ centered at x . The preimage of x ρ in | D C ( x ρ ) | contains the open disk with radius ρ centered at x ρ ∈ D C ( x ρ ) ( C ( x ρ )) . Proof.

Proposition

II.1.11 . Let U = Spa( B, B + ) ⊂ Y C be an aﬃnoid subset. Then for any maximalideal m ⊂ B , the quotient B/ m is a nonarchimedean ﬁeld, inducing an injection Spm( B ) (cid:44) → | U | .This gives a bijection between Spm( B ) and | U | cl := | U | ∩ |Y C | cl ⊂ |Y C | . Proof.

First, if x ∈ | U | cl , then it corresponds to a closed Cartier divisor Spa C (cid:93) (cid:44) → U ⊂ Y C ,and thus deﬁnes a maximal ideal of B , yielding an injection | U | cl (cid:44) → Spm( B ). We need to see thatthis is a bijection.Note that using the tilting map | D C | → |Y C | , one sees that the preimage of U in | D C | hasonly ﬁnitely many connected components (any quasicompact open subset of | D C | has ﬁnitely many Any quasicompact open of | D C (cid:48) | is the base change of a quasicompact open of |Y C (cid:48) × Spa O E Spa O E (cid:48) | for a ﬁniteextension E (cid:48) | E . Passing to the Galois hull of E (cid:48) and taking the orbit of the open subset under the Galois group, itthen follows from the map being a quotient map, as any map of analytic adic spaces. I.1. THE FARGUES–FONTAINE CURVE 53 connected components); we can thus assume that U is connected. In that case, we claim that anynonzero element f ∈ B vanishes only at classical points of | U | . By Proposition II.1.9, it suﬃces tosee that for any nonempty open subset U (cid:48) ⊂ U , the map O ( U ) → O ( U (cid:48) ) is injective. In fact, if V ( f )contains a nonclassical point, it also contains a nonclassical rank 1 point as V ( f ) is generalizing,then after base changing to some C (cid:48) | C , V ( f ) contains an open subset U (cid:48) , and this is impossible if O ( U ) (cid:44) → O ( U (cid:48) ). For this it suﬃces to prove that O ( V ) (cid:44) → O ( V (cid:48) ) where V is a connected componentof U (cid:98) ⊗ O E O E ∞ , and V (cid:48) the intersection of U (cid:48) (cid:98) ⊗ O E O E ∞ with this connected component. Now forany g ∈ O ( V ) \ { } , V ( g ) (cid:54) = V , as perfectoid spaces are uniform (and hence vanishing at all pointsimplies vanishing). We thus have to prove that for any Zariski closed subset Z (cid:40) V , V (cid:48) (cid:54)⊂ Z .By Proposition II.0.2, it suﬃces to prove the similar property for open subsets V (cid:48) ⊂ V ⊂ D C, perf ,with V connected. But then V (cid:48) = W (cid:48) perf and V = W perf for W (cid:48) ⊂ W ⊂ D C , and O ( V ) → O ( V (cid:48) ) istopologically free (with basis t i , i ∈ [0 , ∩ Z [ p − ]) over the corresponding map O ( W ) → O ( W (cid:48) )of classical Tate algebras over C , for which injectivity is classical. (cid:3) The previous proposition implies that, once U is connected, the rings B are principal idealdomains (cf. [ Ked16 ]).

Corollary

II.1.12 . Let U = Spa( B, B + ) ⊂ Y C be an aﬃnoid subset. Then U has ﬁnitely manyconnected components. Assuming that U is connected, the ring B is a principal ideal domain. Proof.

We have already seen in the preceding proof that U has ﬁnitely many connectedcomponents. Moreover, each maximal ideal of B is principal, as it comes from a closed Cartierdivisor on U . Now take any nonzero f ∈ B . We have seen that the vanishing locus of f is containedin | U | cl , and it is also closed in | U | . It it thus a spectral space with no nontrivial specializations,and therefore a proﬁnite set. We claim that it is in fact discrete. For this, let x ∈ V ( f ) be anypoint. We get a generator ξ x ∈ B for the corresponding maximal ideal. We claim that there issome n ≥ f = ξ nx g where g does not vanish at x . Assume otherwise. Note that thespectral norm on U is given by the supremum over ﬁnitely many points, the Shilov boundary of U (cf. proof of Proposition II.1.4). We may normalize ξ x so that its norm at all of these ﬁnitely manypoints is ≥

1. Then for any n , if f = ξ nx g n , one has || g n || ≤ || f || . But inside the open neighborhood U x = {| ξ x | ≤ | [ (cid:36) ] |} of x , this implies that || f || U x ≤ | [ (cid:36) ] | n || f || for all n , and thus || f || U x = 0 as n → ∞ . Thus, f vanishes on all of U x , which is a contradiction.By the above, we can write f = ξ nx g where g does not vanish at x . But then g does not vanishin a neighborhood of x , and therefore x ∈ V ( f ) is an isolated point, and hence V ( f ) is proﬁnite anddiscrete, and thus ﬁnite. Enumerating these points x , . . . , x m , we can thus write f = ξ n x · · · ξ n m x m g where g does not vanish at x , . . . , x m , and thus vanishes nowhere, and hence is a unit. This ﬁnishesthe proof. (cid:3) Remark

II.1.13 . The main new ingredient compared to [

Ked16 ] or [

FF18 , Theorem 2.5.1]that allows us to shorten the proof is Proposition II.0.2, i.e. the use of the fact (proved in [

BS19 ])that “Zariski closed implies strongly Zariski closed” in the terminology of [

Sch15 , Section II.2].Later (cf. Proposition IV.7.3), we will also need the following lemma about non-classical pointsof Y C = Y C × Spa O E Spa E . Lemma

II.1.14 . There is a point x ∈ | Y C | , with completed residue ﬁeld K ( x ) , such that theinduced map Gal( K ( x ) | K ( x )) → I E is surjective, where I E is the inertia subgroup of the absoluteGalois group of E . Note that a priori we have a map Gal( K ( x ) | K ( x )) → Gal( E | E ), but it is clear that its imageis contained in I E , as K ( x ) contains ˘ E . Proof.

In fact, we can be explicit: Looking at the surjection | D ∗ C | → | Y C | from the tilting construction, the image of any Gaußpoint (corresponding to a disc of radius r ,0 < r <

1, around the origin) will have the desired property. This follows from the observationthat this locus of Gaußpoints lifts uniquely to | Y C × Spa ˘ E Spa E (cid:48) | for any ﬁnite extension E (cid:48) | ˘ E . Infact, this cover admits a similar surjection from a punctured open unit disc over C , and there isagain one Gaußpoint for each radius (i.e. the set of Gaußpoints maps isomorphically to (0 , ∞ ) viarad : | Y C | → (0 , ∞ )). (cid:3) II.1.2. The Fargues–Fontaine curve.

Now we can deﬁne the Fargues–Fontaine curve.

Definition

II.1.15 . For any perfectoid space S over F q , the relative Fargues–Fontaine curve is X S = Y S /ϕ Z where Y S = Y S × Spa O E Spa E = Y S \ V ( π ) , which for aﬃnoid S = Spa( R, R + ) with pseudouniformizer (cid:36) is given by Y S = Spa W O E ( R + ) \ V ( π [ (cid:36) ]) . To see that this is well-formed, we note the following proposition.

Proposition

II.1.16 . The action of ϕ on Y S is free and totally discontinuous. In fact, if S = Spa( R, R + ) is aﬃnoid and (cid:36) ∈ R is a pseudouniformizer, one can deﬁne a map rad : | Y S | −→ (0 , ∞ ) taking any point x ∈ Y S with rank- -generalization (cid:101) x to log | [ (cid:36) ]( (cid:101) x ) | / log | π ( (cid:101) x ) | . This factorizesthrough the Berkovich space quotient of | Y S | and satisﬁes rad ◦ ϕ = q · rad .For any interval I = [ a, b ] ⊂ (0 , ∞ ) with rational ends (possibly with a = b ), there is the opensubset Y S,I = {| π | b ≤ | [ (cid:36) ] | ≤ | π | a } ⊂ rad − ( I ) ⊂ Y S which is in fact a rational open subset of Spa W O E ( R + ) and thus aﬃnoid, Y S,I = Spa( B S,I , B + S,I ) , and one can form X S as the quotient of Y S, [1 ,q ] via the identiﬁcation ϕ : Y S, [1 , ∼ = Y S, [ q,q ] . Inparticular, X S is qcqs in case S is aﬃnoid. Proof.

This follows directly from the deﬁnitions. (cid:3)

I.1. THE FARGUES–FONTAINE CURVE 55

In terms of the preceding radius function, the end 0 corresponds to the boundary divisor ( π ),and ∞ to the boundary divisor ([ (cid:36) ]).For each s ∈ S corresponding to a map Spa( K ( s ) , K ( s ) + ) → S , functoriality deﬁnes a morphism X K ( s ) ,K ( s ) + → X S . We way think of X S as the collection of curves ( X K ( s ) ,K ( s ) + ) s ∈ S , the onedeﬁned and studied in [ FF18 ], merged in a “family of curves”. Although X S does not sit over S ,the absolute Frobenius ϕ × ϕ of S × Spd( E ) acts trivially on the topological space and one has | X S | ∼ = | X ♦ S | ∼ = | S × Spd( E ) /ϕ Z × id | ∼ = | S × Spd( E ) / id × ϕ Z | −→ | S | . Thus the topological space | X S | sits over | S | , and for all S the map | X S | → | S | is qcqs. Here, weused the following identiﬁcation of the diamond. Proposition

II.1.17 . There is a natural isomorphism Y ♦ S ∼ = S × Spd( E ) , descending to an isomorphism X ♦ S ∼ = ( S × Spd( E )) /ϕ Z × id . Proof.

This is immediate from Proposition II.1.2. (cid:3)

Moreover, we have the following version of Proposition II.1.4.

Proposition

II.1.18 . The following objects are naturally in bijection. (i)

Sections of Y ♦ S → S ; (ii) Maps S → Spd( E ) ; (iii) Untilts S (cid:93) over E of S .Given such a datum, in particular an untilt S (cid:93) over E of S , there is a natural closed immersion S (cid:93) (cid:44) → Y S presenting S (cid:93) as a closed Cartier divisor in Y S . The composite map S (cid:93) → Y S → X S isstill a closed Cartier divisor, and depends only on the composite S → Spd( E ) → Spd( E ) /ϕ Z . Inthis way, any map S → Spd( E ) /ϕ Z deﬁnes a closed Cartier divisor D ⊂ X S ; this gives an injectionof Spd( E ) /ϕ Z into the space of closed Cartier divisors on X S . Proof.

This is immediate from Proposition II.1.4. (cid:3)

Definition

II.1.19 . A closed Cartier divisor of degree on X S is a closed Cartier divisor D ⊂ X S that arises from a map S → Spd( E ) /ϕ Z . Equivalently, it arises locally on S from anuntilt S (cid:93) over E of S . In particular, we see that the moduli space Div of degree 1 closed Cartier divisors is given byDiv = Spd( E ) /ϕ Z . Note that something strange is happening in the formalism here: Usually the curve itself wouldrepresent the moduli space of degree 1 Cartier divisors!

Remark

II.1.20 . In [

Far18a , D´eﬁnition 2.6] Fargues gives a deﬁnition of a Cartier divisor ofdegree 1 on X S equivalent to the preceding one, similar to the deﬁnition of a relative Cartier divisorin classical algebraic geometry. Proposition

II.1.21 . The map

Div → ∗ is proper, representable in spatial diamonds, andcohomologically smooth. Proof.

First, Spd( E ) → ∗ is representable in locally spatial diamonds and cohomologicallysmooth by [ Sch17a , Proposition 24.5] (for E = Q p , which formally implies the case of E ﬁnite over Q p , and the equal characteristic case is handled in the proof). As | Spd( E ) × S | ∼ = | Y S | → | S | , wesee that ϕ Z acts totally discontinuously with quotient | Spd( E ) /ϕ Z × S | ∼ = | X S | → | S | being qcqsin case | S | is qcqs; thus, Spd( E ) /ϕ Z → ∗ is representable in spatial diamonds, in particular qcqs.Then being proper follows from the valuative criterion [ Sch17a , Proposition 18.3]. (cid:3)

In particular, the map | X S | = | Div × S | −→ | S | is open and closed. We can thus picture X S as being “a proper and smooth family over S ”. Further motivation for Deﬁnition II.1.19 is given by the following.

Definition/Proposition

II.1.22 . The classical points of X C are | X C | cl := | Y C | cl /ϕ Z ⊂| X C | = | Y C | /ϕ Z . They are in bijection with (Spd( E ) /ϕ Z )( C ) = Div ( C ) , i.e. are given untiltsof C over E up to Frobenius, or by degree closed Cartier divisors on X C . For any aﬃnoid opensubset U = Spa( B, B + ) ⊂ X C , the maximal ideals of B are in bijection with | U | cl = | U | ∩ | X C | cl .Any such U has only ﬁnitely many connected components, and if U is connected, then B is aprincipal ideal domain. Proof.

This follows immediately from Proposition II.1.11 and Corollary II.1.12 if U lifts to Y C . In general, Y C → X C is locally split, so the result is true locally on U ; and then it easily followsby gluing in general. (cid:3) II.2. Vector bundles on the Fargues–Fontaine curve

Let us recall a few basic facts about the cohomology of vector bundles. Suppose S = Spa( R, R + )is aﬃnoid perfectoid. Then Y S is ”Stein”, one has Y S = (cid:83) I ⊂ (0 , ∞ ) Y ( R,R + ) ,I where(i) as before I is a compact interval with rational ends(ii) Y ( R,R + ) ,I is aﬃnoid sous-perfectoid(iii) for I ⊂ I , the restriction morphism O ( Y ( R,R + ) ,I ) → O ( Y ( R,R + ) ,I ) has dense image.Let F be a vector bundle on Y S . Point (2) implies that H i ( Y S,I , F | Y ( R,R +) ,I ) = 0 when i >

0. Point(3) implies that R lim ←− I Γ( Y ( R,R + ) ,I , F ) = 0 ([ Gro61 , 0.13.2.4]). We thus have H i ( Y S , F ) = 0 when i > E is a vector bundle on X S , one has R Γ( X S , E ) = (cid:2) H ( Y S , E | Y S ) Id − ϕ −−−−→ H ( Y S , E | Y S )] . In particular, this vanishes in degree > I.2. VECTOR BUNDLES ON THE FARGUES–FONTAINE CURVE 57

Proposition

II.2.1 . Let S be a perfectoid space over F q and E a vector bundle on X S . Thefunctor taking any T ∈ Perf S to R Γ( X T , E| X T ) is a v-sheaf of complexes. In fact, the functor taking any T ∈ Perf S to H ( Y T , E| Y T ) is a v-sheaf,whose cohomology vanishes in case T is aﬃnoid.Moreover, sending S to the groupoid of vector bundles on X S deﬁnes a v-stack. Proof.

By the displayed formula for R Γ( X S , E ) as Frobenius ﬁxed points, it suﬃces to provethe result about Y T . We can assume that S = Spa( R, R + ) is aﬃnoid, pick a pseudouniformizer (cid:36) ∈ R , and one can further reduce to the similar claim for Y T,I for any compact interval I withrational ends. We need to see that for any v-cover T = Spa( R (cid:48) , R (cid:48) + ), the corresponding ˇCechcomplex 0 → O ( Y S,I ) → O ( Y T,I ) → O ( Y T × S T,I ) → . . . of E -Banach spaces is exact. This can be checked after taking a completed tensor product with E ∞ = E ( π /p ∞ ) ∧ . In that case, all algebras become perfectoid, and Y T,I × E E ∞ → Y S,I × E E ∞ is av-cover of aﬃnoid perfectoid spaces, so the result follows from [ Sch17a , Theorem 8.7, Proposition8.8].Similarly, one proves v-descent for the groupoid of vector bundles, cf. [

SW20 , Lemma 17.1.8,Proposition 19.5.3]. (cid:3)

If [ E → E ] is a complex of vector bundles on X S sitting in homological degrees [0 , H ( X T , E | X T ) = 0 for all T ∈ Perf S , we let BC ([ E → E ]) : T (cid:55)→ H ( X T , [ E → E ] | X T )be the corresponding v-sheaf on Perf S . We refer to this as the Banach–Colmez space associatedwith [ E → E ]. We will usually apply this only when either of E and E is zero.Let us also recall the basic examples of vector bundles. Already here it is useful to ﬁx analgebraically closed ﬁeld k | F q , e.g. k = F q . Let ˘ E = W O E ( k )[ π ], the complete unramiﬁed extensionof E with residue ﬁeld k . Recall that, functorially in S ∈ Perf k , there is a natural exact ⊗ -functorIsoc k −→ Bun( X S )( D, ϕ ) (cid:55)−→ E ( D, ϕ )from the category of isocrystals (of a ﬁnite-dimensional ˘ E -vector space D equipped with a σ -linearautomorphism ϕ : D ∼ −→ D ) to the category of vector bundles on X S , deﬁned via descending D ⊗ ˘ E O Y S to X S via ϕ ⊗ ϕ . We denote by O X S ( n ) the image of ( ˘ E, π − n σ ) (note the change of sign— the functor E reverses slopes); more generally, if ( D λ , ϕ λ ) is the simple isocrystal of slope λ ∈ Q in the Dieudonn´e–Manin classiﬁcation, we let O X S ( − λ ) = E ( D λ , ϕ λ ). II.2.1. Lubin–Tate formal groups.

The claim of this paper is that the Fargues–Fontainecurve enables a geometrization of the local Langlands correspondence. As a warm-up, let us recallthe relation between O X S (1) and local class ﬁeld theory in the form of Lubin–Tate theory. Up to isomorphism, there is a unique 1-dimensional formal group G over O ˘ E with action by O E , such that the two induced actions on Lie G coincide; this is “the” Lubin–Tate formal group G = G LT of E . Fixing a uniformizer π ∈ E , we normalize this as follows. First, any Lubin–Tate formal group law over O E is the unique (up to unique isomorphism) lift of a 1-dimensionalformal group over k whose Lie algebra has the correct O E -action. Now, if E is p -adic then G k isclassiﬁed by Dieudonn´e theory by a ﬁnite projective W O E ( k )-module M equipped with a σ -linearisomorphism F : M [ π ] ∼ = M [ π ] such that M ⊂ F ( M ) ⊂ π M . Here, we take M = W O E ( k ) with F = π σ . One can similarly deﬁne G in equal characteristic, but actually we will explain a diﬀerentway to pin down the choice just below; under our normalization, G is already deﬁned over O E .After passing to the generic ﬁbre, G E is isomorphic to the additive group G a , compatibly withthe O E -action, and one can choose a coordinate on G ∼ = Spf O E [[ X ]] so that explicitly, the logarithmmap is given by log G : G E → G a,E : X (cid:55)→ X + π X q + π X q + . . . + π n X q n + . . . . Regarding the convergence of log G , we note that in fact it deﬁnes a map of rigid-analytic varieties(i.e. adic spaces locally of ﬁnite type over E )log G : G ad E ∼ = D E → G ad a,E from the open unit disc G ad E ∼ = Spa O E [[ X ]] × Spa O E Spa E to the adic space corresponding to G a . From the formula, one sees that in small enough discsit deﬁnes an isomorphism, and via rescaling by powers of π (which on the level of G ad E deﬁnesﬁnite ´etale covers of degree q , while it is an isomorphism on G ad a,E ), one sees that one has an exactsequence 0 → G ad E [ π ∞ ] → G ad E → G ad a,E → E , where G ad E [ π ∞ ] ⊂ G ad E is the torsion subgroup.This is, in fact, the generic ﬁbre of G [ π ∞ ] = (cid:83) n G [ π n ] over Spa O E , and each G [ π n ] = Spa A n isrepresented by some ﬁnite O E -algebra A n of degree q n . Inductively, G [ π n − ] ⊂ G [ π n ] giving a map A n → A n − ; after inverting π , this is split, and the other factor is a totally ramiﬁed extension E n | E . Then G ad E = (cid:91) n Spa A n [ π ] = (cid:71) n Spa E n . We also need the “universal cover” of G , deﬁned as (cid:101) G = lim ←− × π G ∼ = Spf O E [[ (cid:101) X /p ∞ ]] , where the inverse limit is over the multiplication by π maps. The isomorphism with Spf O E [[ (cid:101) X /p ∞ ]]is evident modulo π , but as this gives a perfect algebra, we see that in fact the isomorphism liftsuniquely to O E . Explicitly, the coordinate (cid:101) X is given by (cid:101) X = lim n →∞ X q n n As in [

SW20 , p. 99], we renormalize usual covariant Dieudonn´e theory for p -divisible groups by dividing F by p ; and then in the case of π -divisible O E -modules as here, we base change along W ( k ) ⊗ Z p O E → W O E ( k ). I.2. VECTOR BUNDLES ON THE FARGUES–FONTAINE CURVE 59 where X n is the coordinate on the n -th copy of G in the formula (cid:101) G = lim ←− × π G ; in fact, (cid:101) X ≡ X q n n modulo π n . In particular, the logarithm maplog G : (cid:101) G E → G E → G a,E is given by the series (cid:88) i ∈ Z π i (cid:101) X q − i . Note that for any π -adically complete O E -algebra A , one has (cid:101) G ( A ) ∼ = (cid:101) G ( A/π ) = Hom O E ( E/ O E , G ( A/π ))[ π ] . Indeed, the ﬁrst equality follows from O E [[ (cid:101) X /p ∞ ]] being relatively perfect over O E , and the secondequality by noting that any element of G ( A/π ) is π n -torsion for some n . A diﬀerent descriptionbased on (cid:101) G = Spf O E [[ (cid:101) X /p ∞ ]] is (cid:101) G ( A ) = lim ←− x (cid:55)→ x p A ◦◦ = A (cid:91), ◦◦ ⊂ A (cid:91) , the subset of topologically nilpotent elements of the tilt.This is related to the line bundle O X S (1) as follows. Proposition

II.2.2 . Let S = Spa( R, R + ) be an aﬃnoid perfectoid space over F q and let S (cid:93) =Spa( R (cid:93) , R (cid:93) + ) be an untilt of S over E , giving rise to the closed immersion S (cid:93) (cid:44) → X S . Let O X S (1) be the line bundle on X S corresponding to the isocrystal ( E, π − ) . Then the map (cid:101) G ( R (cid:93) + ) ∼ = R ◦◦ → H ( Y S , O Y S ) : X (cid:55)→ (cid:88) i ∈ Z π i [ X q − i ] deﬁnes a natural isomorphism (cid:101) G ( R (cid:93) + ) ∼ = H ( X S , O X S (1)) = H ( Y S , O Y S ) ϕ = π . Under this isomorphism, the map H ( X S , O X S (1)) → H ( S (cid:93) , O S (cid:93) ) = R (cid:93) of evaluation at S (cid:93) is given by the logarithm map log G : (cid:101) G ( R (cid:93) + ) → G ( R (cid:93) + ) → R (cid:93) . Proof.

The compatibility with the logarithm map is clear from the explicit formulas. Assumeﬁrst that E is of characteristic p . Then H ( Y S , O Y S ), where Y = D ∗ S is a punctured open unitdisc over S , can be explicitly understood as certain power series (cid:80) i ∈ Z r i π i with coeﬃcients r i ∈ R (subject to convergence conditions as i → ±∞ ). Then H ( X S , O X S (1)) = H ( Y S , O Y S ) ϕ = π amounts to those series such that r i = r qi +1 for all i ∈ Z . Thus, all r i are determined by r , whichin turn can be any topologically nilpotent element of R . This gives the desired isomorphism H ( X S , O X S (1)) ∼ = R ◦◦ = (cid:101) G ( R + ) = (cid:101) G ( R (cid:93) + )(as (cid:101) G = Spa O E [[ (cid:101) X /p ∞ ]] and R (cid:93) = R ). If E is p -adic, then we argue as follows. First, as in the proof of Proposition II.2.5 below, onecan rewrite H ( X S , O X S (1)) as B ϕ = πR, [1 , ∞ ] where B R, [1 , ∞ ] = O ( Y [1 , ∞ ] ) , for Y [1 , ∞ ] = {| [ (cid:36) ] | ≤ | π | (cid:54) = 0 } ⊂ Spa W O E ( R + ) . By the contracting property of Frobenius, one can also replace B R, [1 , ∞ ] with the crystalline periodring B +crys of R (cid:93) + /π here, and then [ SW13 , Theorem A] gives the desired B ϕ = πR, [1 , ∞ ] = Hom O E ( E/ O E , G ( R (cid:93) + /π ))[ π ] = (cid:101) G ( R (cid:93) + /π ) = (cid:101) G ( R (cid:93) + ) . That this agrees with the explicit formula follows from [

SW13 , Lemma 3.5.1]. (cid:3)

Recall also that the ﬁeld E ∞ obtained as the completion of the union of all E n is perfectoid — infact, one has a closed immersion Spf O E ∞ (cid:44) → (cid:101) G = Spf O E [[ (cid:101) X /p ∞ ]], which induces an isomorphismSpf O (cid:91)E ∞ ∼ = Spf F q [[ X /p ∞ ]]. Over E ∞ , we have an isomorphism O E ∼ = ( T π G )( O E ∞ ) ⊂ (cid:101) G ( O E ∞ ). Bythe proposition, if S (cid:93) lives over E ∞ , we get a nonzero section of O X S (1), vanishing at S (cid:93) ⊂ X S . Proposition

II.2.3 . For any perfectoid space S with untilt S (cid:93) over E ∞ , the above constructionconstruction deﬁnes an exact sequence → O X S → O X S (1) → O S (cid:93) → of O X S -modules. Proof.

The above constructions show that one has a map O X S → I (1) where I ⊂ O X S isthe ideal sheaf of S (cid:93) , which by Proposition II.1.18 is a line bundle. To see that this map is anisomorphism, it suﬃces to check on geometric points, so we can assume that S = Spa C for somecomplete algebraically closed extension C of F q . We have now ﬁxed some nonzero global sectionof O X S (1), which by the above corresponds to some nonzero topologically nilpotent X = (cid:36) ∈ C ,explicitly given by f = (cid:88) i ∈ Z π i [ X q − i ] ∈ H ( Y C , O Y C ) ϕ = π . This is the base change of the function (cid:88) i ∈ Z π i X q − i ∈ O ((Spa O E [[ X /p ∞ ]]) E \ V ( X )) , so it is enough to determine the vanishing locus of this function. But note that under the identiﬁ-cation (cid:101) G = Spf O E [[ X /p ∞ ]], this is precisely the logarithm functionlog G : (cid:101) G ad E \ { } → G ad a,E ;thus, it is enough to determine the vanishing locus of the logarithm function. But this is precisely (cid:71) n Spa E ∞ ⊂ (cid:101) G ad E \ { } , with a simple zero at each of these points. This gives exactly the claimed statement (the Z manycopies are translates under π , which are also translates under ϕ as ϕ ( f ) = πf ). (cid:3) I.2. VECTOR BUNDLES ON THE FARGUES–FONTAINE CURVE 61

Corollary

II.2.4 ([

Far18a , Proposition 2.12]) . There is a well-deﬁned map BC ( O (1)) \ { } → Div sending a nonzero section f ∈ H ( X S , O X S (1)) to the closed Cartier divisor given by V ( f ) .This descends to an isomorphism ( BC ( O (1)) \ { } ) /E × ∼ = Div . Proof.

Note that BC ( O (1)) ∼ = Spd F q [[ X /p ∞ ]] by Proposition II.2.2, and hence BC ( O (1)) \{ } ∼ = Spa F q (( X /p ∞ )) is representable by a perfectoid space. In fact, it is naturally isomorphicto Spd E ∞ = Spa E (cid:91) ∞ , and the previous proposition ensures that the map to Div is well-deﬁnedand corresponds to the projection Spd E ∞ → Spd E → Spd

E/ϕ Z = Div . Here, the ﬁrst mapSpd E ∞ → Spd E is a quotient under O × E , and the second map Spd E → Spd /ϕ Z then correspondsto the quotient by π Z , as ϕ = π on BC ( O (1)). (cid:3) In particular, if one works on Perf k , then Div = Spd ˘ E/ϕ Z , whose π ´et1 is given by the absoluteGalois group of E . On the other hand, the preceding gives a canonical E × -torsor, giving a naturalmap from the absolute Galois group of E to the proﬁnite completion of E × . By comparison withLubin–Tate theory, this is the usual Artin reciprocity map , see [

Far18a , Section 2.3] for moredetails.

II.2.2. Absolute Banach–Colmez spaces.

In this section, we analyze the Banach–Colmezspaces in the case E = E ( D ) for some isocrystal D = ( D, ϕ ). We then sometimes write BC ( D ) and BC ( D [1]) for the corresponding functors on Perf k ; or also BC ( O ( λ )), BC ( O ( λ )[1]) for λ ∈ Q when D = D − λ . These are in fact already deﬁned for all S ∈ Perf F q . Proposition

II.2.5 . Let λ ∈ Q . (i) If λ < , then H ( X S , O X S ( λ )) = 0 for all S ∈ Perf F q . Moreover, the projection from BC ( O ( λ )[1]) : S (cid:55)→ H ( X S , O X S ( λ )) to the point ∗ is relatively representable in locally spatial diamonds, partially proper, and cohomo-logically smooth. (ii) For λ = 0 , the map E → BC ( O ) is an isomorphism of pro-´etale sheaves, and the pro-´etale sheaﬁﬁcation of S (cid:55)→ H ( X S , O X S ) van-ishes. In particular, for all S one gets an isomorphism R Γ pro´et ( S, E ) → R Γ( X S , O X S ) . (iii) For λ > , one has H ( X S , O X S ( λ )) = 0 for all aﬃnoid S ∈ Perf F q , and the projection from BC ( O ( λ )) : S (cid:55)→ H ( X S , O X S ( λ )) to the point ∗ is relatively representable in locally spatial diamonds, partially proper, and cohomo-logically smooth. (iv) If < λ ≤ [ E : Q p ] (resp. for all positive λ if E is of equal characteristic), there is anisomorphism BC ( O ( λ )) ∼ = Spd k [[ x /p ∞ , . . . , x /p ∞ r ]] where λ = r/s with coprime integers r, s > . Proof.

For all statements, we can reduce to the case λ = n ∈ Z by replacing E by itsunramiﬁed extension of degree s . Regarding the vanishing of H ( X S , O X S ( n )) for n > S = Spa( R, R + ) aﬃnoid, pick a pseudouniformizer (cid:36) ∈ R . In terms of the presentation of X S asgluing Y S, [1 ,q ] along ϕ : Y S, [1 , ∼ = Y S, [ q,q ] , it suﬃces to see that ϕ − π n : B R, [1 ,q ] → B R, [1 , is surjective. Any element of B R, [1 , can be written as the sum of an element of B R, [0 , [ π ] and anelement of [ (cid:36) ] B R, [1 , ∞ ] , corresponding to the aﬃnoid subsets Y S, [0 , = {| π | ≤ | [ (cid:36) ] | (cid:54) = 0 } ⊂ Spa W O E ( R + )resp. Y S, [1 , ∞ ] = {| [ (cid:36) ] | ≤ | π | (cid:54) = 0 } ⊂ Spa W O E ( R + ) . If f ∈ B R, [0 , , then the series g = ϕ − ( f ) + π n ϕ − ( f ) + π n ϕ − ( f ) + . . . converges in B R, [0 ,q ] and thus in B R, [1 ,q ] , and f = ϕ ( g ) − π n g . The same then applies to elementsof B R, [0 , [ π ]. On the other hand, if f ∈ [ (cid:36) ] B R, [1 , ∞ ] , then the series g = − π − n f − ϕ ( f ) − π n ϕ ( f ) − . . . converges in B R, [1 ,q ] , and f = ϕ ( g ) − π n g .In fact, the same arguments prove that the map[ B R, [1 , ∞ ] ϕ − π n −−−→ B R, [1 , ∞ ] ] → [ B R, [1 ,q ] ϕ = π n −−−→ B R, [1 , ]is a quasi-isomorphism. Indeed, we have a short exact sequence0 → W O E ( R + )[ π ] → B R, [1 , ∞ ] ⊕ B R, [0 ,q ] [ π ] → B R, [1 ,q ] → W O E ( R + )[ π ] when endowed with the π -adic topology on W O E ( R + )),and similarly 0 → W O E ( R + )[ π ] → B R, [1 , ∞ ] ⊕ B R, [0 , [ π ] → B R, [1 , → . Therefore, it suﬃces to see that the maps B R, [0 ,q ] [ π ] ϕ − π n −−−→ B R, [0 , [ π ]and W O E ( R + )[ π ] ϕ − π n −−−→ W O E ( R + )[ π ]are isomorphisms. In both cases, this follows from convergence of ϕ − + π n ϕ − + π n ϕ − + . . . onthese algebras, giving an explicit inverse.For part (iv), note that in equal characteristic one can describe O ( Y S,I ) = B R,I , for S =Spa( R, R + ) aﬃnoid, explicitly as power series (cid:80) i ∈ Z r i π i with r i ∈ R , satisfying some convergenceconditions as i → ±∞ . Taking the part where ϕ = π n , we require ϕ ( r i ) = r i + n , and we see that wecan freely choose r , . . . , r n . The required convergence holds precisely when all r i are topologicallynilpotent, giving the isomorphism in that case. If E is p -adic, we can reduce to E = Q p (but now λ rational, 0 < λ ≤ X S,E = X S, Q p × Q p E → X S, Q p .In that case, the result follows from the equality H ( X S , O X S ( λ )) = B ϕ r = p s R, [1 , ∞ ] proved above, and[ SW13 , Theorem A, Proposition 3.1.3 (iii)].

I.2. VECTOR BUNDLES ON THE FARGUES–FONTAINE CURVE 63

In particular, for aﬃnoid S we can choose a ﬁbrewise nonzero map O X S → O X S (1), by takinga map S → BC ( O (1)) ∼ = Spd F q [[ x /p ∞ ]] sending x to a pseudouniformizer. By Proposition II.2.3,for any n ∈ Z , we get an exact sequence0 → O X S ( n ) → O X S ( n + 1) → O S (cid:93) → . Applying this for n >

0, we get inductively an exact sequence0 → BC ( O ( n )) | S → BC ( O ( n + 1)) | S → ( A S (cid:93) ) ♦ → . Starting with the base case n = 1 already handled, this allows one to prove part (iii) by induction,using [ Sch17a , Proposition 23.13].Now for part (ii), we use the sequence for n = 0. In that case, for S = Spa( R, R + ), we get anexact sequence 0 → H ( X S , O X S ) → H ( X S , O X S (1)) → R (cid:93) → H ( X S , O X S ) → E , proving (ii).Finally, for part (i), we ﬁrst treat the case n = −

1, where we get an exact sequence0 → E → ( A S (cid:93) ) ♦ → BC ( O ( − | S → H ( X S , O X S ( − E → ( A S (cid:93) ) ♦ is a closed immer-sion, the result follows from [ Sch17a , Proposition 24.2]. Now for n < −

1, the result follows byinduction from the sequence0 → ( A S (cid:93) ) ♦ → BC ( O ( − n )[1]) | S → BC ( O ( − n + 1)[1]) | S → Sch17a , Proposition 23.13]. (cid:3)

II.2.3. The algebraic curve.

We recall the following important ampleness result.

Theorem

II.2.6 ([

KL15 , Proposition 6.2.4]) . Let S = Spa( R, R + ) be an aﬃnoid perfectoidspace over F q and let E be any vector bundle on X S . Then there is an integer n such that for all n ≥ n , the vector bundle E ( n ) is globally generated, i.e. there is a surjective map O mX S → E ( n ) for some m ≥ , and moreover H ( X S , E ( n )) = 0 . Proof.

Pick a pseudouniformizer (cid:36) ∈ R , thus deﬁning a radius function on Y S . Write X S as the quotient of Y S, [1 ,q ] along the isomorphism ϕ : Y S, [1 , ∼ = Y S, [ q,q ] . Correspondingly, E is givenby some ﬁnite projective B R, [1 ,q ] -module M [1 ,q ] , with base changes M [1 , and M [ q,q ] to B R, [1 , and B R, [ q,q ] , and an isomorphism ϕ M : M [ q,q ] ∼ = M [1 , , linear over ϕ : B R, [ q,q ] ∼ = B R, [1 , .For convenience, we ﬁrst reduce to the case that M [1 ,q ] is free (cf. [ KL15 , Corollary 1.5.3]).Indeed, pick a surjection ψ : F [1 ,q ] := B mR, [1 ,q ] → M [1 ,q ] . We want to endow the source with a similar ϕ -module structure ϕ F : F [ q,q ] ∼ = F [1 , (with obvious notation), making ψ equivariant. For this, wewould like to ﬁnd a lift F [ q,q ] ϕ F (cid:47) (cid:47) ψ (cid:15) (cid:15) F [1 , ψ (cid:15) (cid:15) M [ q,q ] ϕ M (cid:47) (cid:47) M [1 , such that ϕ F is an isomorphism. Let N [1 ,q ] = ker( ψ ), with base change N [ q,q ] , N [1 , . Choosing asplitting F [1 ,q ] ∼ = M [1 ,q ] ⊕ N [1 ,q ] , we see that we could ﬁnd ϕ F if there is an isomorphism ϕ ∗ N [ q,q ] ∼ = N [1 , of B R, [1 , -modules. But in the Grothendieck group of B R, [1 , -modules, both are given by[ B mR, [1 , ] − [ M [1 , ]. Thus, after possibly adding a free module (i.e. increasing m ), they are isomorphic,giving the claim.Thus, we can assume that M [1 ,q ] ∼ = B mR, [1 ,q ] is a free B R, [1 ,q ] -module, and then ϕ M = A − ϕ for some matrix A ∈ GL m ( B R, [1 , ). Actually, repeating the above argument starting with thepresentation of X S as the quotient of Y S, [ q − ,q ] via identifying Y S, [ q − , with Y S, [1 ,q ] , one can ensurethat A ∈ GL m ( B R, [ q − , ) . Twisting by O X S ( n ) amounts to replacing A by Aπ n . Let us choose integers N and N (cid:48) such that • the matrix A has entries in π N W O E ( R + ) (cid:104) ( [ (cid:36) ] π ) ± (cid:105)• the matrix A − has entries in π − N (cid:48) W O E ( R + ) (cid:104) π [ (cid:36) ] , [ (cid:36) ] /q π (cid:105) .By twisting, we can replace N and N (cid:48) by N + n and N (cid:48) + n ; we can thus arrange that qN > N (cid:48) , N > r such that 1 < r ≤ q . We will now show that there are m elements v , . . . , v m ∈ ( B mR, [1 ,q ] ) ϕ = A = H ( X S , E )that form a basis of B mR, [ r,q ] . Repeating the above analysis for diﬀerent strips (and diﬀerent choicesof pseudouniformizers (cid:36) ∈ R to get overlapping strips), we can then get global generation of E .In fact, we will choose v i to be of the form [ (cid:36) ] M e i − v (cid:48) i , for some positive integer M chosenlater, where e i ∈ B mR, [1 ,q ] is the i -th basis vector and v (cid:48) i is such that || v (cid:48) i || B R, [ r,q ] ≤ || [ (cid:36) ] M +1 || B R, [ r,q ] = q − M − . Here, we endow all B R,I with the spectral norm, normalizing the norms on all completed residueﬁelds via || [ (cid:36) ] || = q . These v , . . . , v m restrict to a basis of B mR, [ r,q ] since the base change matrixfrom the canonical basis is given by an element of[ (cid:36) M ](Id + [ (cid:36) ] M m ( B ◦ R, [ r,q ] )) ⊂ GL m ( B R, [ r,q ] ) . In order to ﬁnd the v (cid:48) i , it suﬃces to prove that the map ϕ − A : B mR, [1 ,q ] → B mR, [1 , is surjective (yielding H ( X S , E ) = 0), in the following quantitative way: If, for some positiveinteger M chosen later, w ∈ π M W O E ( R + ) (cid:104) ( [ (cid:36) ] π ) ± (cid:105) m ⊂ B mR, [1 , , then there is some v ∈ B mR, [1 ,q ] such that(II.2.1) ( ϕ − A ) v = w and || v || B R, [ r,q ] ≤ q − M − . Indeed, we can then apply this to w i = ( ϕ − A )([ (cid:36) ] M e i ) (since N > A has entries in W O E ( R + ) (cid:104) ( [ (cid:36) ] π ) ± (cid:105) ), getting some v (cid:48) i with w i = ( ϕ − A )( v (cid:48) i ) and || v (cid:48) i || B R, [ r,q ] ≤ q − M − , as desired.Thus, take any w ∈ π M W O E ( R + ) (cid:104) ( [ (cid:36) ] π ) ± (cid:105) m . We can write w = w + w where w ∈ [ (cid:36) ] N − π M − N +1 W O E ( R + ) (cid:104) π [ (cid:36) ] (cid:105) m , w ∈ [ (cid:36) ] N π M − N W O E ( R + ) (cid:104) [ (cid:36) ] π (cid:105) m . Let v = ϕ − ( w ) − A − w ∈ B mR, [1 ,q ] so that w (cid:48) := w − ϕ ( v ) + Av = ϕ ( A − w ) + Aϕ − ( w ) . Note that (as

N > Aϕ − ( w ) ∈ π N [ (cid:36) ] ( N − /q π M − N +1 W O E ( R + ) (cid:104) π [ (cid:36) ] /q (cid:105) m ⊂ π M +1 W O E ( R + ) (cid:104) ( [ (cid:36) ] π ) ± (cid:105) m and also (as qN > N (cid:48) ) ϕ ( A − w ) ∈ π − N (cid:48) [ (cid:36) ] Nq π M W O E ( R + ) (cid:104) [ (cid:36) ] q π (cid:105) m ⊂ π M +1 W O E ( R + ) (cid:104) ( [ (cid:36) ] π ) ± (cid:105) m so that w (cid:48) ∈ π M +1 W O E ( R + ) (cid:104) ( [ (cid:36) ] π ) ± (cid:105) m . If one can thus prove the required bounds on v , this process will converge and prove the desiredstatement. It remains to estimate v . On the one hand, its norm is clearly bounded in terms of thenorm of w (as both w and w are, and ϕ − and A − are bounded operators), and thus, since whenone iterates w goes to zero, v goes to zero, and the process converges by summing to obtain some v such that ( ϕ − A ) v = w . But we need an improved estimate over B R, [ r,q ] to obtain (II.2.1). Notethat the norm of ϕ − ( w ) is bounded above by the norm of [ (cid:36) ] ( N − /q π M − N +1 , which in B R, [ r,q ] isgiven by q − ( N − /q − rM + rN − r . This is at most q − M − once M is large enough. On the other hand, w ∈ π M W O E ( R + ) (cid:104) [ (cid:36) ] π (cid:105) m and so the norm of A − w is bounded by the norm of π − N (cid:48) π M , which in B R, [ r,q ] is given by q rN (cid:48) − rM . Again, this is at most q − M − once M is large enough. Thus, taking M large enough (depending only on N , N (cid:48) and r > (cid:3) We have the following general

GAGA theorem . Its proof is an axiomatization of [

KL15 , The-orem 6.3.9].

Proposition

II.2.7 (GAGA) . Let ( X, O X ) be a locally ringed spectral space equipped with aline bundle O X (1) such that for any vector bundle E on X , there is some n such that for all n ≥ n , the bundle E ( n ) is globally generated. Moreover, assume that for i > , the cohomologygroup H i ( X, E ( n )) = 0 vanishes for all suﬃciently large n .Let P = (cid:76) n ≥ H ( X, O X ( n )) be the graded ring and X alg = Proj( P ) . There is a naturalmap ( X, O X ) → X alg of locally ringed topological spaces, and pullback along this map induces anequivalence of categories between vector bundles on X alg and vector bundles on ( X, O X ) . Moreover,for any vector bundle E alg on X alg with pullback E to X , the map H i ( X alg , E alg ) → H i ( X, E ) is an isomorphism for all i ≥ . Recall that for any graded ring P = (cid:76) n ≥ P n , one can deﬁne a separated scheme Proj( P )by gluing Spec P [ f − ] for all f ∈ P n , n >

0, where P [ f − ] = lim −→ i f − i P in is the degree 0 partof P [ f − ]. In our situation, if n is large enough so that O X ( n ) is globally generated, then itis enough to consider only f ∈ P n for this given n , and in fact only a ﬁnite set of them (as X is quasicompact); in particular, Proj( P ) is quasicompact. Moreover, one sees that there is atautological line bundle O Proj( P ) ( n ) for all suﬃciently large n , compatible with tensor products;thus, there is also a tautological line bundle O Proj( P ) (1), which is an ample line bundle on Proj( P ).The pullback of O Proj( P ) (1) is then given by O X (1). Proof.

The construction of the map f : ( X, O X ) → X alg is formal (and does not rely on anyassumptions): if g ∈ P n , then on the non-vanishing locus U = D ( g ) ⊂ X , there is an isomorphism g | U : O U ∼ −→ O U ( n ). Now, for x = ag k ∈ P [ g − ] , g − k | U ◦ a ∈ O ( U ), and this deﬁnes a morphism ofrings P [ g − ] → O ( U ). One deduces a morphism of locally ringed spaces U → D + ( g ), and thoseglue when g varies to a morphism of locally ringed spaces ( X, O X ) → X alg .We consider the functor taking any vector bundle E on X to the quasicoherent O X alg -module E associated to the graded P -module (cid:76) n ≥ H ( X, E ( n )). This functor is exact as H ( X, E ( n )) = 0for all suﬃciently large n , and it commutes with twisting by O (1). We claim that it takes valuesin vector bundles on X alg . To see this, take a surjection O mX → E ( n ) with kernel F , again a vectorbundle. The map O mX → E ( n ) splits after twisting, i.e. for any f ∈ P n (cid:48) with n (cid:48) large enough, thereis a map E ( n − n (cid:48) ) → O mX such that E ( n − n (cid:48) ) → O mX → E ( n ) is multiplication by f . Indeed,the obstruction to such a splitting is a class in H ( X, H om( E , F )( n (cid:48) )) which vanishes for n (cid:48) largeenough. This implies that E is a vector bundle on Spec P [ f − ] for any such f , and these cover X alg .There is a natural map f ∗ E → E , and the preceding arguments show that this is an isomorphism(on the preimage of any Spec P [ f − ] , and thus globally). It now remains to show that if E alg isany vector bundle on X alg , the map H i ( X alg , E alg ) → H i ( X, E )is an isomorphism for all i ≥

0. By ampleness of O X alg (1), there is some surjection O X alg ( − n ) m → ( E alg ) ∨ , with kernel a vector bundle F . Dualizing, we get an injection E → O X alg ( n ) m with cokernela vector bundle. This already gives injectivity on H by reduction to O X alg ( n ) where it is clear. I.2. VECTOR BUNDLES ON THE FARGUES–FONTAINE CURVE 67

Applying this injectivity also for F , we then get bijectivity on H . This already implies that weget an equivalence of categories (exact in both directions). Finally, picking f , . . . , f m ∈ P n sothat the Spec P [ f − i ] cover X alg , we can look at the corresponding ˇCech complex. Each term is aﬁltered colimit of global sections of vector bundles E ( n ) along multiplication by products of powersof f i ’s. This reduces the assertion to the case of H and the vanishing of H i ( X, E ( n )) for n largeenough. (cid:3) Remark

II.2.8 . One can check that X alg is up to canonical isomorphism independent of thechoice of a line bundle O X (1) satisfying the preceding properties.In particular, for any aﬃnoid perfectoid space S over F q , we can deﬁne the algebraic curve X alg S = Proj (cid:77) n ≥ H ( X S , O X S ( n )) . There is a well-deﬁned map X S → X alg S of locally ringed spectral spaces, pullback along whichdeﬁnes an equivalence of categories of vector bundles, and is compatible with cohomology.Notably, this connects the present discussion to the original deﬁnition of the Fargues–Fontainecurve as given in [ FF18 ], where the case S = Spa( F, O F ) is considered, for a perfectoid ﬁeld F ofcharacteristic p . We will restrict ourselves, as above, to the case that F = C is algebraically closed. Proposition

II.2.9 . Let C be a complete algebraically closed nonarchimedean ﬁeld over F q .Then X alg C is a connected regular noetherian scheme of Krull dimension , and the map | X C | →| X alg C | induces a bijection between | X C | cl and the closed points of | X alg C | . Moreover, for any classicalpoint x ∈ | X C | , the complement X alg C \ { x } is the spectrum of a principal ideal domain. Proof.

Let x ∈ | X C | cl be any classical point, corresponding to some untilt C (cid:93) over E of C .Using Lubin–Tate formal groups, we see that there is an exact sequence0 → O X C → O X C (1) → O C (cid:93) → X C . The corresponding section f ∈ H ( X C , O X C (1)) deﬁnes its vanishing locus in X alg C , whichis then also given by Spec C (cid:93) . In particular, x deﬁnes a closed point of | X alg C | . Now we want toshow that P [ f − ] is a principal ideal domain. Thus, take any nonzero g ∈ H ( X C , O X C ( n )).This has ﬁnitely many zeroes on X C , all at classical points x , . . . , x m . For each x i , we havea section f x i ∈ H ( X C , O X C (1)) as before, and then g = f n · · · f n m m h for some n i ≥

1, andsome h ∈ H ( X C , O X C ( n (cid:48) )) that is everywhere nonzero. In particular, h deﬁnes an isomorphism O X C → O X C ( n (cid:48) ), whence n (cid:48) = 0, and h ∈ E × . This decomposition implies easily that P [ f − ] isindeed a principal ideal domain, and it shows that all maximal ideals arise from classical points of | X C | , ﬁnishing the proof. (cid:3) II.2.4. Classiﬁcation of vector bundles.

At this point, we can recall the classiﬁcation ofvector bundles over X C ; so here we take S = Spa C for a complete algebraically closed nonar-chimedean ﬁeld C over F q . First, one classiﬁes line bundles. Proposition

II.2.10 . The map Z → Pic( X C ) , n (cid:55)→ O X C ( n ) , is an isomorphism. Proof.

By Proposition II.2.9, any line bundle becomes trivial after removing one closed point x ∈ X alg C . As the local rings of X alg C are discrete valuation rings, this implies that any line bundleis of the form O X C ( n [ x ]) for some n ∈ Z . But O X C ([ x ]) ∼ = O X C (1) by Proposition II.2.2, so theresult follows. (cid:3) In particular, one can deﬁne the degree of any vector bundle E on X C viadeg( E ) = deg(det( E )) ∈ Z where det( E ) is the determinant line bundle, and deg : Pic( X C ) ∼ = Z is the isomorphism from theproposition. Of course, one can also deﬁne the rank rk( E ) of any vector bundle, and thus for anynonzero vector bundle its slope µ ( E ) = deg( E )rk( E ) ∈ Q . It is easy to see that this satisﬁes the Harder–Narasimhan axiomatics (for example, rank anddegree are additive in short exact sequences). In particular, one can deﬁne semistable vector bundlesas those vector bundles E such that for all proper nonzero F ⊂ E , one has µ ( F ) ≤ µ ( E ). One saysthat E is stable if in fact µ ( F ) < µ ( E ) for all such F . Example

II.2.11 . For any λ ∈ Q , the bundle O X C ( λ ) is stable of slope λ . Indeed, assume that0 (cid:54) = F (cid:40) O X C ( λ ) is a proper nonzero subbundle, and let r = rk( F ), s = deg( F ). Passing to r -thwedge powers, we get an injectiondet( F ) ∼ = O X C ( s ) (cid:44) → O X C ( rλ ) m , using that (cid:86) r O X C ( λ ) is a direct sum of copies of O X C ( rλ ). This implies that s ≤ rλ . Moreover, ifwe have equality, then r is at least the denominator of λ , which is the rank of O X C ( λ ), i.e. F hasthe same rank as O X C ( λ ). Thus, O X C ( λ ) is stable. Proposition

II.2.12 . Any vector bundle E on X C admits a unique exhaustive separating Q -indexed ﬁltration by saturated subbundles E ≥ λ ⊂ E , called the Harder–Narasimhan ﬁltration, suchthat E λ := E ≥ λ / E >λ , where E >λ = (cid:91) λ (cid:48) >λ E ≥ λ (cid:48) , is semistable of slope λ . The formation of the Harder–Narasimhan ﬁltration is functorial in E . (cid:3) As a preparation for the next theorem, we note that the Harder–Narasimhan ﬁltration is alsocompatible with change of C . Proposition

II.2.13 . Let E be a vector bundle on X C , and let C (cid:48) | C be an extension of completealgebraically closed nonarchimedean ﬁelds, with pullback E (cid:48) of E to X C (cid:48) . Then ( E (cid:48) ) ≥ λ is the pullbackof E ≥ λ .Similarly, if E (cid:48) | E is a ﬁnite separable extension of degree r , and E (cid:48) is the pullback of E along X C,E (cid:48) = X C,E ⊗ E E (cid:48) → X C,E = X C , then ( E (cid:48) ) ≥ λ is the pullback of E ≥ λ/r . Proof.

Consider the case of C (cid:48) | C . By uniqueness of the Harder–Narasimhan ﬁltration, itsuﬃces to see that pullbacks of semistable vector bundles remain semistable. Thus, assume that E is semistable, and assume by way of contradiction that E (cid:48) is not semistable. By induction on the I.2. VECTOR BUNDLES ON THE FARGUES–FONTAINE CURVE 69 rank, we can assume that the formation of the Harder–Narasimhan ﬁltration of E (cid:48) is compatiblewith any base change. Consider the ﬁrst nontrivial piece of the Harder–Narasimhan ﬁltration0 (cid:54) = F (cid:40) E . This is a vector bundle on X C (cid:48) with µ ( F ) > µ ( E ). We claim that F descends to X C .By Proposition II.2.1, it suﬃces to see that the two pullbacks of F to X C (cid:48) (cid:98) ⊗ C C (cid:48) agree. This is trueas there are no nonzero maps from F to E (cid:48) / F after base change to X R for any perfectoid C (cid:48) -algebra R : If there were such a nonzero map, there would also be a nonzero map for some choice of R = C (cid:48)(cid:48) a complete algebraically closed nonarchimedean ﬁeld. But then F is still semistable and all piecesof the Harder–Narasimhan ﬁltration of E (cid:48) / F are of smaller slope, so such maps do not exist.For an extension E (cid:48) | E , the similar arguments work, using Galois descent instead (noting thatone may assume that E (cid:48) | E is Galois by passing to Galois hulls). Note that the pullback of O X C,E (1)is O X C,E (cid:48) ( r ), causing the mismatch in slopes. (cid:3) The main theorem on the classiﬁcation of vector bundles is the following. Our proof followsthe arguments of Hartl-Pink, [

HP04 ], to reduce to Lemma II.2.15 below. However, we give a newand direct proof of this key lemma, which avoids any hard computations by using the geometry ofdiamonds and v-descent. We thus get a new proof of the classiﬁcation theorem. Theorem

II.2.14 . Any vector bundle E on X C is isomorphic to a direct sum of vector bundlesof the form O X C ( λ ) with λ ∈ Q . If E is semistable of slope λ , then E ∼ = O X C ( λ ) m for some m ≥ . Proof.

We argue by induction on the rank n of E , so assume the theorem in rank ≤ n − E ); the case n = 1 has been handled already. By the vanishing of H ( X C , O X C ( λ )) = 0 for λ >

0, the theorem follows for E if E is not semistable. Thus assume E is semistable of slope λ = sr with s ∈ Z and r > O X C ( λ ) → E : Indeed, by stability of O X C ( λ ), the map is necessarily injective (the category of semi-stable vector bundles of slope λ is abelian with simple objects the stable vector bundles of slope λ ), and the quotient will then again be semistable of slope λ , and thus by induction isomorphic to O X C ( λ ) m − . One ﬁnishes by observing that Ext X C ( O X C ( λ ) , O X C ( λ )) = 0 by Proposition II.2.5 (ii).Thus, it suﬃces to ﬁnd a nonzero map O X C ( λ ) → E . Let E (cid:48) | E be the unramiﬁed extensionof degree r , and consider the covering f : X C,E (cid:48) = X C,E ⊗ E E (cid:48) → X C,E = X C . Then O X C ( λ ) = f ∗ O X C,E (cid:48) ( s ), and so it suﬃces to ﬁnd a nonzero map O X C,E (cid:48) ( s ) → f ∗ E . In other words, up tochanging E , we can assume that λ ∈ Z . Then by twisting, we can assume λ = 0.Next, we observe that we are free to replace C by an extension. Indeed, consider the v-sheafsending S ∈ Perf C to the isomorphisms E ∼ = O nX C . This is a v-quasitorsor under GL n ( E ) (usingProposition II.2.5 (ii)). If there is some extension of C where we can ﬁnd a nonzero section of E (and thus also trivialize E ), then it is a v-torsor under GL n ( E ). By v-descent of GL n ( E )-torsors,cf. [ Sch17a , Lemma 10.13], it is then representable by a space pro-´etale over Spa C , and thusadmits a section. First, it has been proven for E of equal characteristic in [ HP04 ] and for p -adic E by Kedlaya in [ Ked04 ];both of these proofs used heavy computations to prove Lemma II.2.15. A more elegant proof was given by Fargues–Fontaine [

FF18 ] (for all E ) by reducing to the description of the Lubin–Tate and Drinfeld moduli spaces of π -divisible O E -modules, and their Grothendieck–Messing period morphisms (which arguably also involve some nontrivialcomputations). Finally, for p -adic E a proof is implicit in Colmez’ work [ Col02 ] on Banach–Colmez spaces.

Let d ≥ O X C ( − d ) (cid:44) → E , possibly after baseenlarging C ; by Theorem II.2.6 some such d exists. We want to see that d = 0, so assume d > d , the quotient F = E / O X C ( − d ) is a vector bundle, and byinduction the classiﬁcation theorem holds true for F .If d ≥

2, then we can by induction ﬁnd an injection O X C ( − d + 2) (cid:44) → F ; taking the pullbackdeﬁnes an extension O X C ( − d ) → G → O X C ( − d + 2) . so by twisting O X C ( − → G ( d − → O X C (1) . By the key lemma, Lemma II.2.15 below, we would, possibly after enlarging C , get an injection O X C (cid:44) → G ( d − O X C ( − d + 1) (cid:44) → G (cid:44) → E , contradicting our choice of d .Thus, we may assume that d = 1. If F is not semistable, then it admits a subbundle F (cid:48) ⊂ F of degree ≥ ≤ n −

2. Applying the classiﬁcation theorem to the pullback0 → O X C ( − → E (cid:48) → F (cid:48) → F (cid:48) , which is of slope ≥

0, we then get that E (cid:48) ⊂ E has a global section.It remains the case that d = − F is semistable, thus necessarily isomorphic to O X C ( n − ). This is the content of the next lemma. (cid:3) Lemma

II.2.15 . Let → O X C ( − → E → O X C ( n ) → be an extension of vector bundles on X C , for some n ≥ . Then there is some extension C (cid:48) | C ofcomplete algebraically closed nonarchimedean ﬁelds such that H ( X C (cid:48) , E| X C (cid:48) ) (cid:54) = 0 . Proof.

Assume the contrary. Passing to Banach–Colmez spaces, we ﬁnd an injection of v-sheaves f : BC ( O X C ( n )) (cid:44) → BC ( O X C ( − . The image cannot be contained in the classical points (as these form a totally disconnected subsetwhile the source is connected), so the image contains some non-classical point. After base change tosome C (cid:48) | C , we thus ﬁnd that the image contains some nonempty open subset of BC ( O X C ( − BC ( O X C ( − A C (cid:93) ) ♦ /E and the similar behaviour of non-classical points of A C (cid:93) . Translating this nonempty open subset tothe origin, we ﬁnd that the image of f contains an open neighborhood of 0, and then by rescalingby the contracting of E × , we ﬁnd that the map f must be surjective, and thus an isomorphism.In particular, this would mean that BC ( O X C ( − E is p -adic, as then the given presentation shows that ( A C (cid:93) ) ♦ is pro-´etale over a perfectoidspace and thus itself a perfectoid space, but A C (cid:93) is clearly not a perfectoid space. In general, we can argue as follows. There is a nonzero map BC ( O X C ( n )) → ( A C (cid:93) ) ♦ We believe that also when E is of equal characteristic, BC ( O X C ( − I.2. VECTOR BUNDLES ON THE FARGUES–FONTAINE CURVE 71 as H ( O X C ( n )) maps nontrivially to its ﬁbre at the chosen untilt Spa C (cid:93) (cid:44) → X C . If f is anisomorphism, we would then get a nonzero map( A C (cid:93) ) ♦ /E ∼ = BC ( O X C ( − f − −−→ BC ( O X C ( n )) → ( A C (cid:93) ) ♦ . On the other hand, one can classify all E -linear maps ( A C (cid:93) ) ♦ → ( A C (cid:93) ) ♦ . The latter are thesame as maps A C (cid:93) → A C (cid:93) if E is p -adic (by [ SW20 , Proposition 10.2.3]), respectively maps A C, perf → A C, perf if E is of equal characteristic. Thus, they are given by some convergent powerseries g ( X ) that is additive, i.e. g ( X + Y ) = g ( X ) + g ( Y ), and satisﬁes g ( aX ) = ag ( X ) for all a ∈ E . (If E is of characteristic p , then g may a priori involve fractional powers X /p i .) Theequation g ( πX ) = πg ( X ) alone in fact shows that only the linear coeﬃcient of g may be nonzero,so g ( X ) = cX for some c ∈ C (cid:93) , and thus g is either an isomorphism or zero. But our given map isnonzero with nontrivial kernel, giving a contradiction. (cid:3) II.2.5. Families of vector bundles.

Using the ampleness of O (1), we can now prove thefollowing result on relative Banach–Colmez spaces. Proposition

II.2.16 . Let S be a perfectoid space over F q . Let E be a vector bundle on X S .Then the Banach–Colmez space BC ( E ) : T (cid:55)→ H ( X T , E| X T ) is a locally spatial diamond, partially proper over S . Moreover, the projectivized Banach–Colmezspace ( BC ( E ) \ { } ) /E × is a locally spatial diamond, proper over S . Proof.

Note that BC ( E ) → S is separated: As it is a group, it suﬃces to see that the zerosection is closed, but being zero can be checked through the vanishing at all untilts T (cid:93) of T , whereit deﬁnes Zariski closed conditions. Using Theorem II.2.6, choose a surjection O X S ( − n ) m → E ∨ .Dualizing, we get an exact sequence0 → E → O X S ( n ) m → F → F . As BC ( F ) is separated, we see that BC ( E ) ⊂ BC ( O X S ( n ) m ) is closed,so the ﬁrst part follows from Proposition II.2.5 (iii). For the second part, we may assume that S is qcqs. It is also enough to prove the similar result for ( BC ( E ) \ { } ) /π Z as the O × E -action is free(so one can apply the last part of [ Sch17a , Proposition 11.24]). This follows from the followinggeneral lemma about contracting group actions on locally spectral spaces, noting that checkingthe conditions formally reduces to the case of BC ( O X S ( n ) m ) and from there to A S (cid:93) by evaluatingsections at some collection of untilts. (cid:3) Lemma

II.2.17 . Let X be a taut locally spectral space such that for any x ∈ X , the set X x ⊂ X of generalizations of x is a totally ordered chain under specialization. Let γ : X ∼ −→ X be anautomorphism of X such that the subset X ⊂ X of ﬁxed points is a spectral space. Moreover,assume that (i) for all x ∈ X , the sequence γ n ( x ) for n → ∞ converges towards X , i.e. for all open neighborhoods U of X , one has γ n ( x ) ∈ U for all suﬃciently positive n ; (ii) for all x ∈ X \ X , the sequence γ n ( x ) for n → −∞ diverges, i.e. for all quasicompact opensubspaces U ⊂ X , one has γ n ( x ) (cid:54)∈ U for all suﬃciently negative n .Then X ⊂ X is a closed subspace, the action of γ on X \ X is free and totally discontinuous(i.e. the action map ( X \ X ) × Z → ( X \ X ) × ( X \ X ) is a closed immersion), and the quotient ( X \ X ) /γ Z is a spectral space. Remark

II.2.18 . For applications of this lemma, we recall the following facts:(i) If X is any locally spatial diamond, then | X | is a locally spectral space such that all for all x ∈ | X | , the set of generalizations of x in | X | is a totally ordered chain under specialization.Indeed, this follows from [ Sch17a , Proposition 11.19] and the similar property for analytic adicspaces.(ii) If X is in addition partially proper over a spatial diamond, then | X | is taut by [ Sch17a , Propo-sition 18.10].This means that the ﬁrst sentence of the lemma is practically always satisﬁed.

Proof.

Let U ⊂ X be some quasicompact open neighborhood of X . First, we claim that onecan arrange that γ ( U ) ⊂ U . Indeed, one has U ⊂ γ − ( U ) ∪ γ − ( U ) ∪ . . . ∪ γ − n ( U ) ∪ . . . , as for any x ∈ U ⊂ X , also γ n ( x ) ∈ U for all suﬃciently large n by assumption, and so x ∈ γ − n ( U )for some n >

0. By quasicompacity of U , this implies that U ⊂ γ − ( U ) ∪ . . . ∪ γ − n ( U ) for some n , and then U (cid:48) = U ∪ γ − ( U ) ∪ . . . ∪ γ − n +1 ( U ) is a quasicompact open neighborhood of X with γ ( U (cid:48) ) ⊂ U (cid:48) .Now ﬁx a quasicompact open neighborhood U of X with γ ( U ) ⊂ U . We claim that X = (cid:92) n ≥ γ n ( U ) . Indeed, if x ∈ X \ X , then by assumption there is some positive n such that γ − n ( x ) (cid:54)∈ U , givingthe result.In particular, for any other quasicompact open neighborhood V of X , there is some n suchthat γ n ( U ) ⊂ V . Indeed, the sequence of spaces γ n ( U ) \ V is a decreasing sequence of spectralspaces with empty inverse limit, and so one of the terms is empty.Consider the closure U ⊂ X of U in X . As X is taut, this is still quasicompact. Repeating theabove argument, we see that for some n >

0, one has U ⊂ γ − ( U ) ∪ . . . ∪ γ − n ( U ) = γ − n ( U ) . This implies that the sequences { γ n ( U ) } n ≥ and { γ n ( U ) } n ≥ are coﬁnal. In particular, X = (cid:92) n ≥ γ n ( U ) = (cid:92) n ≥ γ n ( U )is a closed subset of X .Next, we check that any point x ∈ X \ X has an open neighborhood V such that { γ n ( V ) } n ∈ Z are pairwise disjoint; for this it suﬃces to arrange that V ∩ γ i ( V ) = ∅ for all i >

0. For this,

I.2. VECTOR BUNDLES ON THE FARGUES–FONTAINE CURVE 73 note that if n is chosen such that γ n ( U ) ⊂ U , then up to rescaling by a power of γ , we canassume that x ∈ U \ γ n +1 ( U ). Let V ⊂ U \ γ n +1 ( U ) be a quasicompact open neighborhood of x . Then γ i ( V ) ∩ V = ∅ as soon as i ≥ n + 1. For the ﬁnitely many i = 1 , . . . , n , we can use aquasicompacity argument, and reduce to proving that if X x is the localization of X at x (i.e., theset of all generalizations of x ), then X x ∩ γ i ( X x ) = ∅ for i = 1 , . . . , n . By our assumption on X ,the space X x has a unique generic point η ∈ X x ( X x is pro-constructible in a spectral space thusspectral and by our hypothesis X x is irreducible), which must then also be the unique generic pointof γ i ( X x ) if X x ∩ γ i ( X x ) (cid:54) = ∅ . Thus, if X x ∩ γ i ( X x ) (cid:54) = ∅ , then γ i ( η ) = η , so η ∈ X . But X isclosed, so that x ∈ X , which is a contradiction.In particular, the action of γ on X \ X is free and totally discontinuous, and the quotient X = ( X \ X ) /γ Z is a locally spectral space which is locally isomorphic to X \ X . A basis ofopen neighborhoods of X is given by the image of quasicompact open subsets V ⊂ X \ X forwhich { γ n ( V ) } n ∈ Z are pairwise disjoint; it follows that these are quasicompact open subsets of X .Also, the intersection of two such subsets is of the same form, so the quotient X is quasiseparated.Finally, note that U \ γ ( U ) → X is a bijective continuous map, and the source is a spectralspace (as γ ( U ) ⊂ U is a quasicompact open subspace of the spectral space U ), and in particularquasicompact, and so X is quasicompact. (cid:3) The result on properness of the projectivized Banach–Colmez space enables us to give quickproofs of the main results of [

KL15 ] (including an extension to the case of general E , in particularof equal characteristic). Theorem

II.2.19 ([

KL15 , Theorem 7.4.5, Theorem 7.4.9, Theorem 7.3.7, Proposition 7.3.6]) . Let S be a perfectoid space over F q and let E be a vector bundle over X S of constant rank n . (i) The function taking a geometric point

Spa C → S of S to the Harder–Narasimhan polygon of E| X C is upper semicontinuous. (ii) Assume that the Harder–Narasimhan polygon of E is constant. Then there exists a global (sep-arated exhaustive decreasing) Harder–Narasimhan ﬁltration E ≥ λ ⊂ E specializing to the Harder–Narasimhan ﬁltration at each point. Moreover, after replacing S by apro-´etale cover, the Harder–Narasimhan ﬁltration can be split, and there are isomorphisms E λ ∼ = O X S ( λ ) n λ for some integers n λ ≥ . Proof.

Note that the Harder–Narasimhan polygon can be described as the convex hull of thepoints ( i, d i ) for i = 0 , . . . , n , where d i is the maximal integer such that H ( X C , ( ∧ i E )( − d i ) | X C ) (cid:54) = 0.To prove part (i), it therefore suﬃces to show that for any vector bundle F on X S , the locus ofall geometric points Spa C → S for which H ( X C , F | X C ) (cid:54) = 0 is closed in S . But note that this isprecisely the image of ( BC ( F ) \ { } ) /E × → S. As this map is proper by Proposition II.2.16 (ii), its image is closed. To see that the endpoint ofthe Harder–Narasimhan polygon is locally constant, apply the preceding also to the dual of thedeterminant of E . For part (ii), it is enough to prove that v-locally on S , there exists an isomorphism E ∼ = (cid:76) λ O X S ( λ ) n λ . Indeed, the desired global Harder–Narasimhan ﬁltration will then exist v-locally,and it necessarily descends. The trivialization of each E λ amounts to a torsor under some locallyproﬁnite group, and can thus be done after a pro-´etale cover by [ Sch17a , Lemma 10.13]. Then theability to split the ﬁltration follows from Proposition II.2.5 (i).We argue by induction on the rank of E . Let λ be the maximal slope of E . We claim thatv-locally on S , there is a map O X S ( λ ) → E that is nonzero in each ﬁbre. Indeed, ﬁnding such amap is equivalent to ﬁnding a ﬁbrewise nonzero map O X S → F = H om( O X S ( λ ) , E ). But then BC ( F ) \ { } → ( BC ( F ) \ { } ) /E × → S is a v-cover over which such a map exists: The ﬁrst map is an E × -torsor and thus a v-cover, whilethe second map is proper and surjective on geometric points, thus surjective by [ Sch17a , Lemma12.11]. The dual map E ∨ → O X S ( − λ ) is surjective as can be checked over geometric points (usingthat O X C ( − λ ) is stable), thus the cokernel of O X S ( λ ) → E is a vector bundle E (cid:48) , that again hasconstant Harder–Narasimhan polygon. By induction, one can ﬁnd an isomorphism E (cid:48) ∼ = (cid:77) λ (cid:48) ≤ λ O X S ( λ (cid:48) ) n (cid:48) λ (cid:48) . By Proposition II.2.5 (i)–(ii), the extension0 → O X C ( λ ) → E → (cid:77) λ (cid:48) ≤ λ O X S ( λ (cid:48) ) n (cid:48) λ (cid:48) → (cid:3) Let us explicitly note the following corollary.

Corollary

II.2.20 ([

KL15 , Theorem 8.5.12]) . Let S be a perfectoid space. The category ofpro-´etale E -local systems L is equivalent to the category of vector bundles on X S whose Harder–Narasimhan polygon is constant , via L (cid:55)→ L ⊗ E O X S . Proof.

First, the functor is fully faithful, as we can see by pro-´etale descent (to assume L istrivial) and Proposition II.2.5. Now essential surjectivity follows from Theorem II.2.19. (cid:3) II.3. Further results on Banach–Colmez spaces

We include some further results on Banach–Colmez spaces.

II.3.1. Cohomology of families of vector bundles.

First, we generalize the vanishingresults of Proposition II.2.5 to families of vector bundles. A key tool is given by the followingresult, which is a small strengthening of [

KL15 , Lemma 8.8.13].

Proposition

II.3.1 . Let S be a perfectoid space over F q , and let E be a vector bundle on X S such that all Harder–Narasimhan slopes of E at all geometric points are nonnegative. Then locally(in the analytic topology) on S , there is an exact sequence → O X S ( − d → F → E → where F is semistable of degree at all geometric points. I.3. FURTHER RESULTS ON BANACH–COLMEZ SPACES 75

Proof.

We may assume that the degree of E is constant, given by some d ≥

0. We can assume S = Spa( R, R + ) is aﬃnoid perfectoid and pick d untilts S (cid:93)i = Spa( R (cid:93)i , R (cid:93) + i ) over E , i = 1 , . . . , d ,such that S (cid:93) , . . . , S (cid:93)d ⊂ X S are pairwise disjoint; more precisely, choose d maps S → BC ( O (1)) \ { } .Let W i be the ﬁbre of E over R (cid:93)i , which is a ﬁnite projective R (cid:93)i -module. For any rank 1 quotients W i → R (cid:93)i , we can pull back the sequence0 → O X S ( − d → O dX S → d (cid:77) i =1 O S (cid:93)i → , obtained from Proposition II.2.3, along E → d (cid:77) i =1 E ⊗ O XS O S (cid:93)i = d (cid:77) i =1 W i ⊗ R (cid:93)i O S (cid:93)i → d (cid:77) i =1 O dS (cid:93)i to get an extension 0 → O X S ( − d → E (cid:48) → E → . We claim that one can choose, locally on S , the rank 1 quotients so that E (cid:48) is semistable ofdegree 0. For this, we argue by induction on i = 1 , . . . , d that one can choose (locally on S ) thequotients of W , . . . , W i so that the modiﬁcation0 → O X S ( − i → E i → E → , invoking only the quotients of W , . . . , W i , has the property that the Harder–Narasimhan slopes of E i at all geometric points are nonnegative. It is in fact enough to handle the case i = 1, as theinductive step then reduces to the similar assertion for the bundle E which is of smaller degree.First, we handle the case that S = Spa( K, K + ) for a perfectoid ﬁeld K (not necessarily alge-braically closed). Then E has a Harder–Narasimhan ﬁltration, and look at the subbundle E ≥ λ ⊂ E of maximal slope, where necessarily λ >

0. Also recall that any nonsplit extension0 → O X S ( − → G → E ≥ λ → − E ≥ λ is nonsplit. But if it splits, then the given map E ≥ λ → O S (cid:93) lifts to a map E ≥ λ → O X S ; by consideration of slopes, this map is necessarily trivial. Thus, if we let W (cid:48) ⊂ W be the ﬁbre of E ≥ λ ⊂ E at S (cid:93) , it suﬃces to pick a quotient W → K (cid:93) whose restriction to W (cid:48) isnonzero.Going back to general aﬃnoid S , pick any point s ∈ S . By the preceding argument, we canlocally on S ﬁnd a quotient W → R (cid:93) such that the corresponding extension E has the propertythat the Harder–Narasimhan slopes at s are still nonnegative. By Theorem II.2.19, the same istrue in an open neighborhood, ﬁnishing the proof. (cid:3) In applications, it is often more useful to have the following variant, switching which of thetwo bundles is trivialized, at the expense of assuming strictly positive slopes (and allowing ´etalelocalizations in place of analytic localizations — this is probably unnecessary).

Proposition

II.3.2 . Let S be a perfectoid space over F q and let E be a vector bundle on X S such that at all geometric points of S , all Harder–Narasimhan slopes of E are positive. Then ´etalelocally on S , there is a short exact sequence → G → O mX S → E → where G is semistable of slope − at all geometric points. Proof.

We can assume that E has constant degree d and rank r ; we set m = d + r . Inside BC ( E ) m , we can look at the locus U ⊂ BC ( E ) m of those maps O mX S → E that are surjective andwhose kernel is semistable of slope −

1. This is an open subdiamond of BC ( E ) m : This is clear forthe condition of surjectivity (say, as the cokernel of the universal map O nX T → E| X T is supportedon a closed subset of X T , whose image is then closed in T ), and then the locus where the kernelis semistable of slope − U → S are nonempty. It thus suﬃces to prove that for any geometric point T = Spa( C, C + ) → S , given as a coﬁltered inverse limit of ´etale maps S i = Spa( R i , R + i ) → S , andany section s ∈ BC ( E )( T ), one can ﬁnd a sequence of i ’s and sections s i ∈ BC ( E )( S i ) such that s i | T → s as i → ∞ . Indeed, applying this to E m in place of E and some section of U over T , oneof the s i will then lie in U ( S i ), giving the desired short exact sequence.To prove that one can approximate s , we argue in a way similar to the proof of Theorem II.2.6.To facilitate the estimates, it is useful to assume that all Harder–Narasimhan slopes of E at T areintegral; this can always be achieved through pulling back E to a cover f : X S,E (cid:48) = X S ⊗ E E (cid:48) → X S for some unramiﬁed extension E (cid:48) | E (as this pullback multiplies slopes by [ E (cid:48) : E ]), noting that E is a direct factor of f ∗ f ∗ E = E ⊗ E E (cid:48) . We analyze E in terms of its pullback to Y S, [1 ,q ] and theisomorphism over Y S, [ q,q ] ∼ = Y S, [1 , . Note that as B C, [1 ,q ] is a principal ideal domain, the pullback of E to Y C, [1 ,q ] is necessarily free, and by approximation we can already ﬁnd a basis over some Y S i , [1 ,q ] ;replacing S by S i we can then assume that the pullback of E to Y S, [1 ,q ] is free. The descent datumis then given by A − ϕ for some matrix A ∈ GL n ( B R, [1 , ). After pullback to T , by Theorem II.2.14and the assumption of integral slopes, one can in fact choose a basis so that A is a diagonal matrix D with positive powers of π along the diagonal. Approximating this basis, we can assume that A − D ∈ π N B +( R,R + ) , [1 , for any chosen N > ϕ − D : B rR, [1 ,q ] → B rR, [1 , is surjective by Proposition II.2.5, and in fact there is some M (depending only on D , not on R )such that for any x ∈ ( B +( R,R + ) , [1 , ) r , there is some y ∈ π − M ( B +( R,R + ) , [1 ,q ] ) r with x = ( ϕ − D )( y ).There are two ways to see the existence of M : Either by an explicit reading of the proof ofProposition II.2.5, or as follows. Assume no such M exists; then we can ﬁnd perfectoid algebras R , R , . . . with integral elements R + i ⊂ R i and pseudouniformizers (cid:36) i ∈ R i , and sections x i ∈ ( B +( R i ,R + i ) , [1 , ) r such that there is no y i ∈ π − i ( B +( R i ,R + i ) , [1 ,q ] ) r with x i = ( ϕ − D )( y i ). Let R + be theproduct of all R + i , and R = R + [ (cid:36) ] where (cid:36) = ( (cid:36) i ) i ∈ R + = (cid:81) i R + i . Then all x i deﬁne elementsof ( B +( R,R + ) , [1 , ) r , and x = x + πx + π x + . . . another element. By surjectivity of ϕ − D , thereis some y ∈ ( B ( R,R + ) , [1 ,q ] ) r with x = ( ϕ − D )( y ). But then y ∈ π − i ( B +( R,R + ) , [1 ,q ] ) r for some i , andthen projecting along ( R, R + ) → ( R i , R + i ) contradicts the choice of x i . I.3. FURTHER RESULTS ON BANACH–COLMEZ SPACES 77

Taking

N > M above, one sees that also ϕ − A = ϕ − D + ( D − A ) : B rR, [1 ,q ] → B rR, [1 , is surjective, with the same bound (in particular independent of R ). But now the section s of E over X T can be approximated by a section s (cid:48) i of B R i , [1 ,q ] , so that its image under ϕ − A will besmall. By the preceding surjectivity, we can then replace s (cid:48) i by s i = s (cid:48) i + (cid:15) i for some still small (cid:15) i such that s i ∈ B ϕ = AR i , [1 ,q ] = H ( X S i , E| X Si ) . This gives the desired conclusion. (cid:3)

One can prove the following variants.

Corollary

II.3.3 . Let S be a perfectoid space over F q and let E be a vector bundle on X S . (i) Assume that all Harder–Narasimhan slopes of E are ≥ r . Then locally on S , for some m ≥ there is a short exact sequence −→ O mX S −→ F −→ E −→ , where F is ﬁbrewise semistable of slope r . (ii) Assume that all Harder–Narasimhan slopes of E are ≥ r . Then locally on S , for some m ≥ there is a short exact sequence −→ O X S ( r ) m −→ F −→ E −→ , where F is ﬁbrewise semistable of slope r . (iii) Assume that all Harder–Narasimhan slopes of E are > r . Then ´etale locally on S , for some m ≥ there is a short exact sequence −→ G −→ O X S ( r ) m −→ E −→ , where G is ﬁbrewise semistable of slope . (iv) Assume that all Harder–Narasimhan slopes of E are > r . Then ´etale locally on S , for some m ≥ there is a short exact sequence −→ G −→ O X S ( r ) m −→ E −→ , where G is ﬁbrewise semistable of slope r . Proof.

We can suppose S is aﬃnoid. We start with (i). Let π r : X S,r = Y S /ϕ r Z → X S bethe ﬁnite ´etale cover X S,E r = X S,E ⊗ E E r → X S,E = X S , where E r is the unramiﬁed extensionof degree r of E . We apply Proposition II.3.1 to ( π ∗ r E )( − S , a short exactsequence 0 → O m (cid:48) X S,Er → F (cid:48) → π ∗ r E → F (cid:48) is ﬁberwise semistable of slope 1. Thus, applying π r ∗ , we get a short exact sequence0 → O m (cid:48) rX S → π r ∗ F (cid:48) → π r ∗ π ∗ r E → . Here π r ∗ F (cid:48) is ﬁberwise semistable of slope r . As E is a direct summand of π r ∗ π ∗ r E = E ⊗ E E r , weget via pullback a similar exact sequence. Arguing similarly with ( π ∗ r E )( − (cid:3) Proposition

II.3.4 . Let S ∈ Perf F q , and let E be a vector bundle on X S . (i) If at all geometric points of S , all slopes of E are negative, then H ( X S , E ) = 0 . (ii) If at all geometric points of S , all slopes of E are nonnegative, then there is a pro-´etale cover (cid:101) S → S such that H ( X (cid:101) S , E| X (cid:101) S ) = 0 . (iii) If at all geometric points of S , all slopes of E are positive, then there is an ´etale cover S (cid:48) → S such that for any aﬃnoid perfectoid T over S (cid:48) , one has H ( X T , E| X T ) = 0 . Proof.

Part (i) can be checked on geometric points, where it follows from Theorem II.2.14and Proposition II.2.5 (i). For part (ii), we use Proposition II.3.1 to produce locally on S an exactsequence 0 → O X S ( − d → E (cid:48) → E → E (cid:48) is everywhere semistable of degree 0. By Theorem II.2.19 we can ﬁnd a pro-´etale coverof S over which E (cid:48) ∼ = O rX S . By the vanishing of H , this induces a surjection from H ( X S , O X S ) r onto H ( X, E ). Since H ( X S , O X S ) = H ( S, E ) by Proposition II.2.5 (ii), this vanishes pro-´etalelocally on S .For part (iii), we use Corollary II.3.3 (iii) to produce an ´etale cover of S over which there is anexact sequence 0 → G → O X S ( r ) m → E → . For any aﬃnoid T | S , this induces a surjection from H ( X T , O X T ( r ) m ) onto H ( X T , E| X T ), so weconclude by Proposition II.2.5 (iii). (cid:3) II.3.2. Families of Banach–Colmez spaces.

We can now prove the following strengtheningof Proposition II.2.16.

Proposition

II.3.5 . Let S be a perfectoid space over F q . Let [ E → E ] be a map of vectorbundles on X S such that at all geometric points of S , the bundle E has only negative Harder–Narasimhan slopes. (i) The Banach–Colmez space BC ([ E → E ]) : T (cid:55)→ H ( X T , [ E → E ] | X T ) is a locally spatial diamond, partially proper over S . (ii) The projectivized Banach–Colmez space ( BC ([ E → E ]) \ { } ) /E × is a locally spatial diamond, proper over S . (iii) Assume that all Harder–Narasimhan slopes of E at all geometric points are positive. Then BC ([ E → E ]) → S is cohomologically smooth. I.3. FURTHER RESULTS ON BANACH–COLMEZ SPACES 79

Proof.

All assertions are ´etale local (in fact v-local) on S . For parts (i) and (ii), let us ﬁrstsimplify the form of the complex [ E → E ]. By Theorem II.2.6, we can ﬁnd (for S aﬃnoid) some d > O X S ( − d ) m → E . Let E (cid:48) be the kernel of E ⊕ O X S ( − d ) m → E . Then we ﬁnd a quasi-isomorphism[ E (cid:48) → O X S ( − d ) m ] → [ E → E ] . Note also that E (cid:48) still satisﬁes the assumption on negative slopes. We get an exact sequence0 → BC ([ E (cid:48) → O X S ( − d ) m ]) → BC ( E (cid:48) [1]) → BC ( O X S ( − d ) m [1]) . As BC ( O X S ( − d ) m [1]) is separated by Proposition II.2.5 (i), we see that parts (i) and (ii) reduce tothe case of BC ( E (cid:48) [1]). Now applying Corollary II.3.3 (iv) to the dual of E (cid:48) , we get (´etale locally on S ) an exact sequence 0 → BC ( E (cid:48) [1]) → BC ( O X S ( − r ) m [1]) → BC ( G [1])where G is semistable of slope r everywhere. In particular, BC ( G [1]) is separated over S by pro-´etale descent and Proposition II.2.5 (i). Thus, BC ( E (cid:48) [1]) ⊂ BC ( O X S ( − r ) m [1]) is a closed subfunctor,ﬁnishing the proof of part (i) by applying Proposition II.2.5 (i) again. Part (ii) is then reduced tothe similar assertion for BC ( O X S ( − r ) m [1]). Replacing E by its unramiﬁed extension of degree r ,this reduces to BC ( O X S ( − m [1]). Now, as in the proof of Proposition II.2.16, this follows fromLemma II.2.17, where one checks the required contracting property of multiplication by π by usingthe presentation BC ( O X S ( − A S (cid:93) ) ♦ /E for an untilt S (cid:93) of S over E .It remains to prove part (iii). Note that one has a short exact sequence0 → BC ( E ) → BC ([ E → E ]) → BC ( E [1]) → Sch17a , Proposition 23.13], we can thus handle BC ( E [1]) and BC ( E ) individually. For thecase of BC ( E ), we use Corollary II.3.3 (iv) to get, pro-´etale locally on S , an exact sequence0 → O X S ( r ) m (cid:48) → O X S ( r ) m → E → , inducing a similar sequence of Banach–Colmez spaces. Then the result follows from [ Sch17a ,Proposition 23.13]. For the case of BC ( E [1]), choose a surjection O X S ( − d ) m → E ∨ for some d > → E → O X S ( d ) m → F → F are positive everywhere. This gives an exactsequence 0 → BC ( O X S ( d ) m ) → BC ( F ) → BC ( E [1]) → , so the result follows from [ Sch17a , Proposition 23.13] and the case of positive slopes alreadyestablished. (cid:3)

II.3.3. Punctured absolute Banach–Colmez spaces.

Finally, we analyze punctured ab-solute Banach–Colmez spaces. Recall that, in the situation of Proposition II.2.5 (iv), one has BC ( O ( d )) ∼ = Spd( k [[ x /p ∞ , . . . , x /p ∞ d ]]) , so the v-sheaf BC ( O ( d )) fails to be a perfectoid space, or even a diamond, as it contains thenon-analytic point Spd k . However, passing to the punctured Banach–Colmez space BC ( O ( d )) \ { } ∼ = Spa( k [[ x /p ∞ , . . . , x /p ∞ d ]]) an identiﬁes with the analytic points, which form a perfectoid space; in fact, a qcqs perfectoid space.These objects ﬁrst showed up in [ Far18a ] in the case of positive slopes. It was remarked in[

Far18a ] that the punctured version BC ( O ( d )) \ { } is a diamond for all d ≥

1, that is moreoversimply connected when d >

2. This plays a key role in [

Far18a ] since after base changing fromSpd k to Spa C this is not simply connected anymore. In the above example, BC ( O ( d )) \ { } = Spa( k [[ x /p ∞ , . . . , x /p ∞ d ]]) \ V ( x , · · · , x d )is a qcqs perfectoid space that is simply connected when d >

1. After base changing to Spa( C )this is a punctured n -dimensional open ball over Spa( C ) that is not quasicompact anymore, andnot simply connected. Thus, some new interesting phenomena appear when we consider absoluteBanach–Colmez spaces.Let us ﬁrst continue the discussion with the case of O ( d ) for d ≥

1. In that case, there is arelation to Cartier divisors. Recall that any closed Cartier divisor D ⊂ X S is given by a line bundle I on X S together with an injection I (cid:44) → O X S with closed image. We will only consider the case ofrelative Cartier divisors, so that this map stays injective after base change to any geometric point.Now Theorem II.2.19 implies that after replacing S by an open and closed cover, I is of degree − d for some integer d ≥

0, and that there is an E × -torsor of isomorphisms I ∼ = O X S ( − d ). This showsthat the v-sheaf Div sending any S to the closed relative Cartier divisors is given byDiv = (cid:71) d ≥ Div d , Div d ∼ = ( BC ( O ( d )) \ { } ) /E × . Note that we are implicitly using a diﬀerent deﬁnition of Div here, but Corollary II.2.4 shows thatthey agree.In particular, the moduli space Div d of degree d Cartier divisors is given by the projectivizedBanach–Colmez space for O ( d ). On the other hand, in terms of divisors we can see the followingproposition. Recall that one can take sums of Cartier divisors (by tensoring their ideal sheaves). Proposition

II.3.6 . For any d ≥ , the sum map (Div ) d → Div d : ( D , . . . , D d ) (cid:55)→ D + . . . + D d is a quasi-pro-´etale cover, identifying Div d = (Div ) d / Σ d , where Σ d is the symmetric group. In particular, Div d is a diamond. Proof.

By Proposition II.2.16 (ii), all occuring spaces are proper over ∗ . In particular, the summap is proper. To check surjectivity as v-sheaves, we can then check on geometric points, where it I.3. FURTHER RESULTS ON BANACH–COLMEZ SPACES 81 follows from Proposition II.2.9 (in whose proof we checked that any element of P d is a product ofelements of P ). In fact, we even get bijectivity up to the Σ d -action, and thus the isomorphismDiv d = (Div ) d / Σ d as v-sheaves. But the projection (Div ) d → (Div ) d / Σ d is quasi-pro-´etale by [ Sch17a , Lemma7.19, Deﬁnition 10.1 (i)]. As Div = Spd E/ϕ Z is a diamond, it follows that Div d is a diamond by[ Sch17a , Proposition 11.4, Proposition 11.6]. (cid:3)

Now we can analyze the case of general absolute Banach–Colmez spaces.

Proposition

II.3.7 . Let D be an isocrystal with only negative slopes (resp. with only positiveslopes), and work on Perf k . (i) The punctured Banach–Colmez space BC ( D ) \ { } (resp. BC ( D [1]) \ { } ) is a spatial diamond. (ii) The quotient (cid:0) BC ( D ) \ { } (cid:1) /E × −→ ∗ (cid:0) resp . (cid:0) BC ( D [1]) \ { } (cid:1) /E × −→ ∗ (cid:1) is proper, representable in spatial diamonds, and cohomologically smooth. Proof.

Part (ii) follows from Proposition II.3.5 and (for the cohomological smoothness aftertaking the quotient by E × ) [ Sch17a , Proposition 24.2].For part (i), we are going to apply Lemma II.3.8, so we ﬁrst want to see that BC ( D ) \ { } is aspatial v-sheaf. By the Dieudonn´e–Manin classiﬁcation, we can ﬁnd a basis for D so that ϕ is E -rational and U := ϕ N is a diagonal matrix with entries powers of π for some N >

0; this essentiallymeans that V is decent in the sense of [ RZ96 , Deﬁnition 1.8]. Then BC ( D ) (resp. BC ( D [1])) isalready deﬁned on Perf F q , and the action of U agrees with the action of Frob N . Moreover, theaction of U − (resp. U ) on | BC ( D ) × F q Spa F q (( t /p ∞ )) | (resp. | BC ( D [1]) × F q Spa F q (( t /p ∞ )) | ) stillsatisﬁes the hypotheses of Lemma II.2.17. This implies that( BC ( D ) \ { } ) /ϕ N × F q Spa F q (( t /p ∞ ))is a spatial diamond, which can be translated into( BC ( D ) \ { } ) × F q Spa F q (( t /p ∞ )) /ϕ N being a spatial diamond, as the absolute Frobenius acts trivially on the topological space. ButSpa F q (( t /p ∞ )) /ϕ N → ∗ is qcqs, even proper, and cohomologically smooth. We can thus applypoint (i) of Lemma II.3.8 to conclude that BC ( D ) \ { } (resp. BC ( D [1]) \ { } , for which the sameargument applies) is spatial.It remains to see that it is a diamond. One easily reduces to the case that D is simple, andallowing ourselves to replace E by a ﬁnite unramiﬁed extension, to D of rank 1. The case ofpositive Banach–Colmez spaces now follows from Proposition II.3.6, as it is an E × -torsor over adiamond (so [ Sch17a , Proposition 11.7] applies). It remains to prove that this is a diamond inthe case of a negative absolute Banach–Colmez spaces, i.e. for D = ( E, π n ϕ ) with n >

0. Then D := BC ( D [1]) \ { } classiﬁes extensions0 −→ O X S ( − n ) −→ E −→ O X S −→ that are geometrically ﬁberwise non split on S (remark that those extensions are rigid). We nowapply point (ii) of Lemma II.3.8 using the Harder–Narasimhan stratiﬁcation of D deﬁned by E . Wecan pass to the subsheaf of D where E is, at each geometric point, isomorphic to a given rank 2bundle, necessarily of the form O X S ( − n + i ) ⊕ O X S ( − i ) for some 0 < i ≤ n or to O X S ( − n ).On such a stratum D α ⊂ D there is a global Harder–Narasimhan ﬁltration by Theorem II.2.19,and trivializing the graded piece of lowest slope deﬁnes a pro-´etale morphism (cid:101) D α → D α . For S → (cid:101) D α there is a morphism from O X S ( − n ) to this quotient of E by composing with the inclusion O X S ( − n ) (cid:44) → E . Since the extension is non-split geometrically ﬁberwise on S , this morphism isnon-zero geometrically ﬁberwise. This deﬁnes a morphism from (cid:101) D α → X to a punctured positiveabsolute Banach–Colmez space X , which is a diamond. Thus (cid:101) D α ⊂ (cid:101) D α × X where the latter is a diamond as (cid:101) D α → ∗ is representable in diamonds, so [ Sch17a , Proposition11.10] shows that (cid:101) D α is a diamond. (cid:3) Lemma

II.3.8 . Let F be a small v-sheaf. (i) Suppose there exists a surjective qcqs cohomologically smooth morphism D → F where D is aspatial diamond. Then F is a spatial v-sheaf. (ii) Suppose moreover there is a family of locally closed generalizing subsets ( X α ) α , X α ⊂ |F | , suchthat for each α the associated subsheaf of F is a diamond. Then F is a spatial diamond. Proof.

For point (i), since D is qcqs and D → F qcqs surjective, F is qcqs. Since cohomo-logically smooth implies universally open we can apply [ Sch17a , Lemma 2.10] to conclude it isspatial. For point (ii) we apply [

Sch17a , Theorem 12.18]. Let G α ⊂ F be associated to X α . From[ Sch17a , Lemma 7.6] we deduce that G α (cid:44) → F is quasi-pro-´etale. This implies the result. (cid:3) Remark

II.3.9 . The proof of Proposition II.3.7 for negative absolute Banach–Colmez space goesthe same way as the proof of the fact that Gr ≤ µ is a spatial diamond, [ SW20 , Theorem 19.2.4].One ﬁrst proves this is a spatial v-sheaf and then one stratiﬁes it by locally closed generalizingsubsets that are diamonds.

Remark

II.3.10 . One has to be careful that although the absolute BC ( O ( d )) \ { } is a spatialdiamond, ( BC ( O ( d )) \{ } ) /π Z is not spatial anymore since not quasiseparated, [ Far18a , Remarque2.15]. In this context the good object is the morphism ( BC ( O ( d )) \{ } ) /π Z → ∗ that is representablein spatial diamonds. Remark

II.3.11 . In the equal characteristic case, E = F q (( π )), the structure of punctured posi-tive absolute Banach–Colmez spaces is much simpler since they are perfectoid spaces. Neverthelessthe structure of the punctured negative one is not, they are only spatial diamonds. We will seebelow that BC ( O ( − k (( t )) by the action of a proﬁnite group, and a point Spd k . However, the degeneration to thepoint happens at the boundary of the open unit disc as | t | →

1, not as | t | → k [[ t ]]. Thus BC ( O ( − This example was critical in convincing us to not try to develop a version of the theory of diamonds that wouldallow non-analytic test objects like Spa k [[ t /p ∞ ]] and would thus make BC ( O (1)) itself representable: After all, in the I.3. FURTHER RESULTS ON BANACH–COLMEZ SPACES 83

Example

II.3.12 . The absolute BC ( O ( − \ { } classiﬁes extensions0 −→ O X S ( − −→ E −→ O X S → S . Any such extension is, at each geometric point, isomorphic to O X S ( − ). Parametrizing isomorphisms E ∼ = O X S ( − ) deﬁnes a D × -torsor, where D is the quater-nion algebra over E ; here we use Theorem II.2.19. Remark that if 0 → O X S ( − → O X S ( − ) →L → L ∼ = O X S ,and thus the E × -torsor of isomorphisms between O X S and L is trivial. From this we deduce that BC ( O ( − \ { } (cid:39) ( BC ( O ( )) \ { } ) / SL ( D )with BC ( O ( )) \ { } (cid:39) Spa( k (( x /p ∞ )))the punctured universal cover of a 1-dimensional formal π -divisible O E -module of height 2.Let us compare this with our previous description of BC ( O ( − C ),ﬁxing an untilt C (cid:93) over E and t ∈ H ( X C , O X C (1)) \ { } : the exact sequence0 → O X C ( − → O X C → O C (cid:93) → BC ( O ( − × k Spa( C ) ∼ = ( A C (cid:93) ) ♦ /E. We thus have BC ( O ( − \ { } × k Spa( C ) ∼ = (Ω C (cid:93) ) ♦ /E where Ω = A E \ E is Drinfeld’s upper half plane over E .We deduce an isomorphism (cid:0) ( BC ( O ( )) \ { } ) × k Spa( C ) (cid:1) / SL ( D ) ∼ = (Ω C (cid:93) ) ♦ /E. This isomorphism is in fact deduced from the isomorphism between Lubin–Tate and Drinfeld tow-ers. In fact, [

SW13 ], the Lubin–Tate tower in inﬁnite level, LT ∞ over Spa( C (cid:93) ), is the moduli ofmodiﬁcations O X S (cid:44) → O X S ( ) at the point of the curve deﬁned by the untilt C (cid:93) . From this onededuces a D × -equivariant isomorphismLT (cid:91) ∞ (cid:46) (cid:18) E E × (cid:19) ∼ = ( BC ( O ( )) \ { } ) × k Spa( C ) . Dividing this isomorphism by SL ( D ) one obtains the preceding isomorphism. Example

II.3.13 . The absolute BC ( O ( − \ { } classiﬁes extensions0 −→ O X S ( − −→ E −→ O X S (1) −→ S . There is only one Harder–Narasimhan stratum and geometricallyﬁberwise on S , E is a trivial vector bundle. The moduli of surjections O X S (cid:16) O X S (1) is the opensubset U ⊂ ( BC ( O (1)) \ { } ) × k ( BC ( O (1)) \ { } ) context of absolute Banach–Colmez spaces, BC ( O (1)) = Spd k [[ t /p ∞ ]] and BC ( O ( − equal to U = ( BC ( O (1)) \ { } ) \ ( E × × . ∆where ∆ is the diagonal of ( BC ( O (1)) \{ } ) , that is to say couples ( x, y ) of sections of H ( X S , O X S (1))that are ﬁberwise /S non-zero and linearly independent over E . Here ( E × × . ∆ is a locally proﬁniteunion of copies of ∆.Again, by consideration of determinants, ker( O X S (cid:16) O X S (1)) is canonically identiﬁed with O X S ( − BC ( O ( − \ { } = U/ SL ( E ) . HAPTER III

Bun G Throughout this chapter, we ﬁx a reductive group G over the nonarchimedean local ﬁeld E .As it will be important to study Bun G over a geometric base point, we ﬁx from now on a completealgebraically closed ﬁeld k over F q and work with perfectoid spaces S over Spd k ; write Perf k forthe category. Definition

III.0.1 . Let

Bun G be the prestack taking a perfectoid space S ∈ Perf k to the groupoidof G -bundles on X S . The main results of this chapter are summarized in the following theorem.

Theorem

III.0.2 (Proposition III.1.3; Theorem III.2.2; Theorem III.2.3 and Theorem III.2.7;Theorem III.4.5; Proposition III.5.3) . The prestack

Bun G satisﬁes the following properties. (i) The prestack

Bun G is a small v-stack. (ii) The points | Bun G | are naturally in bijection with Kottwitz’ set B ( G ) of G -isocrystals. (iii) The map ν : | Bun G | → B ( G ) → ( X ∗ ( T ) + Q ) Γ is semicontinuous, and κ : | Bun G | → B ( G ) → π ( G E ) Γ is locally constant. Equivalently, the map | Bun G | → B ( G ) is continuous when B ( G ) is equippedwith the order topology. (iv) The semistable locus

Bun ss G ⊂ Bun G is open, and given by Bun ss G ∼ = (cid:71) b ∈ B ( G ) basic [ ∗ /G b ( E )] . (v) For any b ∈ B ( G ) , the corresponding subfunctor i b : Bun bG = Bun G × | Bun G | { b } ⊂ Bun G is locally closed, and isomorphic to [ ∗ / (cid:101) G b ] , where (cid:101) G b is a v-sheaf of groups such that (cid:101) G b → ∗ isrepresentable in locally spatial diamonds with π (cid:101) G b = G b ( E ) . The connected component (cid:101) G ◦ b ⊂ (cid:101) G b of the identity is cohomologically smooth of dimension (cid:104) ρ, ν b (cid:105) . The hardest part of this theorem is that κ is locally constant. We give two proofs of thisfact. If the derived group of G is simply connected, one can reduce to tori, which are not hardto handle. In general, one approach is to use z-extensions (cid:101) G → G to reduce to the case of simplyconnected derived group. For this, one needs that Bun (cid:101) G → Bun G is a surjective map of v-stacks;

856 III. Bun G we prove this using Beauville–Laszlo uniformization. Alternatively, at least for p -adic E , one canuse the abelianized Kottwitz set of Borovoi [ Bor98 ], which we prove to behave well relatively overa perfectoid space S . III.1. Generalities

There is a good notion of G -torsors in p -adic geometry: Definition/Proposition

III.1.1 ([

SW20 , Proposition 19.5.1] ) . Let X be a sousperfectoidspace over E . The following categories are naturally equivalent. (i) The category of adic spaces T → X with a G -action such that ´etale locally on X , there is a G -equivariant isomorphism T ∼ = G × X . (ii) The category of ´etale sheaves Q on X equipped with an action of G such that ´etale locally, Q ∼ = G . (iii) The category of exact ⊗ -functors Rep E G → Bun( X ) to the category of vector bundles on X .A G -bundle on X is an exact ⊗ -functor Rep E G → Bun( X ); by the preceding, it can equivalently be considered in a geometric or cohomological manner. In particular, G -torsors up to isomorphism are classiﬁed by H ( X, G ). By Proposition II.2.1,the following deﬁnes a v-stack.

Definition

III.1.2 . Let

Bun G be the v-stack taking a perfectoid space S ∈ Perf k to the groupoidof G -bundles on X S . Our goal in this chapter is to analyze this v-stack. Before going on, let us quickly observe thatit is small, i.e there are perfectoid spaces S , R with a v-surjection S → Bun G and a v-surjection R → S × Bun G S . Proposition

III.1.3 . The v-stack

Bun G is small. Proof.

It is enough to prove that if S i = Spa( R i , R + i ), i ∈ I , is an ω -coﬁltered inverse systemof aﬃnoid perfectoid spaces with inverse limit S = Spa( R, R + ), thenBun G ( S ) = lim −→ Bun G ( S i ) . Indeed, then any section of Bun G over an aﬃnoid perfectoid space S = Spa( R, R + ) factors over S (cid:48) =Spa( R (cid:48) , R (cid:48) + ) for some topologically countably generated perfectoid algebra R (cid:48) . But there is only aset worth of such R (cid:48) up to isomorphism, and then taking the disjoint union T = (cid:70) S (cid:48) ,α ∈ Bun G ( S (cid:48) ) S (cid:48) gives a perfectoid space that surjects onto Bun G . Moreover, the equivalence relation T × Bun G T satisﬁes the same limit property, and hence also admits a similar surjection. The reference applies in the case of Z p , but it extends verbatim to O E . II.2. THE TOPOLOGICAL SPACE | Bun G | To see the claim, note ﬁrst that R = lim −→ R i as any Cauchy sequence already lies in some R i .The same applies to B R,I for any interval I , and hence one sees thatBun( X S ) = lim −→ Bun( X S i ) . Now the deﬁnition of G -torsors gives the claim. (cid:3) This proof uses virtually no knowledge about Bun G and shows that any reasonable v-stack issmall. III.2. The topological space | Bun G | III.2.1. Points.

As a ﬁrst step, we recall the classiﬁcation of G -bundles on the Fargues–Fontaine curve over geometric points. This is based on the following deﬁnition of Kottwitz, [ Kot85 ]. Definition

III.2.1 . A G -isocrystal is an exact ⊗ -functor Rep E G → Isoc E . The set of isomorphism classes of G -isocrystals is denoted B ( G ) . By Steinberg’s theorem, the underlying ﬁbre functor to ˘ E -vector spaces is isomorphic to thestandard ﬁbre functor; this shows one can identify B ( G ) with the quotient of G ( ˘ E ) under σ -conjugation.Composing with the exact ⊗ -functorIsoc E → Bun( X S ) : D (cid:55)→ E ( D )any G -isocrystal deﬁnes a G -bundle on X S , for any S ∈ Perf k . Theorem

III.2.2 ([

Far18b ], [

Ans19 ]) . For any complete algebraically closed nonarchimedeanﬁeld C over k , the construction above deﬁnes a bijection B ( G ) → Bun G ( C ) / ∼ = . Proof.

For the convenience of the reader, and as some of the constructions will resurface later,we give a sketch of the proof in [

Ans19 ]. Any G -bundle on X C has its Harder–Narasimhan ﬁltration,and the formation of the Harder–Narasimhan ﬁltration is compatible with tensor products. Thisimplies that any exact ⊗ -functor Rep E G → Bun( X C ) lifts canonically to an exact ⊗ -functorRep E G → FilBun( X C ) to Q -ﬁltered vector bundles. To check exactness, note that if E is p -adic,the category Rep E G is semisimple and thus exactness reduces to additivity, which is clear. If E is of equal characteristic, one needs to argue more carefully, and we refer to the proof of [ Ans19 ,Theorem 3.11].We can now project Rep E G → FilBun( X C ) to the category GrBun( X C ) of Q -graded vectorbundles, and note that the essential image of this functor is landing in the category of bundles (cid:76) λ E λ such that each E λ is semistable of slope λ . This category is in fact equivalent to Isoc E byTheorem II.2.14 and Proposition II.2.5 (ii). Thus, it suﬃces to see that the ﬁltration on the exact ⊗ -functor Rep E G → FilBun( X C ) can be split.Looking at splittings locally on X C , they exist, and form a torsor under a unipotent groupscheme U over X C , where U is parametrizing automorphisms of the ﬁltered ﬁbre functor that are G trivial on the graded pieces. One can then ﬁlter U by vector bundles of positive slopes, and usingthe vanishing of their H , we get the desired splitting. (cid:3) In particular, using [

Sch17a , Proposition 12.7] it follows that the map B ( G ) −→ | Bun G | is a bijection. III.2.2. Harder–Narasimhan stratiﬁcation.

Now we need to recall Kottwitz’s descriptionof B ( G ). This relies on two invariants, the Newton point and the Kottwitz point . Let E be aseparable closure of E and ﬁx a maximal torus inside a Borel subgroup T ⊂ B ⊂ G E ; the set ofdominant cocharacters X ∗ ( T ) + is naturally independent of the choice of T and B , and acquires anaction of Γ = Gal( E | E ) via its identiﬁcation withHom( G mE , G E ) /G ( E ) − conjugacy . The Newton point is a map ν : B ( G ) −→ ( X ∗ ( T ) + Q ) Γ b (cid:55)−→ ν b . When G = GL n , then X ∗ ( T ) ∼ = Z n and the target is the set of nonincreasing sequences of rationalnumbers, which are the slopes of the Newton polygon of the corresponding isocrystal. The Kottwitzpoint is a map κ : B ( G ) −→ π ( G ) Γ b (cid:55)−→ κ ( b ) , where π ( G ) := π ( G Q p ) = X ∗ ( T ) / (coroot lattice) is the Borovoi fundamental group. For G = GL n ,this is naturally isomorphic to Z , and in this case κ ( b ) is the endpoint of the Newton polygon. Ingeneral, this compatibility is expressed by saying that the images of κ ( b ) and ν b in π ( G ) Γ Q agree (using an averaging operation for κ ( b )). However, this means that in general κ ( b ) is notdetermined by ν b , as π ( G E ) Γ may contain torsion.The deﬁnition of κ is done in steps. First, one deﬁnes it for tori, where it is actually a bijection.Then one deﬁnes it for G whose derived group is simply connected; in that case, it is simply done viapassage to the torus G/G der which does not change π . In general, one uses a z -extension (cid:101) G → G such that (cid:101) G has simply connected derived group, observing that B ( (cid:101) G ) → B ( G ) is surjective.Borovoi, [ Bor98 ], gave a more canonical construction of κ as an abelianization map that doesnot use the choice of a z -extension, at least in the case of p -adic E . We will recall the constructionin Section III.2.4.2.Finally, recall that ( ν, κ ) : B ( G ) → ( X ∗ ( T ) + Q ) Γ × π ( G ) Γ is injective.Let v (cid:55)→ v ∗ = w ( − v ) be the involution of the positive Weyl chamber X ∗ ( T ) + Q where w is thelongest element of the Weyl group. The Harder–Narasimhan polygon of E b is ν ∗ b . Its ﬁrst Chernclass is − κ ( b ). II.2. THE TOPOLOGICAL SPACE | Bun G | We need to understand how ν and κ vary on B ( G ). The following result follows from Theo-rem II.2.19 and [ RR96 , Lemma 2.2].

Theorem

III.2.3 ([

SW20 , Corollary 22.5.1]) . The map ν ∗ : | Bun G | ∼ = B ( G ) → ( X ∗ ( T ) + Q ) Γ is upper semicontinuous. We will later prove in Theorem III.2.7 that κ is locally constant on Bun G . III.2.3. Geometrically ﬁberwise trivial G -bundles. Let[ ∗ /G ( E )]be the classifying stack of pro-´etale G ( E )-torsors, andBun G ⊂ Bun G be the substack of geometrically ﬁberwise trivial G -bundles. One has H ( X S , O X S ) = E ( S ) andthus G ( E ) acts on the trivial G -bundle. From this we deduce a morphism[ ∗ /G ( E )] −→ Bun G . We are going to prove that this is an isomorphism. Let us note that, although this is an isomor-phism at the level of geometric points, we can not apply [

SW20 , Lemma 12.5] since it is not clearthat it is qcqs.Theorems III.2.3 and III.2.7 (to follow) taken together imply that the locusBun G ⊂ Bun G is an open substack. One of our proofs of Theorem III.2.7 will however require this statement asan input. Of course, when π ( G ) Γ is torsion free, that is to say H ( E, G ) = { } , Theorem III.2.3is enough to obtain the openness. Theorem

III.2.4 . The substack

Bun G ⊂ Bun G is open, and the map [ ∗ /G ( E )] ∼ −→ Bun G deﬁned above is an isomorphism. Proof.

Let S ∈ Perf k be qcqs with a map to Bun G . We need to see that the subset of | S | overwhich this map is trivial at any geometric point is open; and that if this is all of S , then the datais equivalent to a pro-´etale G ( E )-torsor.Let us check the openness assertion. If T → S is surjective with T qcqs then | T | → | S | isa quotient map. We can thus assume that S is strictly totally disconnected. The locus wherethe Newton point is identically zero is an open subset of S by Theorem III.2.3, so passing tothis open subset, we can assume that the Newton point is zero. In that case, for any algebraicrepresentation ρ : G → GL n , the corresponding rank n vector bundle on X S is trivial. Now,geometrically ﬁberwise on S trivial vector bundles on X S are equivalent to E -local systems on S byCorollary II.2.20. On the other hand, as S is strictly totally disconnected, all E -local systems on S G are trivial (Lemma III.2.6 for G = GL n ), and their category is equivalent to the category of ﬁnitefree modules over C ( | S | , E ). Thus, the preceding discussion deﬁnes a ﬁbre functor on Rep E ( G )with values in C ( | S | , E ) = C ( π ( S ) , E ), i.e. a G -torsor over Spec( C ( π ( S ) , E )). Note that for all s ∈ π ( S ), the local ring lim −→ U (cid:51) s C ( | U | , E ) is henselian as the local ring of the analytic adic space π ( S ) × Spa( E ) at s . This implies that if this G -torsor is trivial at some point of S , then it is trivialin a neighborhood. This concludes the openness assertion.Moreover, the preceding argument shows that the map ∗ → Bun G is a pro-´etale cover. As ∗ × Bun G ∗ = G ( E ), as automorphisms of the trivial G -torsor are given by G ( E ), we thus get thedesired isomorphism. (cid:3) Remark

III.2.5 ([

Sch17a , Lemma 10.13]) . If S is a perfectoid space, and T → S is a pro-´etale G ( E )-torsor then T is representable by a perfectoid space. In fact, T = lim ←− K K \ T where K goesthrough the set of compact open subgroups of G ( E ). By descent of ´etale separated morphisms([ Sch17a , Proposition 9.7]), for each such K , K \ T is represented by a separated ´etale perfectoidspace over S . The transition morphisms in the preceding limit are ﬁnite ´etale. Lemma

III.2.6 . Let S be a strictly totally disconnected perfectoid space. Then any pro-´etale G ( E ) -torsor on S is trivial. Proof.

Let T → S be such a torsor. Fix a compact open subgroup K ⊂ G ( E ). Since K \ T → S is an ´etale cover of perfectoid spaces it has a section and we can assume T → S is in facta K -torsor. Now, T = lim ←− U U \ T where U goes through the set of distinguished open subgroups of K . Each U \ T → S is an ´etale K/U -torsor and is trivial. One concludes using that if U (cid:48) ⊂ U then K/U (cid:48) ( S ) → K/U ( S ) is surjective. (cid:3) III.2.4. Local constancy of the Kottwitz invariant.

A central result is the following.

Theorem

III.2.7 . The map κ : | Bun G | ∼ = B ( G ) → π ( G ) Γ is locally constant. Let us note the following corollary. We give a new proof (and slight strengthening) of a resultof Rapoport–Richartz (when p | | π ( G ) | the original proof used p -adic nearby cycles and relied ona ﬁnite type hypothesis). Corollary

III.2.8 ([

RR96 , Corollary 3.11]) . Let S be an F q -scheme and E an G -isocrystalon S . The map | S | → π ( G ) Γ that sends a geometric point ¯ s → S to κ ( E ¯ s ) is locally constant. Proof.

We can suppose S = Spec( R ) is aﬃne and deﬁned over k . We get a small v-sheafSpd( R, R ), and E deﬁnes a morphism Spd( R, R ) → Bun G . The induced map κ : | Spd(

R, R ) | →| Bun G | → π ( G ) Γ is locally constant by Theorem III.2.7. As open and closed subsets of Spd( R, R )are in bijection with open and closed subschemes of Spec( R ) (by [ SW20 , Proposition 18.3.1] appliedto morphisms to ∗ (cid:116) ∗ ), we can thus assume that κ : | Spd(

R, R ) | → | Bun G | → π ( G ) Γ is constant.But now for any geometric point ¯ s → S , the element κ ( E ¯ s ) ∈ π ( G ) Γ agrees with the image of ∗ = | Spd(¯ s, ¯ s ) | → | Spd(

III.2.9 . There is a natural map | Spd(

R, R ) | → | Spa(

R, R ) | , the latter of which admitstwo natural maps to | Spec( R ) | (given by the support of the valuation, or the prime ideal of allelements of norm < κ deﬁned on | Spd(

R, R ) | and | Spec( R ) | do not makethis diagram commute. Still, there is also the map | Spec( R ) | → | Spd(

R, R ) | , used in the proof,and this is continuous, and commutes with the κ maps.We give two diﬀerent proofs of Theorem III.2.7.III.2.4.1. First proof.

For the ﬁrst proof of Theorem III.2.7, we also need the following lemmathat we will prove in the next section.

Lemma

III.2.10 . Let (cid:101) G → G be a central extension with kernel a torus. Then Bun (cid:101) G → Bun G is a surjective map of v-stacks. In fact, up to correctly interpreting all the relevant structure, if Z ⊂ (cid:101) G is the kernel, then Bun Z is a Picard stack (as for commutative Z one can tensor Z -bundles) which acts on Bun (cid:101) G , and Bun G is the quotient stack. It is in fact clear that it is a quasitorsor, and the lemma ensures surjectivity. First Proof of Theorem III.2.7.

Picking a z -extension, we can by Lemma III.2.10 reduceto the case that G has simply connected derived group. Then we may replace G by G/G der , and soreduce to the case that G is a torus. By a further application of Lemma III.2.10, we can reduce tothe case that G is an induced torus. In that case π ( G ) Γ is torsion-free, and so the Kottwitz mapis determined by the Newton map, so the result follows from Theorem III.2.3, noting that in thecase of tori there are no nontrivial order relations so semicontinuity means local constancy. (cid:3) It remains to prove Lemma III.2.10. This will be done in the next section, using Beauville–Laszlo uniformization.III.2.4.2.

Second proof.

For this proof, we assume that E is p -adic (otherwise certain non-´etaleﬁnite ﬂat group schemes may appear). We deﬁne B ab ( G ) = H ( W E , [ G sc ( ˘ E ) → G ( ˘ E )]) , the abelianized Kottwitz set (cohomology with coeﬃcient in a crossed module, see [ Bor98 ] and[

Lab99 , Appendix B]). There is an abelianization map B ( G ) −→ B ab ( G )deduced from the morphism [1 → G ] → [ G sc → G ]. If T is a maximal torus in G with reciprocalimage T sc in G sc then [ T sc → T ] −→ [ G sc → G ]is a homotopy equivalence. If Z , resp. Z sc , is the center of G , resp. G sc , there is a homotopyequivalence [ Z sc → Z ] −→ [ G sc → G ] . G Lemma

III.2.11 . There is an identiﬁcation B ab ( G ) = π ( G ) Γ through which Kottwitz map κ is identiﬁed with the abelianization map B ( G ) → B ab ( G ) . Proof.

Choose a maximal torus T in G . One has B ab ( G ) = H ( W E , [ T sc ( ˘ E ) → T ( ˘ E )])= coker (cid:0) B ( T sc ) → B ( T ) (cid:1) since H ( W E , T sc ( ˘ E )) = 0 (use [ Ser94 , Chapter II.3.3 example (c)] and [

Ser94 , Chapter III.2.3Theorem 1’]). The result is deduced using Kottwitz description of B ( T sc ) and B ( T ) = X ∗ ( T ) Γ . (cid:3) For S ∈ Perf k there is a morphism of sites τ : ( X S ) ´et −→ S ´et deduced from the identiﬁcations ( X S ) ´et = ( X ♦ S ) ´et = (Div S ) ´et and the projection Div S → S . Equivalently, τ ∗ takes any ´etale T → S to X T → X S , which is again´etale.We now interpret some ´etale cohomology groups of the curve as Galois cohomology groups, asin [ Far18b ] where this type of computation was done for the schematical curve attached to analgebraically closed perfectoid ﬁeld.

Proposition

III.2.12 . Let S ∈ Perf k . (i) Let F be a locally constant sheaf of ﬁnite abelian groups on Spa( E ) ´et . One has Rτ ∗ F | X S = R Γ ´et (Spa( E ) , F ) as a constant complex on S ´et . (ii) If D is a diagonalizable algebraic group over E , the pro-´etale sheaf associated to T /S (cid:55)−→ H ( X T , D ) is the constant sheaf with value H ( W E , D ( ˘ E )) . Proof.

Let us note G = F | X S . There is a natural morphism R Γ ´et (Spa( E ) , F ) → Rτ ∗ G . Themorphism Div S → S is proper and applying [ Sch17a , Corollary 16.10 (ii)], we are reduced to provethat H • ´et (Spa( E ) , F ) ∼ −→ H • ´et ( X C,C + , G )when C is an algebraically closed ﬁeld. Since X C,C + is quasicompact quasiseparated H • ´et ( X C,C + (cid:98) ⊗ E (cid:98) E, G ) = lim −→ E (cid:48) | E ﬁnite H • ( X C,C + ⊗ E E (cid:48) , G ) , and, using Galois descent, it thus suﬃces to prove that the left member vanishes in positive degrees,and equals F | E in degree 0. II.2. THE TOPOLOGICAL SPACE | Bun G | Let K = F q (( T )) and X ( C,C + ) ,K the equal characteristic Fargues–Fontaine curve over Spa( K ).Identifying (cid:98) E (cid:91) with (cid:91) K sep , one has( X ( C,C + ) ,E (cid:98) ⊗ E (cid:98) E ) (cid:91) = X ( C,C + ) ,K (cid:98) ⊗ (cid:91) K sep . Using this we are reduced to prove that for any prime number n , for i > H i ´et ( X ( C,C + ) ,K (cid:98) ⊗ K (cid:91) K sep , Z /n Z ) = 0 . This is reduced, as above, to prove that any class in H i ´et ( X ( C,C + ) ,K , Z /n Z ) is killed by pullback toa ﬁnite separable extension of K .When n (cid:54) = p one has R Γ ´et ( X ( C,C + ) ,K , Z /n Z ) = R Γ( ϕ Z , R Γ ´et ( D ∗ C,C + , Z /n Z ))where T is the coordinate on the open punctured disk D ∗ C,C + = Spa( C, C + ) × Spa( K ) . One has H k ( D ∗ C,C + , Z /n Z )) = 0 for k >

1, and this is equal, via Kummer theory, to Z /n Z ( −

1) for k = 1. The Kummer covering of D ∗ C,C + induced by T (cid:55)→ T n kills any class in H ( D ∗ C,C + , Z /n Z ).Also H ( D ∗ C,C + , Z /n Z ) = Z /n Z and the class in H ( ϕ Z , Z /n Z ) = Z /n Z is killed by passing upalong an unramiﬁed extension of K of degree n .When n = p we use Artin-Schreier theory. Since C is an algebraically closed ﬁeld we have H i ( X ( C,C + ) ,K , O ) = 0 when i >

0. Since the adic space X ( C,C + ) ,K is noetherian we deduce that H i ´et ( X ( C,C + ) ,K , O ) = 0 for i >

0. Thus, H i ´et ( X ( C,C + ) ,K , Z /n Z ) is 0 for i > K F − Id −−−→ K )when i = 1, which is killed by pullback to an Artin-Schreier extension of K . This ﬁnishes the proofof point (1).For point (2). There is a natural morphism H ( W E , D ( ˘ E )) → H ( X S , D )(see just after the proof of this proposition). Suppose ﬁrst that D is a torus. Then point (2) is thecomputation of the coarse moduli space of Bun D as a pro-´etale stack. This itself is a consequenceof Theorem III.2.4 using a translation argument from 1 to any [ b ] ∈ B ( D ) (use the Picard stackstructure on Bun D ).For any D we use the exact sequence1 −→ D −→ D −→ π ( D ) −→ . For

T /S there is a diagram H ( E, π ( D )) B ( D ) H ( W E , D ( ˘ E )) H ( E, π ( D )) 0 H ( X T , π ( D )) H ( X T , D ) H ( X T , D ) H ( X T , π ( D ))since H ( W E , D ( ˘ E )) = 0 and H • ( W E , π ( D )( ˘ E )) = H • ( W E , π ( D )( E )). The result is then de-duced from part (1) and the torus case. (cid:3) G For S ∈ Perf k there is a natural morphism of groups B ab ( G ) −→ H ( X S , [ G sc → G ]) . This is deduced from the natural continuous morphism of sites( X S ) ´et −→ { discrete W E -sets } . Proposition

III.2.13 . For S ∈ Perf k , the pro-´etale sheaf on S associated with T /S (cid:55)−→ H ( X T , [ G sc → G ]) is the constant sheaf with value B ab ( G ) . Proof.

We use the homotopy equivalence [ Z sc → Z ] → [ G sc → G ]. There is a diagram H ( W E , Z sc ( ˘ E )) (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) H ( W E , Z ( ˘ E )) (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) H ( W E , [ G sc ( ˘ E ) → G ( ˘ E )]) (cid:47) (cid:47) (cid:15) (cid:15) H ( W E , Z sc ( ˘ E )) (cid:39) (cid:15) (cid:15) H ( X T , Z sc ) (cid:47) (cid:47) H ( X T , Z ) (cid:47) (cid:47) H ( X T , [ Z sc → Z ]) (cid:47) (cid:47) H ( X T , Z sc ) (cid:47) (cid:47) H ( W E , Z ( ˘ E )) (cid:16) (cid:39) (cid:47) (cid:47) (cid:127) (cid:95) (cid:15) (cid:15) H ( E, π ( Z )) (cid:17) (cid:39) (cid:15) (cid:15) (cid:47) (cid:47) H ( X T , Z ) (cid:16) (cid:47) (cid:47) H ( X T , π ( Z )) (cid:17) . Using Proposition III.2.12 and some diagram chasing we conclude. (cid:3)

Second Proof of Theorem III.2.7.

The theorem is now deduced from the preceding Propo-sition III.2.13 and the abelianization map H ( X S , G ) → H ( X S , [ G sc → G ]). (cid:3) Remark

III.2.14 . Let X alg C be the schematical curve associated to C | F q algebraically closed.The results of [ Far18b ] for the ´etale cohomology of torsion local systems, [

Far18b , Theorem 3.7]and the vanishing of the H ( X C , T ) for a torus T , [ Far18b , Theorem 2.7], can be stated in a moreuniform way; if D is a diagonalizable group over E then H i ( W E , D ( ˘ E )) ∼ −→ H i ´et ( X C , D ) for 0 ≤ i ≤ Weil cohomology of E is the natural cohomology theory that corresponds to ´etale cohomologyof the curve. For example Theorem III.2.2 can be restated as H ( W E , G ( ˘ E )) ∼ −→ H ( X C , G ) for areductive group G . III.2.5. The explicit description of | Bun G | . Theorem III.2.3 and Theorem III.2.7 implythat the map | Bun G | → B ( G )is continuous when the target is endowed with the topology induced by the order on ( X ∗ ( T ) + Q ) Γ and the discrete topology on π ( G ) Γ . Conjecture

III.2.15 . The map | Bun G | → B ( G ) is a homeomorphism. II.3. BEAUVILLE–LASZLO UNIFORMIZATION 95

In other words, whenever b, b (cid:48) ∈ B ( G ) such that b > b (cid:48) , there should be a specialization from b to b (cid:48) in | Bun G | .The conjecture is known for G = GL n by work of Hansen, [ Han17 ], based on [

BFH + ]. Theargument has been extended to some other classical groups in unpublished work of Hamann. Whileﬁnishing our manuscript, a proof for general G has been given by Viehmann [ Vie21 ].We will later prove some weak form of the conjecture in Corollary IV.1.23, determining theconnected components of Bun G using simple geometric considerations. III.3. Beauville–Laszlo uniformization

Recall from [

SW20 , Lecture XIX] the B +dR -aﬃne GrassmannianGr G of G over Spd E , sending an aﬃnoid perfectoid S = Spa( R, R + ) over Spd E to the G -torsorsover Spec( B +dR ( R (cid:93) )) with a trivialization over B dR ( R (cid:93) ); here R (cid:93) /E is the untilt of R given by S → Spd( E ). In [ SW20 ] this was considered over Spa( C (cid:91) ) for some C | E algebraically closed butwe want now to consider it in a more “absolute” way over Spd( E ).Since any G -torsor over Spec( B +dR ( R (cid:93) )) is trivial locally on Spa( R, R + ) ´et , this coincides withthe ´etale sheaf associated to the presheaf ( R, R + ) (cid:55)→ G ( B dR ( R (cid:93) )) /G ( B +dR ( R (cid:93) )). This has an interpretation as a Beilinson–Drinfeld type aﬃne Grassmannian. If E , E (cid:48) ∈ Bun G ( S ) and D ∈ Div ( S ), a modiﬁcation between E and E (cid:48) at D is an isomorphism E| X S \ D ∼ −−→ E (cid:48) | X S \ D that is meromorphic along D . The latter means that for any representation in Rep E ( G ), theassociated isomorphism between vector bundles, F | X S \ D ∼ −−→ F (cid:48) | X S \ D extends to a morphism F → F (cid:48) ( kD ) for k (cid:29) F (cid:48) (cid:44) → F (cid:48) ( kD ). Beauville–Laszlo gluing then identiﬁesGr G /ϕ Z −→ Div with the moduli of D ∈ Div ( S ), E ∈

Bun G ( X S ), and a modiﬁcation between the trivial G -bundleand E at D , cf. [ SW20 , Proposition 19.1.2]. This deﬁnes a morphism of v-stacksGr G −→ Bun G . Proposition

III.3.1 . The Beauville–Laszlo morphism Gr G −→ Bun G is a surjective map of v-stacks; in fact, of pro-´etale stacks. Proof.

Pick any S = Spa( R, R + ) ∈ Perf k aﬃnoid perfectoid with a map to Bun G , givenby some G -bundle E on X S . Fix an untilt S (cid:93) of S over Spa( E ). To prove surjectivity as pro-´etale stacks, we can assume that S is strictly totally disconnected. By [ Far18b , Th´eor`eme 7.1](in case G quasisplit) and [ Ans19 , Theorem 6.5] (for general G ), for any connected componentSpa( C, C + ) ⊂ S of S , the map Gr G ( C ) → Bun G ( C ) is surjective, so in particular, we can pick amodiﬁcation E (cid:48) C of E| X C,C + at Spa( C (cid:93) , C (cid:93), + ) (cid:44) → X C,C + such that E (cid:48) C is trivial. G Now, since S is strictly totally disconnected, we can trivialize E at the completion at S (cid:93) ; asGr G ( S ) → Gr G ( C ) is surjective (Lemma III.3.2), we can lift E (cid:48) C to a modiﬁcation E (cid:48) of E . NowTheorem III.2.4 implies that E (cid:48) is trivial in a neighborhood of the given point, and as S is strictlytotally disconnected, the corresponding G ( E )-torsor is trivial (Lemma III.2.6), so we can trivialize E (cid:48) in a neighborhood of the given point. This shows that locally on S , the bundle E is in the imageof Gr G → Bun G , as desired. (cid:3) Lemma

III.3.2 . For S = Spa( R, R + ) a strictly totally disconnected perfectoid space over Spa( E ) ,and s ∈ S , the map Gr G ( R ) → Gr G ( K ( s )) is surjective. Proof.

Set C = K ( s ). First note that R → C is surjective, as any connected component of S is an intersection of open and closed subsets, and thus Zariski closed.Since C is algebraically closed, Gr G ( C ) = G ( B dR ( C )) /G ( B +dR ( C )). For any ﬁnite degree exten-sion E (cid:48) | E in C , since S ⊗ E E (cid:48) → S is ´etale and S strictly totally disconnected, E (cid:48) can be embeddedin R . Fix such an E (cid:48) ⊂ R that splits G and a pair T ⊂ B inside G E (cid:48) . Let ξ ∈ W O E ( R (cid:91), + ) be agenerator of the kernel of θ . The Cartan decomposition G ( B dR ( C )) = (cid:97) µ ∈ X ∗ ( T ) + G ( B +dR ( C )) µ ( ξ ) G ( B +dR ( C ))shows then that we only need to prove that G ( B +dR ( R )) → G ( B +dR ( C )) is surjective.Let us ﬁrst remark that G ( R ) → G ( C ) is surjective. In fact, since S is totally disconnected itsuﬃces to check that G ( O S,s ) → G ( C ) is surjective. But this is a consequence of the smoothnessof G and the fact that O S,s is Henselian (with residue ﬁeld C ).Moreover G ( B +dR ( R )) = lim ←− n ≥ G ( B +dR ( R ) / Fil n ). Using the surjectivity of Lie( G ) ⊗ R → Lie( G ) ⊗ C the result is then deduced by an approximation argument. (cid:3) Remark

III.3.3 . In the “classical case” of the moduli of G -bundles over a proper smoothalgebraic curve over a ﬁeld k , G/k , Proposition III.3.1 is true only when G is semi-simple ([ DS95 ]).Typically this is false for GL n in general. The main reason why it is true in our situation is thatPic ( X C,C + ) is trivial, equivalently that X alg C \ { x } is the spectrum of a principal ideal domain inProposition II.2.9. Lemma

III.3.4 . If G is split then Gr G = lim −→ µ ∈ X ∗ ( T ) + Gr G, ≤ µ as a v-sheaf, where the index set is a partially ordered set according to the dominance order ( µ ≤ µ (cid:48) if µ (cid:48) − µ is a nonnegative integral sum of positive coroots). Proof.

Consider a morphism S → Gr G with S quasicompact quasiseparated. Fix an embed-ding G (cid:44) → GL n such that the image of T lies in the standard maximal torus of GL n . This inducesan embedding X ∗ ( T ) + (cid:44) → Z n . One checks easily that the image of | S | → | Gr GL n | lies in a ﬁniteunion of aﬃne Schubert cells. Since the ﬁbers of X ∗ ( T ) + → Z n / S n are ﬁnite we deduce that thereis a ﬁnite collection ( µ i ) i , µ i ∈ X ∗ ( T ) + , such that the image of | S | → | Gr G | lies in ∪ i | Gr G, ≤ µ i | . By[ SW20 , Proposition 19.2.3], S i := S × Gr G Gr G, ≤ µ i is closed in S and thus quasicompact. Since the II.3. BEAUVILLE–LASZLO UNIFORMIZATION 97 morphism (cid:96) i S i → S is surjective at the level of points with quasicompact source it is quasicompactand thus a v-cover. This allows us to conclude. (cid:3) Lemma

III.3.5 . Suppose (cid:101) G → G is a central extension with kernel a torus. Then Gr (cid:101) G → Gr G is a surjective map of v-sheaves. Proof.

Up to replacing E by a ﬁnite degree extension we can suppose G and (cid:101) G are split. Fix (cid:101) T → T inside (cid:101) G → G . According to Lemma III.3.4 it is enough to prove that for any (cid:101) µ ∈ X ∗ ( (cid:101) T ) + ,if µ ∈ X ∗ ( T ) + is its image in G , then Gr (cid:101) G, ≤ (cid:101) µ → Gr G, ≤ µ is surjective. This is clearly surjectiveat the level of points since, if D is the kernel of (cid:101) G → G , then H (Spec( B dR ( C )) , D ) = 0, andthus (cid:101) G ( B dR ( C )) → G ( B dR ( C )) is surjective. Since Gr (cid:101) G, ≤ (cid:101) µ is quasicompact over Spd( E ) andGr G, ≤ µ quasiseparated over Spd( E ) (both are proper according to [ SW20 , Proposition 19.2.3]),Gr (cid:101) G, ≤ (cid:101) µ → Gr G, ≤ µ is quasicompact and thus a v-cover by [ Sch17a , Lemma 12.11]. (cid:3)

Using Proposition III.3.1, we thus have now a proof of Lemma III.2.10.

Let us record a few facts we can deduce from the preceding results.

Proposition

III.3.6 . Suppose G is split. (i) There is a locally constant map | Gr G | → π ( G ) inducing a decomposition in open/closed sub-sheaves Gr G = (cid:97) α ∈ π ( G ) Gr αG characterized by Gr G,µ ⊂ Gr µ (cid:93) G . (ii) The composite | Gr G | Beauville–Laszlo −−−−−−−−−−→ |

Bun G | κ −−→ π ( G ) is the opposite of the preceding map. (iii) For each α ∈ π ( G ) , Gr αG = lim −→ µ ∈ X ∗ ( T ) + ,µ (cid:93) = α Gr αG, ≤ µ as a ﬁltered colimit of v-sheaves. Proof.

Point (1) is reduced to the case when G der is simply connected using Lemma III.3.5and a z -extension. Now, if G der is simply connected, for µ , µ ∈ X ∗ ( T ) + , µ ≤ µ implies µ (cid:93) = µ (cid:93) .The result is then deduced from the fact that Gr G, ≤ µ ⊂ Gr G is closed for any µ .Point (2) is can be similarly reduced ﬁrst to the case that G der is simply connected by passageto a z-extension; then to the case of a torus by taking the quotient by G der ; then to the case of aninduced torus by another z-extension; and then to G m by changing E . In that case, it follows fromProposition II.2.3.Point (3) is deduced from Lemma III.3.4 and the fact that for α ∈ π ( G ), { µ ∈ X ∗ ( T ) + | µ (cid:93) = α } is a ﬁltered ordered set. (cid:3) G When G is not split, choosing E (cid:48) | E Galois of ﬁnite degree splitting G , using the formulaGr G × Spd( E ) Spd( E (cid:48) ) = Gr G E (cid:48) , one deduces:(i) There is a decomposition Gr G = (cid:96) ¯ α ∈ Γ \ π ( G ) Gr ¯ αG and a formula Gr G = lim −→ ¯ µ ∈ Γ \ X ∗ ( T ) + Gr G, ≤ ¯ µ such that Gr G, ≤ ¯ µ ⊂ Gr ¯ µ (cid:93) G .(ii) The composite | Gr G | BL −−→ | Bun G | κ −→ π ( G ) Γ is induced by the opposite of Γ \ π ( G ) → π ( G ) Γ and the preceding decomposition.This description of the Kottwitz map, together with Proposition III.3.1, in fact gives anotherproof of Theorem III.2.7. III.4. The semistable locusIII.4.1. Pure inner twisting.

Recall the following. In the particular case of non abeliangroup cohomology this is called “torsion au moyen d’un cocycle” in [

Ser94 , I.5.3].

Proposition

III.4.1 . Let X be a topos, H a group in X and T an H -torsor. Let H T = Aut( T ) as a group in X . Then: (i) H T is the “pure inner twisting” of H by T , H T = H H ∧ T where H acts by conjugation on H .In particular [ H T ] ∈ H ( X, H ad ) is the image of [ T ] via H ( X, H ) → H ( X, H ad ) . (ii) The morphism of stacks in X , [ ∗ /H ] → [ ∗ /H T ] , that sends an H -torsor S to Isom(

S, T ) , is anequivalence. In the following we use the cohomological description of G -bundles on the curve as G -torsorson the ´etale site of the sous-perfectoid space X S (here G is seen as an E -adic group, for ( R, R + ) asous-perfectoid E -algebra its Spa( R, R + )-points being G ( R )). Proposition

III.4.2 . Let S ∈ Perf k , b ∈ B ( G ) basic and E b → X S the associated ´etale G -torsor.Then the ´etale sheaf of groups G b × Spa( E ) X S over X S is the pure inner twisting of G × Spa( E ) X S by E b . Proof.

One has E b = ( G ˘ E × Spa( ˘ E ) Y S ) / (( bσ ) × ϕ ) Z −→ X S where bσ acts on G ˘ E by translation on the right. The G × X S = ( G ˘ E × Spa( ˘ E ) Y S ) / ( Id × ϕ ) Z -torsorstructure is given by multiplication on the left on G ˘ E . The group G b × X S = ( G b × Y S ) / ( Id × ϕ ) Z acts on this torsor on the right via the morphism G b → G ˘ E , which gives a morphism G b × X S → Aut( E b ) . After pullback via the ´etale cover Y S → X S and evaluation on T = Spa( R, R + ) → Y S aﬃnoidsous-perfectoid, this is identiﬁed with the map G b ( R ) = { g ∈ G ( ˘ E ⊗ E R ) | g · ( bσ ⊗

1) = ( bσ ⊗ · g } −→ G ( R )deduced from the ˘ E -algebra structure of R . But as b is basic, the natural map G b × E ˘ E → G × E ˘ E II.4. THE SEMISTABLE LOCUS 99 is an isomorphism, cf. [

RZ96 , Corollary 1.14]. (cid:3)

Thus, extended pure inner forms, as deﬁned by Kottwitz, become pure inner forms, as deﬁnedby Vogan, when pulled back to the curve.

Corollary

III.4.3 . For b basic there is an isomorphism of v-stacks Bun G (cid:39) Bun G b that induces an isomorphism Bun bG (cid:39) Bun G b . Example

III.4.4 . Take G = GL n and ( D, ϕ ) an isoclinic isocrystal of height n . Let B =End( D, ϕ ) be the associated simple algebra over E . Since ( D, ϕ ) is isoclinic the action of B on E ( D, ϕ ) induces an isomorphism B ⊗ E O X S ∼ −→ End( E ( D, ϕ )) for any S ∈ Perf k . The stackBun B × is identiﬁed with the stack S (cid:55)→ { rank 1 locally free B ⊗ E O X S -modules } . There is then anisomorphism (Morita equivalence)Bun GL n ∼ −−→ Bun B × E (cid:55)−→

Hom O XS ( E , E ( D, ϕ )) . III.4.2. Description of the semi-stable locus.

Recall from [

Far18b ] that a G -bundle over X C,C + is semistable if and only if it corresponds to some basic element of B ( G ). Also recall thatfor any b ∈ B ( G ), the automorphism group of the corresponding G -isocrystal deﬁnes an algebraicgroup G b over G . If G is quasisplit, then G b is an inner form of a Levi subgroup of G , and it is aninner form of G precisely when b is basic. More generally G b is an inner form of a Levi subgroupof the quasisplit inner form of G . Theorem

III.4.5 . The semistable locus

Bun ss G ⊂ Bun G is open, and there is a canonical decomposition as open/closed substacks Bun ss G = (cid:97) b ∈ B ( G ) basic Bun bG . For b basic there is an isomorphism [ ∗ /G b ( E )] ∼ −→ Bun bG . Proof.

Theorem III.2.3 implies that Bun ss G is open. Recall that the basic elements of B ( G )map isomorphically to π ( G ) Γ via the Kottwitz map. Thus Theorem III.2.7 gives a disjoint decom-position Bun ss G = (cid:97) b ∈ B ( G ) basic Bun bG . The result is then a consequence of Proposition III.4.2 and Theorem III.2.4. (cid:3)

Example

III.4.6 . For a torus T , Bun T = Bun ssT and there is an exact sequence of Picard stacks0 −→ [ ∗ /T ( E )] −→ Bun T −→ X ∗ ( T ) Γ −→ ,

00 III. Bun G where we recall that X ∗ ( T ) Γ = B ( T ). The ﬁber of Bun T → B ( T ) over β is a gerbe bandedby T ( E ) over ∗ via the action Bun T . This gerbe is neutralized after choosing some b such that[ b ] = β . In case there is a section of T ( ˘ E ) (cid:16) B ( T ), for example if B ( T ) is torsion free, thenBun T (cid:39) [ ∗ /T ( E )] × X ∗ ( T ) Γ as a Picard stack. III.4.3. Splittings of the Harder–Narasimhan ﬁltration.

We can also consider the fol-lowing moduli problem, parametrizing G -bundles with a splitting of their Harder–Narasimhanﬁltration. Proposition

III.4.7 . Consider the functor

Bun HN - split G taking each S ∈ Perf k to the groupoidof exact ⊗ -functors from Rep E G to the category of Q -graded vector bundles E = (cid:76) λ E λ on X S such that E λ is everywhere semistable of slope λ for all λ ∈ Q . For any b ∈ B ( G ) , the bundle E b naturally reﬁnes to a Q -graded bundle E gr b , using the Q -grading on isocrystals, and for S aﬃnoidthe natural map G b × E X alg S → Aut( E gr b ) of group schemes over X alg S is an isomorphism. In particular, we get a natural map (cid:71) b ∈ B ( G ) [ ∗ /G b ( E )] → Bun HN - split G , and this is an isomorphism. Proof.

Recall that the natural map G b × E ˘ E → G × E ˘ E , recording the map of underly-ing ˘ E -vector spaces, is a closed immersion identifying G b × E ˘ E with the centralizer of the slopehomomorphism ν b : D → G × E ˘ E , cf. [ RZ96 , Corollary 1.14]. This implies that the natural map G b × E X alg S → Aut( E gr b )is an isomorphism.We get the evident functor from (cid:70) b ∈ B ( G ) [ ∗ /G b ( E )] to this moduli problem, and it is clearly fullyfaithful. To see that it is surjective, take any strictly totally disconnected S and an exact ⊗ -functor E gr from Rep E G to such Q -graded vector bundles. For any point s ∈ S , note that Q -graded vectorbundles of the given form on X K ( s ) are equivalent to Isoc E , so at s ∈ S there is an isomorphismwith some E gr b . The type of the Q -ﬁltration is locally constant, so after replacing S by an openneighborhood of s , we can assume that Isom( E gr b , E gr )deﬁnes an Aut( E gr b )-torsor over X alg S , i.e. a G b -torsor over S . This deﬁnes a map S → Bun G b ,taking s into Bun G b , and by Theorem III.2.4 and Lemma III.2.6 it follows that after replacing S by an open neighborhood, we can assume that the torsor is trivial. This concludes the proof ofsurjectivity. (cid:3) II.5. NON-SEMISTABLE POINTS 101

III.5. Non-semistable pointsIII.5.1. Structure of

Aut( E b ) . Next, we aim to describe the non-semi-stable strata Bun bG .Thus, ﬁx any b ∈ B ( G ) and consider the associated G -bundle E b on X S . For any algebraic repre-sentation ρ : G → GL n , the corresponding vector bundle ρ ∗ E b has its Harder–Narasimhan ﬁltration( ρ ∗ E b ) ≥ λ ⊂ ρ ∗ E b , λ ∈ Q . If G is quasisplit and we ﬁx a Borel B ⊂ G , then this deﬁnes a reductionof E b to a parabolic P ⊂ G containing B .Now inside the automorphism v-sheaf (cid:101) G b = Aut( E b )(which necessarily preserves the Harder–Narasimhan ﬁltration of ρ ∗ E b for any ρ ∈ Rep E ( G )) onecan consider for any λ > (cid:101) G ≥ λb ⊂ (cid:101) G b of all automorphisms γ : E b ∼ −→ E b such that( γ − ρ ∗ E b ) ≥ λ (cid:48) ⊂ ( ρ ∗ E b ) ≥ λ (cid:48) + λ for all λ (cid:48) and all representations ρ of G . We also set (cid:101) G >λb = (cid:83) λ (cid:48) >λ (cid:101) G ≥ λ (cid:48) b , noting that this union iseventually constant.As G b ( E ) is the automorphism group of the isocrystal corresponding to b , and H ( X S , O X S ) = E ( S ), we have a natural injection G b ( E ) (cid:44) → (cid:101) G b . Now, for any automorphism γ of E b and any representation ρ , γ induces an automorphism of the Q -graded vector bundle (cid:77) λ ∈ Q Gr λ ( ρ ∗ E b ) . Using Proposition III.4.7, we deduce that the preceding injection has a section and (cid:101) G b = (cid:101) G > b (cid:111) G b ( E ) . For a G -bundle E on X S we note ad E for its adjoint bundle deduced by pushforward by the adjointrepresentation G → GL(Lie( G )). This is in fact a Lie algebra bundle. Proposition

III.5.1 . One has (cid:101) G b = (cid:101) G > b (cid:111) G b ( E ) , and for any λ > , there is a natural isomorphism (cid:101) G ≥ λb / (cid:101) G >λb ∼ −−→ BC ((ad E b ) ≥ λ / (ad E b ) >λ ) , the Banach–Colmez space associated to the slope − λ isoclinic part of (Lie( G ) ⊗ E ˘ E, Ad( b ) σ ) .In particular, (cid:101) G b is an extension of G b ( E ) by a successive extension of positive Banach–Colmezspaces, and thus (cid:101) G b → ∗ is representable in locally spatial diamonds, of dimension (cid:104) ρ, ν b (cid:105) (whereas usual ρ is the half-sum of the positive roots). We refer to [

SR72 , Section IV] and [

Zie15 ] for some general discussion of ﬁltered and gradedﬁbre functors.

02 III. Bun G Proof.

We already saw the ﬁrst part. For the second part, suppose S = Spa( R, R + ) is aﬃnoid.Let X alg R be the schematical curve. We use the GAGA correspondence, Proposition II.2.7. Now, weapply Proposition III.5.2 to X alg R and the G -bundle E b associated to b on X alg R . Let H be the innertwisting of G × X alg R by E b as a reductive group scheme over X alg R . It is equipped with a ﬁltration( H ≥ λ ) λ ≥ satisfying • H ≥ /H > ∼ = G b × E X alg R , • for λ > H ≥ λ /H >λ = (ad E b ) ≥ λ / (ad E b ) >λ , • (cid:101) G ≥ λb ( S ) = H ≥ λ ( X alg R ),functorially in ( R, R + ); the ﬁrst part uses Proposition III.4.7. Since H ( X alg R , O ( µ )) = 0 as soonas µ >

0, we deduce by induction on µ >

0, starting with µ (cid:29) H ≥ µ /H >µ , that H ( X alg R , H ≥ µ ) = 0 for µ >

0. From this we deduce that H ≥ λ ( X alg R ) /H >λ ( X alg R ) = ( H ≥ λ /H >λ )( X alg R ) . Finally, it remains to compute the dimension. This is given by (cid:88) λ> λ · dim(ad E b ) ≥ λ / (ad E b ) >λ , which is given by (cid:104) ρ, ν b (cid:105) . (cid:3) Proposition

III.5.2 . Let G be a reductive group over a ﬁeld K , and let X be a schemeover K . Let E be a G -bundle on X with automorphism group scheme H/X (an inner formof G × K X , cf. Proposition III.4.1). Consider a Q -ﬁltration on the ﬁbre functor Rep K ( G ) →{ Vector bundles on X } associated with E . Deﬁning groups H ≥ λ ⊂ H for λ ≥ as before, they aresmooth group schemes, H ≥ is a parabolic subgroup with unipotent radical H > , the Lie algebra of H ≥ λ is given by (ad E ) ≥ λ ⊂ Lie ad E = Lie (cid:101) G , and the quotient H ≥ λ /H >λ is a vector group, thus H ≥ λ /H >λ ∼ = (ad E ) ≥ λ / (ad E ) >λ Proof.

The Lie algebra of H is ad E . All statements can be checked ´etale locally on X .According to [ Zie15 , Theorem 1.3] the Q -ﬁltration on the ﬁber functor is split locally on X .Moreover E is split ´etale locally on X . We can thus suppose that E is the trivial G -bundle and theﬁltration given by some ν : D /X → G × K X . Then the statement is easily checked, see [ SR72 ]. (cid:3) III.5.1.1.

The quasi-split case.

Suppose now that G is moreover quasi-split. Fix A ⊂ T ⊂ B with A a maximal split torus and T a maximal torus of G inside a Borel subgroup B . Up to σ -conjugating b one can suppose that ν b : D → A and ν b ∈ X ∗ ( A ) + Q . Let M b be the centralizer of ν b and P + b the parabolic subgroup associated to ν b , the weights of ν b in Lie( P + b ) are ≥

0. One has B ⊂ P + b , P + b is a standard parabolic subgroup with standard Levi subgroup M b . Let P − b be theopposite parabolic subgroup, the weights of ν b in Lie( P − b ) are ≤

0. One has b ∈ M b ( ˘ E ) and wedenote it b M as an element of M b ( ˘ E ). II.5. NON-SEMISTABLE POINTS 103

Then, if Q = E b M M b × P − b ,R u Q = E b M M b × R u P − b as X alg R -group-schemes, one has (cid:101) G b ( R, R + ) = Q ( X alg R ) (cid:101) G > b ( R, R + ) = R u Q ( X alg R ) . III.5.2. Description of Harder–Narasimhan strata.

Now we can describe the structureof the stratum Bun bG . Proposition

III.5.3 . Let b ∈ B ( G ) be any element given by some G -isocrystal. The inducedmap x b : ∗ → Bun bG is a surjective map of v-stacks, and ∗ × Bun bG ∗ ∼ = (cid:101) G b , so that Bun bG ∼ = [ ∗ / (cid:101) G b ] is the classifying stack of (cid:101) G b -torsors. In particular, the map (cid:101) G b → π (cid:101) G b ∼ = G b ( E ) induces a map Bun bG → [ ∗ /G b ( E )] that admits a splitting. Proof.

Let S = Spa( R, R + ) ∈ Perf k be strictly totally disconnected and let E be a G -bundleon X alg R , the schematical curve, all of whose geometric ﬁbers are isomorphic to E b . In particular,the Harder–Narasimhan polygon of ρ ∗ E is constant for all representations ρ : G → GL n , and thusby Theorem II.2.19, the vector bundle ρ ∗ E admits a relative Harder–Narasimhan ﬁltration. Thisdeﬁnes a Q -ﬁltration on the ﬁber functor Rep E ( G ) → { vector bundles on X alg R } deﬁned by E , andexactness can be checked on geometric points where it holds by the classiﬁcation of G -bundles.Since for any ρ , the Harder–Narasimhan polygon of ρ ∗ E b and the one of ρ ∗ E are equal, the twoﬁltered ﬁber functors on Rep E ( G ) deﬁned by E and E b are of the same type. Thus, ´etale locally on X alg R those two ﬁltered ﬁber functors are isomorphic. Let H = Aut( E b ) and H ≥ = Aut ﬁltered ( E b )as group schemes over X alg R , cf. Proposition III.5.1. Now, look at T = Isom ﬁltered ( E b , E ) . This is an H ≥ -torsor over X alg R that is a reduction to H ≥ of the H -torsor Isom( E b , E ). Let us lookat the image of [ T ] ∈ H ( X alg R , H ≥ ) in H ( X alg R , H ≥ /H > ), that is to say the H ≥ /H > -torsor T /H > . This parametrizes isomorphisms of graded ﬁber functors between the two obtained bysemi-simplifying the ﬁltered ﬁber functors attached to E b and E . By Proposition III.4.7, this torsoris locally trivial. Now the triviality of T follows from the vanishing of H ( X alg R , H > ). In fact, for λ > H ( X alg R , H ≥ λ /H >λ ) = 0 since H ( X alg R , O ( λ )) = 0.It is clear that ∗ × Bun bG ∗ is given by (cid:101) G b , so the rest follows formally. (cid:3)

04 III. Bun G Remark

III.5.4 (Followup to Remark III.2.5) . From the vanishing of H v ( S, (cid:101) G > b ) for S aﬃnoidperfectoid one deduces that for such S , any (cid:101) G b -torsor is of the form T × (cid:101) G > b where T → S is a G b ( E )-torsor. Here the action of g (cid:111) g ∈ G b ( E ) (cid:111) (cid:101) G > b on T × (cid:101) G > b is given by ( x, y ) (cid:55)→ ( g · x, g g yg − ).In particular any (cid:101) G b -torsor is representable in locally spatial diamonds. HAPTER IV

Geometry of diamonds

In this chapter, we extend various results on schemes to the setting of diamonds, showing thatmany advanced results in ´etale cohomology of schemes have analogues for diamonds.In Section IV.1, we introduce a notion of Artin v-stacks, and discuss some basic properties; inparticular, we show that Bun G is a cohomologically smooth Artin v-stack. Moreover, we can deﬁnea notion of dimension for Artin v-stacks, which we use to determine the connected components ofBun G . In Section IV.2, we develop the theory of universally locally acyclic sheaves. In Section IV.3,we introduce a notion of formal smoothness for maps of v-stacks. In Section IV.4, we use theprevious sections to prove the Jacobian criterion for cohomological smoothness, by establishingﬁrst formal smoothness, and universal local acyclicity. In Section IV.5, we prove a result on thevanishing of certain partially compactly supported cohomology groups, ensuring that for exampleSpd k [[ x , . . . , x d ]] behaves like a strictly local scheme for D ´et . In Section IV.6, we establish Braden’stheorem on hyperbolic localization in the world of diamonds. Finally, in Section IV.7, we establishseveral version of Drinfeld’s lemma in the present setup. The theme here is the idea π ((Div ) I ) = W IE . Unfortunately, we know no deﬁnition of π making this true, but for example it becomes truewhen considering Λ-local systems for any Λ. IV.1. Artin stacksIV.1.1. Generalities.

IV.1.1.1.

Deﬁnition and basic properties.

In this paper, we consider many small v-stacks likeBun G as above. However, they are stacky in some controlled way, in that they are Artin v-stacksin the sense of the following deﬁnition. Definition

IV.1.1 . An Artin v-stack is a small v-stack X such that the diagonal ∆ X : X → X × X is representable in locally spatial diamonds, and there is some surjective map f : U → X from a locally spatial diamond U such that f is separated and cohomologically smooth. Remark

IV.1.2 . We are making the assumption that f is separated, because only in this casewe have deﬁned cohomological smoothness. This means that we are imposing some (probablyunwanted) very mild separatedness conditions on Artin v-stacks. In particular, it implies that ∆ X is quasiseparated : Let f : U → X be as in the deﬁnition, and assume without loss of generalitythat U is a disjoint union of spatial diamonds (replacing it by an open cover if necessary), so inparticular U is quasiseparated. As f is separated, the map U × X U → U is separated, and inparticular U × X U is again quasiseparated. This is the pullback of ∆ X : X → X × X along thesurjection U × U → X × X , so ∆ X is quasiseparated. Remark

IV.1.3 . The stack

Bun G is not quasiseparated. In fact, [ ∗ /G ( E )] is already not qua-siseparated since the sheaf of automorphisms of the trivial G -bundle, G ( E ), is not quasicompact.This is diﬀerent from the “classical situation” of the stack of G -bundles on a proper smooth curve,this one being quasiseparated (although not separated). In the “classical schematical case” of Artinstacks it is a very mild assumption to suppose that Artin stacks are quasiseparated. In our situationthis would be a much too strong assumption, but it is still a very mild assumption to suppose thatthe diagonal is quasiseparated. Remark

IV.1.4 . By Remark IV.1.2, for any Artin v-stack X , the diagonal ∆ X is quasisep-arated. Conversely, let X be any small v-stack, and assume that there is some surjective map U → X from a small v-sheaf. Then:(i) If U is quasiseparated, then ∆ X is quasiseparated, by the argument of Remark IV.1.2.(ii) If U is a locally spatial diamond and U → X is representable in locally spatial diamonds, then∆ X is quasiseparated (as we may without loss of generality assume that U is quasiseparated, sothat (i) applies), and to check that ∆ X is representable in locally spatial diamonds, it suﬃces tosee that ∆ X is representable in diamonds. Indeed, [ Sch17a , Proposition 13.4 (v)] shows that if ∆ X is quasiseparated and representable in diamonds, then representability in locally spatial diamondscan be checked v-locally on the target. But the pullback of ∆ X along U × U → X × X is U × X U ,which is a locally spatial diamond as we assumed that U → X is representable in locally spatialdiamonds.(iii) Finally, in the situation of (ii), checking whether ∆ X is representable in diamonds can be doneafter pullback along a map V → X × X that is surjective as a map of pro-´etale stacks, by [ Sch17a ,Proposition 13.2 (iii)].In particular, if there is a map f : U → X from a locally spatial diamond U such that f isseparated, cohomologically smooth, representable in locally spatial diamonds, and surjective as amap of pro-´etale stacks, then X is an Artin v-stack. If one only has a map f : U → X from a locallyspatial diamond such that f is separated, cohomologically smooth, representable in locally spatialdiamonds, and surjective as a map of v-stacks, then it remains to prove that ∆ X is representablein diamonds, which can be done after pullback along a map V → X × X that is surjective as amap of pro-´etale stacks. Remark

IV.1.5 . Since cohomologically smooth morphisms are open, to prove that a separated,representable in locally spatial diamonds, cohomologically smooth morphism U → X is surjective,it suﬃces to verify it on geometric points. Remark

IV.1.6 . If X is a small v-stack with a map g : X → S to some “base” small v-stack S ,one might introduce a notion of an “Artin v-stack over S ”, asking instead that ∆ X/S : X → X × S X is representable in locally spatial diamonds; note that the condition on the chart f : U → X willevidently remain the same as in the absolute case. We note that as long as the diagonal of S isrepresentable in locally spatial diamonds (for example, S is an Artin v-stack itself), this agreeswith the absolute notion. Indeed, if ∆ X/S and ∆ S are representable in locally spatial diamonds,then also ∆ X is representable in locally spatial diamonds, as X × S X → X × X is a pullback of∆ S and thus representable in locally spatial diamonds, so ∆ X is the composite of the two maps X → X × S X → X × X both of which are representable in locally spatial diamonds. Conversely,assume that X and S are such that their diagonals are representable in locally spatial diamonds. V.1. ARTIN STACKS 107

Then both X and X × S X are representable in locally spatial diamonds over X × X , thus any mapbetween them is. Example

IV.1.7 . Any locally spatial diamond is an Artin v-stack.Before giving other examples let us state a few properties.

Proposition

IV.1.8 . (i) Any ﬁbre product of Artin v-stacks is an Artin v-stack. (ii)

Let S → ∗ be a pro-´etale surjective, representable in locally spatial diamonds, separated andcohomologically smooth morphism of v-sheaves. The v-stack X is an Artin v-stack if and only if X × S is an Artin v-stack. (iii) If X is an Artin v-stack and f : Y → X is representable in locally spatial diamonds, then Y isan Artin v-stack. Proof.

For point (1), if X = X × X X is such a ﬁbre product and f i : U i → X i are separated,representable in locally spatial diamonds, and cohomologically smooth surjective maps from locallyspatial diamonds U i , then U = ( U × X U ) × U ( U × X U ) is itself a locally spatial diamond(using that ∆ X is representable in locally spatial diamonds), and the projection f : U → X is a separated, representable in locally spatial diamonds, and cohomologically smooth surjection.For the diagonal, since ∆ X and ∆ X are representable in locally spatial diamonds, ∆ X × ∆ X : X × X → ( X × X ) × ( X × X ) is representable in locally spatial diamonds. Since ∆ X isrepresentable in locally spatial diamonds, its pullback by X × X → X × X , that is to say u : X × X X → X × X , is representable in locally spatial diamonds. Thus, ∆ X × X X is a mapbetween stacks that are representable in locally spatial diamonds over ( X × X ) × ( X × X ), andthus is representable in locally spatial diamonds.For point (2), suppose X × S is an Artin v-stack. If U is a locally spatial diamond and U → X × S is separated, representable in locally spatial diamonds, cohomologically smooth, and surjective, thenthe composite U → X × S → X is too. It remains to see that ∆ X is representable in locally spatialdiamonds. By Remark IV.1.4 it suﬃces to prove that the pullback of ∆ X by X × X × S → X × X is representable in locally spatial diamonds. But this pullback is the composite of ∆ X × S with X × X × S × S → X × X × S , and we conclude since the projection S × S → S is representable inlocally spatial diamonds for evident reasons.For point (3), if U is a locally spatial diamond and U → X is surjective, separated, representablein locally spatial diamonds, and cohomologically smooth, then V = U × X Y is a locally spatialdiamond, and V → Y is surjective, separated, representable in locally spatial diamonds, andcohomologically smooth. It remains to see that ∆ Y is representable in locally spatial diamonds.By Remark IV.1.4, it suﬃces to see that ∆ Y is representable in diamonds. But we can write ∆ Y as the composite Y → Y × X Y → Y × k Y . The ﬁrst map is 0-truncated and injective and thusrepresentable in diamonds by [ Sch17a , Proposition 11.10], while the second map is a pullback of∆ X . (cid:3) We can now give more examples.

Example

IV.1.9 .

08 IV. GEOMETRY OF DIAMONDS (i) According to point (2) of Proposition IV.1.8, the v-stack X is an Artin v-stack if and only if X × Spd E , resp. X × Spa( F q (( t /p ∞ ))), is an Artin v-stack. To check that X is an Artin v-stackwe can thus replace the base point ∗ by Spd E , resp. Spa F q (( t /p ∞ )) . (ii) For example, any small v-sheaf X such that X → ∗ is representable in locally spatial diamondsis an Artin v-stack ; e.g. X = ∗ .(iii) Using point (3) of Proposition IV.1.8 and [ Sch17a , Proposition 11.20] we deduce that anylocally closed substack of an Artin v-stack is an Artin v-stack. (iv) Let G be a locally proﬁnite group that admits a closed embedding into GL n ( E ) for some n .Then the classifying stack [ ∗ /G ] is an Artin v-stack. For this it suﬃces to see that [Spd

E/G ] =Spd E × [ ∗ /G ] is an Artin v-stack. Now let H = GL ♦ n,E ; then there is a closed immersion G × Spd

E (cid:44) → H . The map H → Spd E is representable in locally spatial diamonds, separated, andcohomologically smooth; hence so is H/G → [Spd E/G ] (by [

Sch17a , Proposition 13.4 (iv), Propo-sition 23.15]), and

H/G is a locally spatial diamond (itself cohomologically smooth over Spd E by[ Sch17a , Proposition 24.2] since this becomes cohomologically smooth over the separated ´etalecover

H/K → H/G for some compact open pro- p subgroup K of G ). It is clear that the diagonalof [ ∗ /G ] is representable in locally spatial diamonds. Remark

IV.1.10 . If G is a smooth algebraic group over the ﬁeld k then Spec( k ) → [Spec( k ) /G ]is a smooth presentation of the Artin stack [Spec( k ) /G ]. However, in the situation of point (4)of Example IV.1.9 the map f : ∗ → [ ∗ /G ] is not cohomologically smooth since for its pullback (cid:101) f : G → ∗ , the sheaf (cid:101) f ! Λ is the sheaf of distributions on G with values in Λ.IV.1.1.2. Smooth morphisms of Artin v-stacks.

Notions that can be checked locally with respectto cohomologically smooth maps can be extended to Artin v-stacks (except possibly for subtletiesregarding separatedness). In particular:

Definition

IV.1.11 . Let f : Y → X be a map of Artin v-stacks. Assume that there is someseparated, representable in locally spatial diamonds, and cohomologically smooth surjection g : V → Y from a locally spatial diamond V such that f ◦ g : V → X is separated. Then f is cohomologicallysmooth if for any (equivalently, one) such g , the map f ◦ g : V → X (which is separated byassumption, and automatically representable in locally spatial diamonds) is cohomologically smooth. In the preceding deﬁnition the “equivalently, one” assertion is deduced from [

Sch17a , Propo-sition 23.13] that says that cohomological smoothness is “cohomologically smooth local on thesource”. More precisely, if checked for one then for all g : V → X separated cohomologicallysmooth (not necessarily surjective) from a locally spatial diamond V , f ◦ g is separated cohomo-logically smooth. Convention

IV.1.12 . In the following, whenever we say that a map f : Y → X of Artinv-stacks is cohomologically smooth, we demand that there is some separated, representable inlocally spatial diamonds, and cohomologically smooth surjection g : V → Y from a locally spatialdiamond V such that f ◦ g : V → X is separated. Note that this condition can be tested aftertaking covers U → X by separated, representable in locally spatial diamonds, and cohomologicallysmooth surjections; i.e. after replacing X by X × Y U and Y by U . If X and Y have the propertythat one can ﬁnd a cover U → X , V → Y , as above with U and V perfectoid spaces, and ∆ X isrepresentable in perfectoid spaces, then the condition is automatic, as all maps of perfectoid spaces V.1. ARTIN STACKS 109 are locally separated. That being said there is no reason that this is true in general since there aremorphisms of spatial diamonds that are not locally separated.We will not try to give a completely general 6-functor formalism that includes functors Rf ! and Rf ! for stacky maps f (this would require some ∞ -categorical setting). However, we can extendthe functor Rf ! to cohomologically smooth maps of Artin v-stacks. Let Λ be a ring killed by someinteger n prime to p , or an adic ring as in [ Sch17a , Section 26].

Definition

IV.1.13 . Let f : Y → X be a cohomologically smooth map of Artin v-stacks. Thedualizing complex Rf ! Λ ∈ D ´et ( Y, Λ) is the invertible object equipped with isomorphisms Rg ! ( Rf ! Λ) ∼ = R ( f ◦ g ) ! Λ for all separated, representable in locally spatial diamonds, and cohomologically smooth maps g : V → Y from a locally spatial diamond V , such that for all cohomologically smooth maps h : V (cid:48) → V between such g (cid:48) : V (cid:48) → Y and g : V → Y , the composite isomorphism R ( g (cid:48) ) ! ( Rf ! Λ) ∼ = R ( f ◦ g (cid:48) ) ! Λ ∼ = R ( f ◦ g ◦ h ) ! Λ ∼ = Rh ! ( R ( f ◦ g ) ! Λ) ∼ = Rh ! ( Rg ! ( Rf ! Λ)) ∼ = R ( g (cid:48) ) ! ( Rf ! Λ) is the identity. As Rf ! Λ is locally concentrated in one degree, it is easy to see that Rf ! Λ is unique up tounique isomorphism. Let us be more precise. Let C be the category whose objects are separatedcohomologically smooth morphisms V → Y with V a locally spatial diamond, and morphisms( V (cid:48) g (cid:48) −→ Y ) → ( V g −→ Y ) are couples ( h, α ) where h : V (cid:48) → V is separated cohomologically smoothand α : g ◦ h ⇒ g (cid:48) is a 2-morphism. Then the rule( V g −→ Y ) (cid:55)−→ R H om Λ ( Rg ! Λ , R ( f ◦ g ) ! Λ)deﬁnes an element of2- lim ←− ( V → Y ) ∈C { invertible objects in D ´et ( V, Λ) } ∼ = { invertible objects in D ´et ( Y, Λ) } . Remark

IV.1.14 . If g : V → Y is a compactiﬁable representable in locally spatial diamondsmorphism of small v-stacks with dim . trg g < ∞ such that f ◦ g satisﬁes the same hypothesis, itis not clear that Rg ! ( Rf ! Λ) ∼ = R ( f ◦ g ) ! Λ. This is a priori true only when V is a locally spatialdiamond and g is separated cohomologically smooth, the only case we will need. Definition

IV.1.15 . Let f : Y → X be a cohomologically smooth map of Artin v-stacks. Thefunctor Rf ! : D ´et ( X, Λ) → D ´et ( Y, Λ) is given by Rf ! = Rf ! Λ ⊗ L Λ f ∗ . Remark

IV.1.16 . Checking after a cohomologically smooth cover, one sees that Rf ! preservesall limits (and colimits) and hence admits a left adjoint Rf ! . Definition

IV.1.17 . Let f : Y → X be a cohomologically smooth map of Artin v-stacks and let (cid:96) (cid:54) = p be a prime. Then f is pure of (cid:96) -dimension d ∈ Z if Rf ! F (cid:96) sits locally in homological degree d .

10 IV. GEOMETRY OF DIAMONDS As Rf ! F (cid:96) is locally constant, any cohomologically smooth map f : Y → X of Artin v-stacksdecomposes uniquely into a disjoint union of f d : Y d → X that are pure of (cid:96) -dimension d . A priorithis decomposition may depend on (cid:96) and include half-integers d , but this will not happen in anyexamples that we study. IV.1.2. The case of

Bun G . IV.1.2.1.

Smooth charts on

Bun G . One important example is the following. We use Beauville–Laszlo uniformization to construct cohomologically smooth charts on Bun G . More reﬁned chartswill be constructed in Theorem V.3.7. For ¯ µ ∈ X ∗ ( T ) + / Γ we note Gr G, ¯ µ for the subsheaf of Gr G such that Gr G, ¯ µ × Spd( E ) Spd( E (cid:48) ) = (cid:96) µ (cid:48) ≡ µ Gr G,µ (cid:48) where E (cid:48) | E is a ﬁnite degree Galois extensionsplitting G . We will use the following simple proposition. Proposition

IV.1.18 . For any µ ∈ X ∗ ( T ) + , the open Schubert cell Gr G,µ / Spd E (cid:48) is cohomo-logically smooth of (cid:96) -dimension (cid:104) ρ, µ (cid:105) . We defer the proof to Proposition VI.2.4 as we do not want to make a digression on Gr G here. Theorem

IV.1.19 . The stack

Bun G is a cohomologically smooth Artin v-stack of (cid:96) -dimension . The Beauville–Laszlo map deﬁnes a separated cohomologically smooth cover (cid:97) ¯ µ ∈ X ∗ ( T ) + / Γ [ G ( E ) \ Gr G, ¯ µ ] −→ Bun G . Proof.

We check ﬁrst that ∆

Bun G is representable in locally spatial diamonds. For this, itsuﬃces to see that for a perfectoid space S with two G -bundles E , E on X S , the functor ofisomorphisms between E and E is representable by a locally spatial diamond over S . By theTannakian formalism, one can reduce to vector bundles. For example, according to Chevalley, onecan ﬁnd a faithful linear representation ρ : G → GL n , a representation ρ (cid:48) : GL n → GL( W ), and aline D ⊂ W such that G is the stabilizer of D inside GL( V ). Then G -bundles on X S are the sameas a rank n vector bundle E together with a sub-line bundle L inside ρ (cid:48)∗ E . In terms of those data,isomorphisms between ( E , L ) and ( E , L ) are given by a couple ( α, β ) where α : E ∼ −→ E , and β : L ∼ −→ L satisfy ( ρ (cid:48)∗ α ) |L = β . Since the category of locally spatial diamonds is stable underﬁnite projective limits we are reduced to the case of the linear group. Now the result is given byLemma IV.1.20.It remains to construct cohomologically smooth charts for Bun G . We ﬁrst prove that themorphism π : (cid:97) ¯ µ ∈ X ∗ ( T ) + / Γ [ G ( E ) \ Gr G, ¯ µ ] −→ Bun G × k Spd ˘ E is separated cohomologically smooth. Since this is surjective at the level of geometric points wededuce that it is a v-cover, cf. Remark IV.1.5.To verify this, note that for a perfectoid space S mapping to Bun G × k Spd ˘ E corresponding toa G -bundle E on X S as well as a map S → Spd E inducing an untilt S (cid:93) /E and a closed immersion i : S (cid:93) → X S , the ﬁbre of π over S parametrizes modiﬁcations of E of locally constant type that aretrivial at each geometric point. This is open in the space of all modiﬁcations of E of locally constant V.1. ARTIN STACKS 111 type, which is v-locally isomorphic to (cid:70) ¯ µ Gr G, ¯ µ,E × Spd E S → S . Thus, Proposition IV.1.18 givesthe desired cohomological smoothness.Moreover, the preceding argument shows that when restricted to [ G ( E ) \ Gr G, ¯ µ ], the map π has (cid:96) -dimension equal to (cid:104) ρ, µ (cid:105) . Thus, it now suﬃces to see that [ G ( E ) \ Gr G, ¯ µ ] is an (cid:96) -cohomologicallysmooth Artin v-stack of (cid:96) -dimension equal to (cid:104) ρ, µ (cid:105) . But the map[ G ( E ) \ Gr G, ¯ µ ] → [Spd E/G ( E )]is representable in locally spatial diamonds and (cid:96) -cohomologically smooth of (cid:96) -dimension equal to (cid:104) ρ, µ (cid:105) , as Gr G, ¯ µ → ∗ is by Proposition IV.1.18. We conclude by using that [ ∗ /G ( E )] → ∗ is anArtin v-stack, cohomologically smooth of (cid:96) -dimension 0, by Example IV.1.9 (4). (cid:3) Lemma

IV.1.20 . For E , E vector bundles on X S , the sheaf T /S (cid:55)→ { surjections E | X T (cid:16) E | X T } , resp. T /S (cid:55)→

Isom( E | X T , E | X T ) , is representable by an open subdiamond of BC ( E ∨ ⊗ E ) .In particular, those are locally spatial diamonds. Proof.

The case of isomorphisms is reduced to the case of surjections since a morphism u of vector bundles is an isomorphisms if and only if u and u ∨ are surjective. For any morphism g : E → E , the support of its cokernel is a closed subset of | X S | , whose image in | S | is thus closed;this implies the result. (cid:3) Remark

IV.1.21 . It would be tempting to study D ´et (Bun G , Λ) using the preceding charts.But, contrary to the sheaves coming from the geometric Satake correspondence, the sheaves onGr G obtained via pullback from Bun G are not locally constant on open Schubert strata. We willprefer other smooth charts to study D ´et (Bun G , Λ), see Theorem V.3.7.Moreover, each HN stratum Bun bG gives another example. Proposition

IV.1.22 . For every b ∈ B ( G ) , the stratum Bun bG is a cohomologically smoothArtin v-stack of (cid:96) -dimension −(cid:104) ρ, ν b (cid:105) . Proof.

Under the identiﬁcation Bun bG ∼ = [ ∗ / (cid:101) G b ], note that we have a map [ ∗ / (cid:101) G b ] → [ ∗ /G b ( E )]where the target is a cohomologically smooth Artin v-stack of dimension 0, while the ﬁbre admits acohomologically smooth surjection from ∗ (as positive Banach–Colmez spaces are cohomologicallysmooth) of (cid:96) -dimension (cid:104) ρ, ν b (cid:105) . This gives the result. (cid:3) IV.1.2.2.

Connected components of

Bun G . A consequence is that we can classify the connectedcomponents of Bun G . Corollary

IV.1.23 . The Kottwitz map induces a bijection κ : π (Bun G ) → π ( G ) Γ . Proof.

The Kottwitz map is well-deﬁned and surjective. It remains to see that it is injective.To see this, recall that the basic elements of B ( G ) biject via κ to π ( G ) Γ . Thus, it suﬃces to seethat any nonempty open subsheaf U of Bun G contains a basic point. Note that the topologicalspace ( X ∗ ( T ) + Q ) Γ × π ( G ) Γ equipped with the product topology given by the order on ( X ∗ ( T ) + Q ) Γ and the discrete topology on π ( G ) Γ , is (T0), and an increasing union of ﬁnite open subspaces; and | Bun G | maps continuously to it. Pick some ﬁnite open V ⊂ ( X ∗ ( T ) + Q ) Γ × π ( G ) Γ such that its

12 IV. GEOMETRY OF DIAMONDS preimage in U is a nonempty open U (cid:48) ⊂ U . Then U (cid:48) is a nonempty ﬁnite (T0) space, and thus hasan open point by Lemma IV.1.24.Thus, there is some b ∈ B ( G ) such that Bun bG ⊂ U ⊂ Bun G is open. Combining Theo-rem IV.1.19 and Proposition IV.1.22, this forces −(cid:104) ρ, ν b (cid:105) = 0, i.e. ν b is central, which means that b is basic. (cid:3) Lemma

IV.1.24 . If X is a nonempty ﬁnite spectral space, that is to say a ﬁnite (T ) topologicalspace, there exists an open point x ∈ X . Proof.

Take x maximal for the specialization relation, i.e. x is a maximal point. Then, since X is (T0), X \ { x } = ∪ y (cid:54) = x { y } , a ﬁnite union of closed spaces thus closed. (cid:3) IV.2. Universally locally acyclic sheavesIV.2.1. Deﬁnition and basic properties.

In many of our results, and in particular in the(formulation and) proof of the geometric Satake equivalence, a critical role is played by the notionof universally locally acyclic (ULA) sheaves. Roughly speaking, for a morphism f : X → S ofschemes, these are constructible complexes of ´etale sheaves A on X whose relative cohomology isconstant in all ﬁbres of S , even locally. Technically, one requires that for all geometric points x of X mapping to a geometric point s of S and a generization t of s in S , the natural map R Γ( X x , A ) → R Γ( X x × S s t, A )is an isomorphism, where X x is the strict henselization of X at x (and S s is deﬁned similarly).Moreover, the same property should hold universally after any base change along S (cid:48) → S . By[

Ill06 , Corollary 3.5], universal local acyclicity is equivalent to asking that, again after any basechange, the map R Γ( X x , A ) → R Γ( X x × S s S t , A )is an isomorphism; we prefer the latter formulation as strict henselizations admit analogues for adicspaces, while the actual ﬁbre over a point is only a pseudo-adic space in Huber’s sense [ Hub96 ].In the world of adic spaces, there are not enough specializations to make this an interestingdeﬁnition; for example, there are no specializations from Gr

G,µ into Gr G, ≤ µ \ Gr G,µ . Thus, we needto adapt the deﬁnition by adding a condition on preservation of constructibility that is automaticin the scheme case under standard ﬁniteness hypothesis, but becomes highly nontrivial in the caseof adic spaces.

Definition

IV.2.1 . Let f : X → S be a compactiﬁable map of locally spatial diamonds withlocally dim . trg f < ∞ and let A ∈ D ´et ( X, Λ) for some Λ with n Λ = 0 with n prime to p . (i) The sheaf of complexes A is f -locally acyclic if (a) For all geometric points x of X with image s in S and a generization t of s , the map R Γ( X x , A ) → R Γ( X x × S s S t , A ) is an isomorphism. Recently, Gabber proved that this is automatic when S is noetherian and f is of ﬁnite type, cf. [ LZ19 , Corollary6.6].

V.2. UNIVERSALLY LOCALLY ACYCLIC SHEAVES 113 (b)

For all separated ´etale maps j : U → X such that f ◦ j is quasicompact, the complex R ( f ◦ j ) ! ( A | U ) ∈ D ´et ( S, Λ) is perfect-constructible. (ii) The sheaf of complexes A is f -universally locally acyclic if for any map S (cid:48) → S of locally spatialdiamonds with base change f (cid:48) : X (cid:48) = X × S S (cid:48) → S (cid:48) and A (cid:48) ∈ D ´et ( X (cid:48) , Λ) the pullback of A , thesheaf of complexes A (cid:48) is f (cid:48) -locally acyclic. Recall that if (

K, K + ) is an aﬃnoid Huber ﬁeld, S = Spa( K, K + ), then | S | = | Spec( K + /K ) | as a topological spectral space, that is identiﬁed with the totally ordered set of open prime idealsin K + . For any s ∈ S , S s ⊂ S is pro-constructible generalizing. For example, the maximalgeneralization is Spa( K, O K ) = ∩ a ∈O K {| a | ≤ } . Remark

IV.2.2 . In the setup of condition (a), note that X x is a strictly local space, i.e. of theform Spa( C, C + ) where C is algebraically closed and C + ⊂ C is an open and bounded valuationsubring; thus, R Γ( X x , A ) = A x is just the stalk of A . Moreover, S t ⊂ S s is a quasicompactpro-constructible generalizing subspace, and thus X x × S s S t ⊂ X x is itself a quasicompact pro-constructible generalizing subset that is strictly local. Its closed point y is the minimal generizationof x mapping to t , and R Γ( X x × S s S t , A ) = A y is the stalk at y . Thus, condition (a) means that A is “overconvergent” along the horizontal lifts of generizations of S . Remark

IV.2.3 . (i) Another way to phrase the “relative overconvergence condition” (a), is to say that if ¯ s is ageometric point of S , ¯ t a generization of ¯ s , j : X × S S ¯ t (cid:44) → X × S S ¯ s , a pro-constructible generalizingimmersion, and B = A | X × S S ¯ s , then B = Rj ∗ j ∗ B (use quasicompact base change).(ii) Still another way to phrase it is to say that for any Spa( C, C + ) → S , if B = A | X × S Spa(

C,C + ) ,and j : X × S Spa( C, O C ) (cid:44) → X × S Spa(

C, C + ), then B = Rj ∗ j ∗ B .(iii) Still another way is to say that if ¯ x (cid:55)→ ¯ s and f ¯ x : X ¯ x → S ¯ s then Rf ¯ x ∗ A | X ¯ x is overconvergenti.e. constant.In fact, asking for condition (a) universally, i.e. after any base change, amounts to asking that A is overconvergent. Proposition

IV.2.4 . Let f : X → S be a compactiﬁable map of locally spatial diamondswith locally dim . trg f < ∞ and let A ∈ D ´et ( X, Λ) for some Λ with n Λ = 0 with n prime to p .The condition (a) of Deﬁnition IV.2.1 holds after any base change S (cid:48) → S if and only if A isoverconvergent, i.e. for any specialization y (cid:32) x of geometric points of X , the map A x → A y is anisomorphism. Proof.

The condition is clearly suﬃcient. For necessity, take the base change along X x → S .Then x lifts to a section x (cid:48) : X x → X × S X x , and applying the relative overconvergence conditionto x (cid:48) (cid:55)→ x and the generization y of x , we see that A x → A y is an isomorphism. (cid:3) Proposition

IV.2.5 . Local acyclicity descends along v-covers of the target. More precisely, inthe setup of Deﬁnition IV.2.1, if S (cid:48) → S is a v-cover and A (cid:48) is f (cid:48) -locally acyclic, then automatically A is f -locally acyclic. Proof.

Condition (a) follows by lifting geometric points, and condition (b) descends by [

Sch17a ,Proposition 20.13]. (cid:3)

14 IV. GEOMETRY OF DIAMONDS

Proposition

IV.2.6 . Let Y be a spatial diamond. (i) If F is a constructible ´etale sheaf of Λ -modules on Y , then F is locally constant if and only if F is overconvergent. (ii) If A ∈ D ´et ,pc ( Y, Λ) , then A is overconvergent if and only if it is locally a constant perfect complexof Λ -modules. Proof.

For a geometric point y of Y , writing Y y = Spa( C, C + ) = lim ←− y → U U as a limit of the´etale neighborhoods, according to [ Sch17a , Proposition 20.7],2- lim −→ y → U Cons( U, Λ) = Cons( Y y , Λ) . An ´etale sheaf on Y y = Spa( C, C + ) is locally constant if and only if it is constant if and only if itis overconvergent. This gives point (1). Point (2) goes the same way using [ Sch17a , Proposition20.15]. (cid:3)

Remark

IV.2.7 . The preceding argument shows that if F is constructible on Y then F islocally constant in a neighborhood of any maximal point of Y . For example, if Y = X ♦ with X a K -rigid space, then any constructible sheaf on Y is locally constant in a neighborhood of allclassical Tate points of X . Thus, the diﬀerence between constructible and locally constant sheavesshows up at rank > Example

IV.2.8 . Let j : B K \ { } (cid:44) → B K be the inclusion of the punctured disk inside the disk.Then j ! Λ is not constructible since not locally constant around { } . Nevertheless, if R ∈ | K × | and x is the coordinate on B K , j R : { R ≤ | x | ≤ } (cid:44) → B K , j R ! Λ is constructible and j ! Λ = lim −→ R → j R ! Λ.The category of ´etale sheaves of Λ-modules on a spatial diamond is the Ind-category of constructible´etale sheaves, cf. [

Sch17a , Proposition 20.6].

Proposition

IV.2.9 . Assume that f : S → S is the identity. Then A ∈ D ´et ( S, Λ) is f -locallyacyclic if and only if it is locally constant with perfect ﬁbres. Proof.

Applying part (b) of the deﬁnition, we see that A is perfect-constructible. On theother hand, part (a) says that A is overconvergent. This implies that A is locally constant byProposition IV.2.6. (cid:3) Let us ﬁnish with a basic example of universally locally acyclic sheaves relevant to the smoothbase change theorem. A more general result will be given in Proposition IV.2.13.

Proposition

IV.2.10 . Assume that f : X → S is a separated map of locally spatial diamondsthat is (cid:96) -cohomological smooth for all divisors (cid:96) of n , where n Λ = 0 . If A ∈ D ´et ( X, Λ) is locallyconstant with perfect ﬁbres, then A is f -universally locally acyclic. Proof.

It is enough to show that A is f -locally acyclic, as the hypotheses are stable under basechange. Condition (a) follows directly from A being locally constant. Condition (b) follows fromthe preservation of constructible sheaves of complexes under Rf ! if f is quasicompact, separatedand cohomologically smooth, see [ Sch17a , Proposition 23.12 (ii)]. (cid:3)

V.2. UNIVERSALLY LOCALLY ACYCLIC SHEAVES 115

IV.2.2. Proper push-forward, smooth pull-back.

In the “classical algebraic case”, if Y g −→ X f −→ S are morphisms of ﬁnite type between noetherian schemes, using proper and smooth basechange:(i) if g is proper and A ∈ D bc ( Y, Λ) is f ◦ g -locally acyclic then Rg ∗ A is f -locally acyclic;(ii) if g is smooth and A ∈ D bc ( X, Λ) is f -locally acyclic then g ∗ A is f ◦ g -locally acyclic. Moreoverif g is surjective then A ∈ D bc ( X, Λ) is f -locally acyclic if and only if g ∗ A is f ◦ g -locally acyclic.We are going to see that the same phenomenon happens in our context. The fact that localacyclicity is smooth local on the source is essential to deﬁne local acyclicity for morphisms of Artinv-stacks, cf. Deﬁnition IV.2.31. Proposition

IV.2.11 . Let g : Y → X , f : X → S be maps of locally spatial diamonds where g is proper and f is compactiﬁable and locally dim . trg g, dim . trg f < ∞ . Assume that A ∈ D ´et ( Y, Λ) is f ◦ g -locally acyclic (resp. f ◦ g -universally locally acylic). Then Rg ∗ A ∈ D ´et ( X, Λ) is f -locallyacyclic (resp. f -universally locally acyclic). Proof.

It is enough to consider the locally acyclic case, as the hypotheses are stable underbase change. For condition (a), we use Remark IV.2.3 (2). Let ¯ s be a geometric point of S , with S ¯ s = Spa( C, C + ). Let us look at the cartesian diagram Y × Spa(

C,C + ) Spa( C, O C ) YX × Spa(

C,C + ) Spa( C, O C ) X k gj one has by local acyclicity of f ◦ g , A = Rk ∗ k ∗ A . Applying Rg ∗ , this gives the desired Rg ∗ A = Rj ∗ j ∗ ( Rg ∗ A ) . For condition (b), take any separated ´etale map j : U → X such that f ◦ j is quasicompact,and set j (cid:48) : V = U × X Y → Y , which is an ´etale map such that f ◦ g ◦ j (cid:48) is quasicompact. Let g (cid:48) : V → U denote the pullback of g . Using proper base change and Rg ∗ = Rg ! , we see that R ( f ◦ j ) ! j ∗ Rg ∗ A = R ( f ◦ j ) ! Rg (cid:48) ! j (cid:48)∗ A = R ( f ◦ g ◦ j (cid:48) ) ! j (cid:48)∗ A, which is perfect-constructible by the assumption that A is f ◦ g -locally acyclic. (cid:3) In particular we have the following that generalizes the “proper and smooth case”.

Corollary

IV.2.12 . Let f : X → S be a proper map of locally spatial diamonds with dim . trg f < ∞ and A ∈ D ´et ( X, Λ) that is f -locally acyclic. Then Rf ∗ A is locally a constant perfect complex of Λ -modules. Next proposition says that local acyclicity is “cohomologically smooth local” on the source.

Proposition

IV.2.13 . Let f : X → S be a compactiﬁable map of locally spatial diamonds withlocally dim . trg f < ∞ . For the statements in the locally acyclic case below, assume that S is spatialand that the cohomological dimension of U ´et for all quasicompact separated ´etale U → S is ≤ N for some ﬁxed integer N .

16 IV. GEOMETRY OF DIAMONDS

Let A ∈ D ´et ( X, Λ) where n Λ = 0 for some n prime to p and let g : Y → X be a separated mapof locally spatial diamonds that is (cid:96) -cohomologically smooth for all (cid:96) dividing n . (i) If A is f -locally acyclic (resp. f -universally locally acyclic), then g ∗ A is f ◦ g -locally acyclic(resp. f ◦ g -universally locally acyclic). (ii) Conversely, if g ∗ A is f ◦ g -locally acyclic (resp. f ◦ g -universally locally acyclic) and g is surjective,then A is f -locally acyclic (resp. f -universally locally acyclic). Proof.

It is enough to handle the locally acyclic case with the assumption on S ; then theuniversally locally acyclic case follows by testing after pullbacks to strictly totally disconnectedspaces. Let us treat point (i). We can assume that X and Y are qcqs, i.e. spatial. In fact, thisis clear for condition (a). For condition (b), if j : V → Y is separated ´etale such that f ◦ g ◦ j isquasicompact, up to replacing S by an open cover we can suppose that S is spatial and thus V is spatial (since f , g , and j are separated, f ◦ g ◦ j is separated quasicompact, and thus S spatialimplies X spatial). Then one can replace Y , resp. X , by the quasicompact open subsets j ( V ),resp. ( g ◦ j )( V ), that are separated over S and thus spatial too.Condition (a) follows as pullbacks preserve stalks. For condition (b), let j : V → Y be anyquasicompact separated ´etale map. Then by the projection formula for g ◦ j , one has R ( f ◦ g ◦ j ) ! j ∗ g ∗ A = Rf ! ( A ⊗ L Λ R ( g ◦ j ) ! Λ) . As g ◦ j : V → X is a quasicompact separated (cid:96) -cohomologically smooth map, it follows that R ( g ◦ j ) ! Λ ∈ D ´et ( X, Λ) is perfect-constructible by [

Sch17a , Proposition 23.12 (ii)]. Thus, thedesired result follows from Lemma IV.2.14.In the converse direction, i.e. for part (ii), condition (a) of A being f -locally acyclic followsby lifting geometric points from X to Y and noting that stalks do not change. For condi-tion (b), we may replace X by U to reduce to the assertion that Rf ! A ∈ D ´et ( S, Λ) is perfect-constructible. Consider the thick triangulated subcategory C of D ´et ( X, Λ) of all B ∈ D ´et ( X, Λ)such that Rf ! ( A ⊗ L Λ B ) ∈ D ´et ( S, Λ) is perfect-constructible. We have to see that Λ ∈ C . We knowthat for all perfect-constructible C ∈ D ´et ( Y, Λ), the perfect-constructible complex Rg ! C lies in C .Indeed, using [ Sch17a , Proposition 20.17], this reduces to the case C = j ! Λ where j : U → Y is aquasicompact separated ´etale map, and then Rf ! ( A ⊗ L Λ R ( g ◦ j ) ! Λ) = R ( f ◦ g ) ! ( g ∗ A ⊗ L Λ Rj ! Λ) , which is perfect-constructible as g ∗ A is f ◦ g -locally acyclic. Thus, it is enough to show that theset of Rg ! C ∈ D ´et ( X, Λ) with C ∈ D ´et ( Y, Λ) perfect constructible form a set of compact genera-tors of D ´et ( X, Λ). Equivalently, for any complex B ∈ D ´et ( X, Λ) with R Hom D ´et ( X, Λ) ( Rg ! C, B ) =0 for all perfect-constructible C ∈ D ´et ( Y, Λ), then B = 0. The hypothesis is equivalent to R Hom D ´et ( Y, Λ) ( C, Rg ! B ) = 0 for all such C . By [ Sch17a , Proposition 20.17] and the standingassumptions on ﬁnite cohomological dimension (on S , f and g ), this implies that Rg ! B = 0. As g is (cid:96) -cohomologically smooth, this is equivalent to g ∗ B = 0, which implies B = 0 as g is surjective. (cid:3) Lemma

IV.2.14 . Let f : X → S be a compactiﬁable map of locally spatial diamonds withlocally dim . trg f < ∞ . Suppose there exists an integer N such that the cohomological dimensionof U ´et is bounded by N for all U → X separated ´etale. Let A ∈ D ´et ( X, Λ) be f -locally acyclicand B ∈ D ´et ( X, Λ) be perfect-constructible. Then A ⊗ L Λ B satisﬁes condition (ii) of Deﬁnition V.2. UNIVERSALLY LOCALLY ACYCLIC SHEAVES 117

IV.2.1: for any j : U → X separated ´etale such that f ◦ j is quasicompact, R ( f ◦ j ) ! j ∗ ( A ⊗ L Λ B ) isperfect-constructible. Proof.

We can suppose X is spatial. According to [ Sch17a , Proposition 20.17], B lies in thetriangulated subcategory generated by j (cid:48) ! Λ where j (cid:48) : U (cid:48) → X is separated quasicompact ´etale.For such a B , using the projection formula, A ⊗ L Λ Rj (cid:48) ! Λ = Rj (cid:48) ! j (cid:48)∗ A . Thus, if V = U × X U (cid:48) withprojection k : U × X U (cid:48) → U , j ∗ Rj (cid:48) ! j (cid:48)∗ A = Rk ! k ∗ j ∗ A. We thus have R ( f ◦ j ) ! j ∗ ( A ⊗ L Λ Rj (cid:48) ! B ) = R ( f ◦ j ◦ k ) ! ( j ◦ k ) ∗ ( A )and we can conclude. (cid:3) IV.2.3. Local acyclicity and duality.

In this section, we prove that universal local acyclicitybehaves well with respect to Verdier duality.IV.2.3.1.

Compatibility with base change.

We note that for f -ULA sheaves, the formation ofthe (relative) Verdier dual D X/S ( A ) := R H om Λ ( A, Rf ! Λ)commutes with base change in S . Proposition

IV.2.15 . Let f : X → S be a compactiﬁable map of locally spatial diamonds withlocally dim . trg f < ∞ and let A ∈ D ´et ( X, Λ) be f -universally locally acyclic. Let g : S (cid:48) → S bea map of locally spatial diamonds with pullback f (cid:48) : X (cid:48) = X × S S (cid:48) → S (cid:48) , (cid:101) g : X (cid:48) → X . Then thecomposite (cid:101) g ∗ D X/S ( A ) → D X (cid:48) /S (cid:48) ( (cid:101) g ∗ A ) of the natural maps (cid:101) g ∗ R H om Λ ( A, Rf ! Λ) → R H om Λ ( (cid:101) g ∗ A, (cid:101) g ∗ Rf ! Λ) → R H om Λ ( (cid:101) g ∗ A, Rf (cid:48) ! Λ) is an isomorphism.More generally, for any B ∈ D ´et ( S, Λ) , the map (cid:101) g ∗ R H om Λ ( A, Rf ! B ) → R H om Λ ( (cid:101) g ∗ A, Rf (cid:48) ! B ) is an isomorphism. Proof.

The assertion is local, so we may assume that X , S and S (cid:48) are spatial. By choosing astrictly totally disconnected cover S (cid:48)(cid:48) of S (cid:48) , one reduces the result for S (cid:48) → S to the cases of S (cid:48)(cid:48) → S (cid:48) and S (cid:48)(cid:48) → S , so we may assume that S (cid:48) is strictly totally disconnected. In that case, by [ Sch17a ,Proposition 20.17], whose hypothesis apply as X (cid:48) → S (cid:48) is of ﬁnite dim . trg and S (cid:48) is strictly totallydisconnected, it suﬃces to check on global sections over all quasicompact separated ´etale maps V (cid:48) → X (cid:48) . According to Lemma IV.2.16 we can write S (cid:48) as a coﬁltered limit of quasicompact opensubsets S (cid:48) i of ﬁnite-dimensional balls over S . Then V (cid:48) comes via pullback from a quasicompact

18 IV. GEOMETRY OF DIAMONDS separated ´etale map V i → X × S S (cid:48) i for i large enough by [ Sch17a , Proposition 11.23]. We thushave a diagram with cartesian squares

V V (cid:48) X (cid:48) X × S S (cid:48) i XS (cid:48) S (cid:48) i S. h i The result we want to prove is immediate when S (cid:48) → S is cohomologically smooth. Up to replacing X → S by V (cid:48) → S (cid:48) i we are thus reduced to prove that R Γ( X (cid:48) , (cid:101) g ∗ D X/S ( A )) ∼ −→ R Γ( X (cid:48) , D X (cid:48) /S (cid:48) ( (cid:101) g ∗ A )).Thus, it suﬃces to check the result after applying Rf (cid:48)∗ . In that case, Rf (cid:48)∗ (cid:101) g ∗ R H om Λ ( A, Rf ! B ) = g ∗ Rf ∗ R H om Λ ( A, Rf ! B )= g ∗ R H om Λ ( Rf ! A, B ) , using [ Sch17a , Proposition 17.6, Theorem 1.8 (iv)]. On the other hand, Rf (cid:48)∗ R H om Λ ( (cid:101) g ∗ A, Rf (cid:48) ! B ) = R H om Λ ( Rf (cid:48) ! (cid:101) g ∗ A, B )= R H om Λ ( g ∗ Rf ! A, B )using [

Sch17a , Theorem 1.8 (iv), Theorem 1.9 (ii)]. But by condition (b) of being f -locally acyclic,the complex Rf ! A ∈ D ´et ( S, Λ) is perfect-constructible, and thus the formation of R H om Λ ( Rf ! A, B )commutes with any base change by Lemma IV.2.17. (cid:3)

Lemma

IV.2.16 . Let S be a spatial diamond and X → S be a morphism from an aﬃnoidperfectoid space to S . Then one can write X = lim ←− i U i where U i is a quasicompact open subsetinside a ﬁnite dimensional ball over S , and the projective limit is coﬁltered. Proof. If I = O ( X ) + , one has a closed immersion over S deﬁned by elements of I , X (cid:44) → B IS where B IS is the spatial diamond over S that represents the functor T /S (cid:55)→ ( O ( T ) + ) I (an “inﬁnitedimensional perfectoid ball over S ” when S is perfectoid). Now, B IS = lim ←− J ⊂ I B JS where J goes through the set of ﬁnite subsets of I and B IS → B JS is the corresponding projection.For each such J the composite X (cid:44) → B IS (cid:16) B JS is a spatial morphism of spatial diamonds. Its imageis a pro-constructible generalizing subset of B JS and can thus be written as (cid:84) α ∈ A J U α where U α isa quasicompact open subset of B JS . Then one has X = lim ←− J ⊂ I lim ←− α ∈ A J U α . (cid:3) Lemma

IV.2.17 . Let X be a spatial diamond and A ∈ D ´et ( X, Λ) perfect-constructible, and let B ∈ D ´et ( X, Λ) be arbitrary. Then the formation of R H om Λ ( A, B ) commutes with any base change. V.2. UNIVERSALLY LOCALLY ACYCLIC SHEAVES 119

Proof.

Using [

Sch17a , Proposition 20.16 (iii)] this is reduced to the case when A = j ! ( L| Z )where j : U → X is separated quasicompact ´etale, Z ⊂ U closed constructible, and L ∈ D ´et ( U, Λ)locally constant with perfect ﬁbres. If j (cid:48) : U \ Z → X , that is again quasicompact (since Z isconstructible inside U ) separated ´etale, using the exact sequence 0 → j (cid:48) ! L → j ! L → j ! ( L | Z ) → A of the form j ! L . In this case R H om Λ ( A, − ) is given by Rj ∗ ( L ∨ ⊗ L Λ − ),and this commutes with any base change by quasicompact base change, [ Sch17a , Proposition17.6]. (cid:3)

One has to be careful that, in general, the naive dual of a perfect constructible complex is notconstructible.

The following lemma says that in fact it is overconvergent, so never constructibleunless locally constant.

Lemma

IV.2.18 . For X a spatial diamond and A ∈ D ´et ( X, Λ) perfect constructible, R H om Λ ( A, Λ) is overconvergent. Proof.

Using Lemma IV.2.17 this is reduced to the case when X = Spa( C, C + ). More-over, one can assume that A = j ! Λ for some quasicompact open immersion j : U → X . Then R H om Λ ( A, Λ) = Rj ∗ Λ = Λ, which is overconvergent. (cid:3)

IV.2.3.2.

Twisted inverse images.

Also, if A is f -ULA, then one can relate appropriately A -twisted versions of f ∗ and Rf ! . Proposition

IV.2.19 . Let f : X → S be a compactiﬁable map of locally spatial diamondswith locally dim . trg f < ∞ and let A ∈ D ´et ( X, Λ) be f -universally locally acyclic. Then for all B ∈ D ´et ( S, Λ) , the natural map D X/S ( A ) ⊗ L Λ f ∗ B → R H om Λ ( A, Rf ! B ) given as the composite R H om Λ ( A, Rf ! Λ) ⊗ L Λ f ∗ B → R H om Λ ( A, Rf ! Λ ⊗ L Λ f ∗ B ) → R H om Λ ( A, Rf ! B ) is an isomorphism. Proof.

First, we note that both sides commute with any base change, by Proposition IV.2.15.It suﬃces to check that we get an isomorphism on stalks at all geometric points Spa(

C, C + )of X . For this, we may base change along the associated map Spa( C, C + ) → S to reduce to thecase that S = Spa( C, C + ) is strictly local, and we need to check that we get an isomorphismat the stalk of a section s : S → X . We may also assume that X is spatial, in which case X is of bounded cohomological dimension, so [ Sch17a , Proposition 20.17] applies, and perfect-constructible complexes are the same thing as compact objects in D ´et ( X, Λ) = D ( X ´et , Λ).Next, we note that the functor B (cid:55)→ R H om Λ ( A, Rf ! B ) is right adjoint to A (cid:48) (cid:55)→ Rf ! ( A ⊗ L Λ A (cid:48) ).The latter functor preserves perfect-constructible complexes, i.e. compact objects, by condition(b), see Lemma IV.2.14. Thus, B (cid:55)→ R H om Λ ( A, Rf ! B ) commutes with arbitrary direct sums, seeLemma IV.2.20. Obviously, the functor B (cid:55)→ D X/S ( A ) ⊗ L Λ f ∗ B also commutes with arbitrary directsums, so it follows that it suﬃces to check the assertion for B = j ! Λ for some quasicompact openimmersion j : S (cid:48) = Spa( C, C (cid:48) + ) → S (the shifts of those compact objects generate D ´et ( S, Λ)). If S (cid:48) = S , then B = Λ and the result is clear. Otherwise, the stalk of j ! Λ at the closed point is zero,

20 IV. GEOMETRY OF DIAMONDS and thus the stalk of the left-hand side D X/S ( A ) ⊗ L Λ f ∗ B at our ﬁxed section is zero. It remains tosee that the stalk of R H om Λ ( A, Rf ! j ! Λ)at the (closed point of) the section s : S → X is zero. This stalk is given by the ﬁltered colimitover all quasicompact open neighborhoods U ⊂ X of s ( S ) of R Hom D ´et ( U, Λ) ( A | U , Rf ! j ! Λ | U ) = R Hom D ´et ( S, Λ) ( Rf U ! A | U , j ! Λ) , where f U : U → S denotes the restriction of f (the possibility to use only open embeddings in placeof general ´etale maps results from the observation that the intersection of all these open subsetsis the strictly local space S already; the set of open neighborhoods of s ( S ) is coﬁnal among ´etaleneighborhoods of s ( S )).Now we claim that the inverse systems of all such U and of the compactiﬁcations U /S arecoﬁnal. Note that the intersection of all U /S (taken inside X /S ) is simply s ( S ): Indeed, given anypoint x ∈ X /S \ s ( S ), there are disjoint open neighborhoods x ∈ V and s ( S ) ⊂ U . In fact, themaximal Hausdorﬀ quotient | X /S | B is compact Hausdorﬀ by [ Sch17a , Proposition 13.11] and itspoints can be identiﬁed with rank 1 points of X /S , which are the same as rank 1 points of X . Butas s ( S ) ⊂ X /S is closed, no point outside s ( S ) admits the same rank 1 generalization, so x and s ( S )deﬁne distinct points of the Hausdorﬀ spaces | X /S | , so that the desired disjoint open neighborhoodsexist x ∈ V and s ( S ) ⊂ U exist. Then x (cid:54)∈ U /S . Thus, s ( S ) = (cid:92) U ⊃ s ( S ) U /X . Now given any open neighborhood U of s ( S ), the complement | X /S | \ U is quasicompact, whichthen implies that there is some U (cid:48) such that U (cid:48) /S ⊂ U . It follows that the direct systems of Rf U ! A | U and Rf U /S ∗ Ri ! U /S A are equivalent, where i U /S : U /S → X and f U /S : U /S → S are the evident maps. Now observe thatif j (cid:48) X : X η = X × S Spa( C, O C ) (cid:44) → X denotes the proconstructible generalizing immersion, thencondition (a) in being f -locally acyclic implies that A = Rj (cid:48) X ∗ A | X η (see Remark IV.2.3), and then Rf U /S ∗ Ri ! U /S A = Rf U /S ∗ Ri ! U /S Rj (cid:48) X ∗ A | X η = Rf U /S ∗ Rj (cid:48) U /S ∗ Ri ! U /Sη A | X η = Rj (cid:48) S ∗ Rf U /Sη ∗ Ri ! U /Sη A | X η with hopefully evident notation; in particular, j (cid:48) S : Spa( C, O C ) → S = Spa( C, C + ) denotes thepro-open immersion of the generic point on the base. V.2. UNIVERSALLY LOCALLY ACYCLIC SHEAVES 121

In summary, we can rewrite the stalk of R H om Λ ( A, Rf ! j ! Λ) at s ( S ) as the ﬁltered colimit of R Hom D ´et ( S, Λ) ( Rj (cid:48) S ∗ Rf U /Sη ∗ Ri ! U /Sη A, j ! Λ) , and we need to prove that this vanishes. This follows from the observation that for all M ∈ D ´et (Spa( C, O C ) , Λ) = D (Λ), one has R Hom D ´et ( S, Λ) ( Rj (cid:48) S ∗ M, j ! Λ) = 0 . For this, note that Rj (cid:48) S ∗ M = M is just the constant sheaf given by the complex of Λ-modules M ,and one has a triangle R Hom D ´et ( S, Λ) ( M, j ! Λ) → R Hom D ´et ( S, Λ) ( M, Λ) → R Hom D ´et ( S, Λ) ( M, i ∗ Λ) , where i denotes the complementary closed immersion. Both the second and last term are given by R Hom Λ ( M, Λ), ﬁnishing the proof. (cid:3)

We used the following classical lemma, cf. [

Nee96 , Theorem 5.1].

Lemma

IV.2.20 . Let C and D be triangulated categories such that C is compactly generated. Let F : C → D and G : D → C be such that G is right adjoint to F . If F sends compact objects tocompact objects then G commutes with arbitrary direct sums. Proof.

Since C is compactly generated it suﬃces to prove that for any compact object A in C and any collection ( B i ) i of objects of D , Hom( A, ⊕ i G ( B i )) ∼ −→ Hom(

A, G ( ⊕ i B i )). By compact-ness of A , Hom( A, ⊕ i G ( B i )) = ⊕ i Hom(

A, G ( B i )), by adjunction this is equal to ⊕ i Hom( F ( A ) , B i ),since F ( A ) is compact this is equal to Hom( F ( A ) , ⊕ i B i ), and by adjunction this is Hom( A, G ( ⊕ i B i )). (cid:3) Remark

IV.2.21 . (i) In fact, outside of the overconvergence condition (a) in Deﬁnition IV.2.1, the property of Propo-sition IV.2.19 characterizes locally acyclic complexes under the assumption that locally on X theexists an integer N such that for any U → X quasicompact separated ´etale the cohomologicaldimension of X ´et is bounded by N . More precisely, if j : U → X is separated ´etale with f ◦ j quasicompact then R Hom Λ ( R ( f ◦ j ) ! A, B ) = R Γ( U, R H om Λ ( A, Rf ! B )). Thus, if for all B one has D X/S ( A ) ⊗ L Λ f ∗ B ∼ −→ R H om Λ ( A, Rf ! B ) then R ( f ◦ j ) ! A is compact since R Γ( U, − ) commutes witharbitrary direct sums.(ii) Outside of the overconvergence condition (a) in Deﬁnition IV.2.1, the property of Proposi-tion IV.2.19 universally on S characterize universally locally acyclic objects. In fact, using [ Sch17a ,Proposition 20.13], the constructibility property is reduced to the case when the base is strictlytotally disconnected, in which case we can apply point (1).Finally, let us note that all the previous results extend to the setting where the base S is ageneral small v-stack, taking the following deﬁnition. Definition

IV.2.22 . Let f : X → S be a map of small v-stacks that is compactiﬁable andrepresentable in locally spatial diamonds with locally dim . trg f < ∞ . Let A ∈ D ´et ( X, Λ) . Then A is f -universally locally acyclic if for any map S (cid:48) → S from a locally spatial diamond S (cid:48) withpullback f (cid:48) : X (cid:48) = X × S S (cid:48) → S (cid:48) , the complex A | X (cid:48) ∈ D ´et ( X (cid:48) , Λ) is f (cid:48) -locally acyclic.

22 IV. GEOMETRY OF DIAMONDS

IV.2.3.3.

Dualizability.

From the previous two propositions, we can deduce an analogue of arecent result of Lu-Zheng, [

LZ20 ], characterizing universal local acyclicity in terms of dualizabilityin a certain monoidal category. We actually propose a diﬀerent such characterization closer to howdualizability will appear later in the discussion of geometric Satake. In terms of applications toabstract properties of universal local acyclicity, such as its preservation by Verdier duality, it leadsto the same results.Fix a base small v-stack S , and a coeﬃcient ring Λ (killed by some integer prime to p ). Wedeﬁne a 2-category C S as follows. The objects of C S are maps f : X → S of small v-stacks thatare compactiﬁable, representable in locally spatial diamonds, with locally dim . trg f < ∞ . For any X, Y ∈ C S , the category of maps Fun C S ( X, Y ) is the category D ´et ( X × S Y, Λ). Note that any such A ∈ D ´et ( X × S Y, Λ) deﬁnes in particular a functor D ´et ( X, Λ) → D ´et ( Y, Λ) : B (cid:55)→ Rπ ( A ⊗ L Λ π ∗ B )with kernel A , where π : X × S Y → X , π : X × S Y → Y are the two projections. The compositionin C S is now deﬁned to be compatible with this association. More precisely, the compositionFun C S ( X, Y ) × Fun C S ( Y, Z ) → Fun C S ( X, Z )is deﬁned to be the functor D ´et ( X × S Y, Λ) × D ´et ( Y × S Z, Λ) → D ´et ( X × S Z, Λ) : (

A, B ) (cid:55)→ A (cid:63) B = Rπ ( π ∗ A ⊗ L Λ π ∗ B )where π ij denotes the various projections on X × S Y × S Z . It follows from the projection formulathat this indeed deﬁnes a 2-category C S . The identity morphism is given by R ∆ ! Λ = R ∆ ∗ Λ ∈ D ´et ( X × S X, Λ), where ∆ :

X (cid:44) → X × S X is the diagonal. We note that C S is naturally equivalentto C op S . Indeed, D ´et ( X × S Y, Λ) is invariant under switching X and Y , and the deﬁnition ofcomposition (and coherences) is compatible with this switch.Recall that in any 2-category C , there is a notion of adjoints. Namely, a morphism f : X → Y is a left adjoint of g : Y → X if there are maps α : id X → gf and β : f g → id Y such that thecomposites f fα −−→ f gf βf −→ f, g αg −→ gf g gβ −→ g are the identity. If a right adjoint g of f exists, it is (together with the accompanying data)moreover unique up to unique isomorphism. As is clear from the deﬁnition, any functor of 2-categories preserves adjunctions. In particular, this applies to pullback functors C S → C S (cid:48) for maps S (cid:48) → S of small v-stacks, or to the functor from C S to triangulated categories taking X to D ´et ( X, Λ)and A ∈ Fun C S ( X, Y ) to the functor Rπ ( A ⊗ L Λ π ∗ − ) with kernel A . Theorem

IV.2.23 . Let S be a small v-stack and X ∈ C S , and A ∈ D ´et ( X, Λ) . The followingconditions are equivalent. (i) The complex A is f -universally locally acyclic. (ii) The natural map p ∗ D X/S ( A ) ⊗ L Λ p ∗ A → R H om Λ ( p ∗ A, Rp !2 A ) is an isomorphism, where p , p : X × S X → X are the two projections. (iii) The object A ∈ Fun C S ( X, S ) is a left adjoint in C S . In that case, its right adjoint is given by D X/S ( A ) ∈ D ´et ( X, Λ) = Fun C S ( S, X ) . V.2. UNIVERSALLY LOCALLY ACYCLIC SHEAVES 123

Proof.

That (i) implies (ii) follows from Proposition IV.2.15 and Proposition IV.2.19. For (ii)implies (iii), we claim that A ∈ Fun C S ( X, S ) is indeed a left adjoint of D X/S ( A ) ∈ Fun C S ( A ). Thecomposites are given by A (cid:63) D X/S ( A ) = Rf ! ( D X/S ( A ) ⊗ L Λ A ) ∈ D ´et ( S, Λ) = Fun C S ( S, S )and D X/S ( A ) (cid:63) A = p ∗ A ⊗ L Λ p ∗ D X/S ( A ) ∈ D ´et ( X × S X, Λ) = Fun C S ( X, X ) . Then we take β : A (cid:63) D X/S ( A ) → id S to be given by the map Rf ! ( D X/S ( A ) ⊗ L Λ A ) → Λ adjointto the map D X/S ( A ) ⊗ L Λ A → Rf ! Λ which is just the tautological pairing. On the other hand, for α : id X → D X/S ( A ) (cid:63) A , we have to produce a map R ∆ ! Λ → p ∗ A ⊗ L Λ p ∗ D X/S ( A ) . Using (ii), the right-hand side is naturally isomorphic to R H om Λ ( p ∗ A, Rp !2 A ). Now maps from R ∆ ! Λ are adjoint to sections of R ∆ ! R H om Λ ( p ∗ A, Rp !2 A ) ∼ = R H om Λ ( A, A )(using [

Sch17a , Theorem 1.8 (v)]), which has the natural identity section. It remains to prove thatcertain composites are the identity. This follows from a straightforward diagram chase.Finally, it remains to prove that (iii) implies (i). We can assume that S is strictly totallydisconnected. It follows that the functor Rf ! ( A ⊗ L Λ − ) admits a right adjoint that commutes withall colimits. This implies that condition (b) in Deﬁnition IV.2.1 is satisﬁed. In fact, more preciselywe see that the right adjoint R H om Λ ( A, Rf ! − ) is given by A (cid:48) ⊗ L Λ f ∗ − for some A (cid:48) ∈ D ´et ( X, Λ), andby using the self-duality of C op S , we also see that R H om Λ ( A (cid:48) , Rf ! − ) is given by A ⊗ L Λ f ∗ − . Appliedto the constant sheaf, this shows in particular that A ∼ = R H om Λ ( A (cid:48) , Rf ! Λ) is a Verdier dual.For condition (a), we can assume that S = Spa( C, C + ) and reduce to checking overconvergencealong sections s : S → X . In fact, using part (2) of Remark IV.2.3, let j : Spa( C, O C ) → Spa(

C, C + ) be the pro-open immersion, with pullback j X : X × Spa(

C,C + ) Spa( C, O C ) → X , and f η : X × Spa(

C,C + ) Spa( C, O C ) → Spa( C, O C ) the restriction of f . To see the overconvergence, it isenough to see that A ∼ = Rj X ∗ A for some A . But A ∼ = R H om Λ ( A (cid:48) , Rf ! Λ) ∼ = R H om Λ ( A (cid:48) , Rf ! Rj ∗ Λ) ∼ = R H om Λ ( A (cid:48) , Rj X ∗ Rf ! η Λ) ∼ = Rj X ∗ R H om Λ ( j ∗ X A (cid:48) , Rf ! η Λ) , giving the desired overconvergence. (cid:3) Before moving on, let us observe the following relative variant.

Proposition

IV.2.24 . Let S be a small v-stack and X, Y ∈ C S . If Y /S is proper and A ∈ Fun C S ( X, Y ) = D ´et ( X × S Y, Λ) is p -universally locally acyclic, then it is a left adjoint and theright adjoint is given by D X × S Y/Y ( A ) ∈ D ´et ( X × S Y, Λ) ∼ = D ´et ( Y × S X, Λ) = Fun C S ( Y, X ) . The assumption that

Y /S is proper is important here. Already if X = S and A = Λ ∈ D ´et ( Y, Λ),which is always id Y -universally locally acyclic, being a left adjoint in C S implies that there is some B ∈ D ´et ( Y, Λ) for which Rf ∗ ∼ = Rf ! ( B ⊗ L Λ − ).

24 IV. GEOMETRY OF DIAMONDS

Proof.

We need to produce the maps α and β again. Let us give the construction of α , whichis the harder part. First, using the various projections π ij on X × S Y × S X , we have D X × S Y/Y ( A ) (cid:63) A ∼ = Rπ ( π ∗ D X × S Y/Y ( A ) ⊗ L Λ π ∗ A ) ∼ = Rπ ∗ R H om Λ ( π ∗ A, Rπ !23 A )using that A is p -universally locally acyclic, and properness of π (which is a base change of Y → S ). Now giving a map R ∆ ! Λ → D X × S Y/Y ( A ) (cid:63) A , for ∆ = ∆ X/S , amounts to ﬁnding a sectionof R ∆ ! Rπ ∗ R H om Λ ( π ∗ A, Rπ !23 A ) ∼ = Rp ∗ R ∆ ! X × S Y/Y R H om Λ ( π ∗ A, Rπ !23 A ) ∼ = Rp ∗ R H om Λ ( A, A ) , where we can take the identity. (cid:3) Theorem IV.2.23 has the following notable consequences.

Corollary

IV.2.25 . Let f : X → S be a compactiﬁable map of locally spatial diamonds withlocally dim . trg f < ∞ and let A ∈ D ´et ( X, Λ) be f -universally locally acyclic. Then D X/S ( A ) isagain f -universally locally acyclic, and the biduality map A → D X/S ( D X/S ( A )) is an isomorphism.If f i : X i → S for i = 1 , are compactiﬁable maps of small v-stacks that are representable inlocally spatial diamonds with locally dim . trg f i < ∞ and A i ∈ D ´et ( X i , Λ) are f i -universally locallyacyclic, then also A (cid:2) A ∈ D ´et ( X × S X , Λ) is f × S f -universally locally acyclic, and the naturalmap D X /S ( A ) (cid:2) D X /S ( A ) → D X × S X /S ( A (cid:2) A ) is an isomorphism. Proof.

By Theorem IV.2.23, the object D X/S ( A ) ∈ Fun C S ( S, X ) is a right adjoint of A ∈ Fun C S ( X, S ). But C S ∼ = C op S ; under this equivalence, this means that D X/S ( A ) ∈ Fun C S ( S, X ) is aleft adjoint of A ∈ Fun C S ( S, X ). Thus, applying Theorem IV.2.23 again, the result follows.For the second statement, note that A i ∈ D ´et ( X i , Λ) deﬁne left adjoints, hence so does A (cid:63) A = A (cid:2) A ∈ D ´et ( X × S X , Λ) = Fun C S ( X × S X , S ) . Its right adjoint is the similar composition, giving the claim. (cid:3)

The ﬁnal statement admits the following generalization concerning “compositions” of univer-sally locally acyclic sheaves.

Proposition

IV.2.26 . Let g : Y → X and f : X → S be compactiﬁable maps of smallv-stacks representable in locally spatial diamonds with locally dim . trg f, dim . trg g < ∞ , and let A ∈ D ´et ( X, Λ) be f -universally acyclic and B ∈ D ´et ( Y, Λ) be g -universally locally acyclic. Then g ∗ A ⊗ L Λ B is f ◦ g -universally locally acyclic, and there is natural isomorphism D Y/S ( g ∗ A ⊗ L Λ B ) ∼ = g ∗ D X/S ( A ) ⊗ L Λ D Y/X ( B ) . V.2. UNIVERSALLY LOCALLY ACYCLIC SHEAVES 125

Proof.

It is easy to see that condition (a) of being locally acyclic holds universally, so it suﬃcesto identify the functor R H om Λ ( g ∗ A ⊗ L Λ B, R ( f ◦ g ) ! − ). We compute: R H om Λ ( g ∗ A ⊗ L Λ B, R ( f ◦ g ) ! − ) = R H om Λ ( B, R H om Λ ( g ∗ A, Rg ! Rf ! − ))= R H om Λ ( B, Rg ! R H om Λ ( A, Rf ! − ))= D Y/X ( B ) ⊗ L Λ g ∗ R H om( A, Rf ! − )= D Y/X ( B ) ⊗ L Λ g ∗ D X/S ( A ) ⊗ L Λ g ∗ f ∗ − , implying that it commutes with colimits, hence its left adjoint preserves perfect-constructible com-plexes (after reduction to strictly totally disconnected S and X and Y spatial). Moreover, evaluatingthis functor at Λ gives the identiﬁcation of the Verdier dual. (cid:3) Let us also note another corollary of Theorem IV.2.23 concerning retracts.

Corollary

IV.2.27 . Let f : X → S and g : Y → S be maps of small v-stacks that arecompactiﬁable and representable in locally spatial diamonds with locally dim . trg f, dim . trg g < ∞ .Assume that f is a retract of g over S , i.e. there are maps i : X → Y , r : Y → X over S such that ri = id X . If Λ is g -universally locally acyclic, then Λ is f -universally locally acyclic. Proof.

One can check this directly from the deﬁnitions, or note that the map in C S given byΛ ∈ D ´et ( X, Λ) = Fun C S ( X, S ) is a retract of the map given by Λ ∈ D ´et ( Y, Λ) = Fun C S ( Y, S ), fromwhich one can easily obtain adjointness. (cid:3)

Moreover, in some cases the converse to Proposition IV.2.11 holds.

Proposition

IV.2.28 . Let g : Y → X , f : X → S be maps of locally spatial diamonds where g isproper and quasi-pro-´etale and f is compactiﬁable and locally dim . trg f < ∞ . Then A ∈ D ´et ( Y, Λ) is f ◦ g -universally locally acyclic if and only if Rg ∗ A ∈ D ´et ( X, Λ) is f -universally locally acyclic. Proof.

One direction is given by Proposition IV.2.11. For the converse, assume that Rg ∗ A is f -universally locally acylic. To see that A is h = f ◦ g -universally locally acyclic, it suﬃces byTheorem IV.2.23 that the map p ∗ ,Y R H om( A, Rh ! Λ) ⊗ L Λ p ∗ ,Y A → R H om( p ∗ ,Y A, Rp !2 ,Y A )in D ´et ( Y × S Y, Λ) is an isomorphism, where p ,Y , p ,Y : Y × S Y → Y are the two projections. As g × S g : Y × S Y → X × S X is proper and quasi-pro-´etale, pushforward along g × S g is conservative:By testing on stalks, this follows from the observation that for a proﬁnite set T , the global sectionsfunctor R Γ( T, − ) is conservative on D ( T, Λ) (as one can write any stalk as a ﬁltered colimit offunctors that are direct summands of the global sections functor). Applying R ( g × S g ) ∗ = R ( g × S g ) ! to the displayed map, we get the map p ∗ ,X R H om( Rg ∗ A, Rf ! Λ) ⊗ L Λ p ∗ ,Y Rg ∗ A → R H om( p ∗ ,X Rg ∗ A, Rp !2 ,X Rg ∗ A )where p ,X , p ,X : X × S X → X are the two projections. This is an isomorphism precisely when Rg ∗ A is f -universally locally acyclic. (cid:3) The following corollary shows that smooth base change generalizes to universally locally acyclicmaps. The general version of this corollary was suggested by David Hansen.

26 IV. GEOMETRY OF DIAMONDS

Corollary

IV.2.29 (ULA base change) . Consider a cartesian diagram of small v-stacks X (cid:48) (cid:101) g (cid:47) (cid:47) f (cid:48) (cid:15) (cid:15) X f (cid:15) (cid:15) S (cid:48) g (cid:47) (cid:47) S with f representable in locally spatial diamonds, compactiﬁable, locally dim . trg f < ∞ . Assumethat Λ is f -universally locally acyclic. Then the base change map f ∗ Rg ∗ A → R (cid:101) g ∗ f (cid:48)∗ A is an isomorphism. More generally, if B ∈ D ´et ( X, Λ) is f -universally locally acyclic and A ∈ D ´et ( S (cid:48) , Λ) , then ( f ∗ Rg ∗ A ) ⊗ L Λ B ∼ −→ R (cid:101) g ∗ ( f (cid:48)∗ A ⊗ L Λ (cid:101) g ∗ B ) . Proof.

We apply Proposition IV.2.19 to the universally locally acyclic D X/S ( B ), so that byCorollary IV.2.25, we get f ∗ Rg ∗ A ⊗ L Λ B ∼ = R H om Λ ( D X/S ( B ) , Rf ! Rg ∗ A ) . By [

Sch17a , Theorem 1.9 (iii)], Rf ! Rg ∗ A ∼ = R (cid:101) g ∗ Rf (cid:48) ! A , and then one can rewrite further as R H om Λ ( D X/S ( B ) , R (cid:101) g ∗ Rf (cid:48) ! A ) ∼ = R (cid:101) g ∗ R H om Λ ( (cid:101) g ∗ D X/S ( B ) , Rf (cid:48) ! A ) . Now another application of Proposition IV.2.19 gives the result. (cid:3)

Finally let us note the following consequence of Theorem IV.2.23 and [

LZ20 ]. Proposition

IV.2.30 . Let K be a complete non-archimedean ﬁeld with residue characteristic p , f : X → S a separated morphism of K -schemes locally of ﬁnite type, and A ∈ D bc ( X, Λ) . Then A is f -universally acyclic if and only if its analytiﬁcation A ad is f ad, ♦ -universally locally acyclic,where f ad, ♦ : X ad, ♦ → S ad, ♦ . Proof.

The criterion of Theorem IV.2.23 (2) applies similarly in the algebraic case by [

LZ20 ],and all operations are compatible with passing to analytic adic spaces (and diamonds). (cid:3)

For example, if S = Spec K then any A is f -universally locally acyclic and thus A ad is f ad, ♦ -universally acyclic. This gives plenty of examples of ULA sheaves. IV.2.4. Local acyclicity for morphisms of Artin v-stacks.

Using the descent resultsRemark IV.2.2 and Proposition IV.2.13, one can extend the previous deﬁnition and results to thecase of maps of Artin v-stacks as follows.

Definition

IV.2.31 . Let f : X → S be a map of Artin v-stacks and assume that there issome separated, representable in locally spatial diamonds, and cohomologically smooth surjection g : U → X from a locally spatial diamond U such that f ◦ g : U → S is compactiﬁable with locally dim . trg( f ◦ g ) < ∞ . Then A ∈ D ´et ( X, Λ) is f -universally locally acyclic if g ∗ A is f ◦ g -universallylocally acyclic. V.2. UNIVERSALLY LOCALLY ACYCLIC SHEAVES 127

All previous results concerning universally locally acyclic complexes also hold in this setting(assuming that the relevant operations are deﬁned in the case of interest – we did not deﬁne Rf ! and Rf ! for general stacky maps), and follow by the reduction to the case when S and X are locallyspatial diamonds. In particular, the characterization in terms of dualizability gives the following. Proposition

IV.2.32 . Let f : X → S be a cohomologically smooth map of Artin v-stacks, andlet A ∈ D ´et ( X, Λ) . Consider X × S X with its two projections p , p : X × S X → X . Then A is f -universally locally acyclic if and only if the natural map p ∗ R H om Λ ( A, Λ) ⊗ L Λ p ∗ A → R H om Λ ( p ∗ A, p ∗ A ) is an isomorphism. Proof.

Taking a chart for S , we can assume that S is a locally spatial diamond, and thentaking a presentation for X we can assume that also X is a locally spatial diamond, noting thatthe condition commutes with smooth base change. In that case, replacing some occurences of p ∗ by Rp !2 using cohomological smoothness, the result follows from Theorem IV.2.23. (cid:3) There is a simple characterization of (cid:96) -cohomological smoothness in terms of universal localacyclicity.

Proposition

IV.2.33 . Let f : X → S be a compactiﬁable map of v-stacks that is representablein locally spatial diamonds with locally dim . trg f < ∞ . Then f is (cid:96) -cohomologically smooth if andonly if F (cid:96) is f -universally locally acyclic and its Verdier dual Rf ! F (cid:96) is invertible. Note that in checking whether F (cid:96) is f -universally locally acyclic, condition (a) of Deﬁni-tion IV.2.1 is automatic. Also, by Theorem IV.2.23, the condition that F (cid:96) is f -universally locallyacyclic is equivalent to the condition that the natural map p ∗ Rf ! F (cid:96) → Rp !2 F (cid:96) is an isomorphism, where p , p : X × S X → X are the two projections. Thus, f is (cid:96) -cohomologicallysmooth if and only if Rf ! F (cid:96) is invertible and its formation commutes with any base change . Proof.

The conditions are clearly necessary. For the converse, we may assume that S is strictlytotally disconnected. By Proposition IV.2.19, the natural transformation of functors Rf ! Λ ⊗ L Λ f ∗ → Rf ! is an equivalence. As Rf ! Λ is assumed to be invertible (it commutes with base change by Proposi-tion IV.2.15), this shows that the condition of [

Sch17a , Deﬁnition 23.8] is satisﬁed. (cid:3)

In particular, we can resolve a question from [

Sch17a ]. Corollary

IV.2.34 . The map f : Spd O E → Spd F q is (cid:96) -cohomologically smooth for all (cid:96) (cid:54) = p . Proof.

This is clear if E is of equal characteristic, so assume that E is p -adic. First, we provethat F (cid:96) is f -ULA. This follows from f (cid:48) : Spd O (cid:101) E ∼ = Spd F q [[ t /p ∞ ]] → Spd F q being (cid:96) -cohomologicallysmooth, where (cid:101) E/E is some totally ramiﬁed Z p -extension, by the argument of the proof of [ Sch17a ,Proposition 24.3] (in essence, the compactly supported pushforward for any base change of f are

28 IV. GEOMETRY OF DIAMONDS the Z p -invariants inside the compactly supported pushforward for the corresponding base changeof f (cid:48) , so constructibility of the latter implies constructibility of the former).It remains to show that Rf ! F (cid:96) is invertible. If j : Spd E → Spd O E is the open immersion withcomplement i : Spd F q → Spd O E , we have Ri ! Rf ! F (cid:96) = F (cid:96) by transitivity, and j ∗ Rf ! F (cid:96) ∼ = F (cid:96) (1)[2]by [ Sch17a , Proposition 24.5], so we get a triangle i ∗ F (cid:96) → Rf ! F (cid:96) → Rj ∗ F (cid:96) (1)[2] . Using this, one computes Rf ! F (cid:96) ∼ = F (cid:96) (1)[2], as desired. (cid:3) IV.3. Formal smoothnessIV.3.1. Deﬁnition.

A key step in the proof of Theorem IV.4.2, the Jacobian criterion ofcohomological smoothness, is the following notion of formal smoothness.

Definition

IV.3.1 . Let f : Y → X be a map of v-stacks. Then f is formally smooth if forany aﬃnoid perfectoid space S of characteristic p with a Zariski closed subspace S ⊂ S , and anycommutative diagram S g (cid:47) (cid:47) (cid:15) (cid:15) Y f (cid:15) (cid:15) S h (cid:47) (cid:47) X, there is some ´etale map S (cid:48) → S containing S in its image and a map g : S (cid:48) → Y ﬁtting in acommutative diagram S (cid:48) × S S (cid:47) (cid:47) (cid:15) (cid:15) S g (cid:47) (cid:47) (cid:15) (cid:15) Y f (cid:15) (cid:15) S (cid:48) g (cid:53) (cid:53) (cid:47) (cid:47) S h (cid:47) (cid:47) X. This kind of formal smoothness is closely related to the notion of absolute neighborhood retracts ([ Bor67 ], [

Dol80 ]). In fact, suppose Y → X is formally smooth with Y and X aﬃnoid perfectoid.Choose a Zariski closed embedding Y (cid:44) → B IX for some set I . Then there exists an ´etale neighborhood U → B IX of Y such that the closed embedding i : Y × B IX U (cid:44) → U admits a retraction r : U → Y × B IX U , r ◦ i = Id . Thus, Y /X is a retract of an (´etale) neighborhoodinside a ball /X . IV.3.2. Examples and basic properties.

We will see that formally smooth morphismsshare analogous properties to cohomologically smooth morphisms. Let’s begin with the followingobservations:(i) The composite of two formally smooth morphisms is formally smooth,(ii) The formally smooth property is stable under pullback: if Y → X is formally smooth and X (cid:48) → X is any map then Y × X X (cid:48) → X (cid:48) is formally smooth. V.3. FORMAL SMOOTHNESS 129 (iii) Separated ´etale maps are formally smooth.(iv) For morphisms of locally spatial diamonds, formal smoothness is ´etale local on the source andthe target.Let us observe that, like cohomologically smooth morphisms, formally smooth morphisms areuniversally open.

Proposition

IV.3.2 . Formally smooth maps are universally open.

Proof.

Let Y → X be formally smooth. We can suppose X is aﬃnoid perfectoid. Since anyopen subsheaf of Y is formally smooth over X we are reduced to prove that the image of Y → X is open. Let S → Y be a morphism with S aﬃnoid perfectoid. Choose a Zariski closed embedding S (cid:44) → B IX for some set I . The formal smoothness assumption implies that there exists an ´etaleneighborhood U → B IX of S ⊂ B IX such that S × B IX U → S → Y extends to a map U → Y ; inparticular, the image of S → Y → X is contained in the image of U → Y → X , and it suﬃces toprove that the latter is open. We can suppose U is quasicompact and separated over B IX . Writing B IX = lim ←− J B JX where J goes through the set of ﬁnite subsets of I , there exists a some J ⊂ I ﬁniteand V → B JX such that U → B IX is the pullback of V → B JX via the projection B IX → B JX , cf.[ Sch17a , Proposition 6.4]. Since B IX → B JX is a v-cover,Im( U → X ) = Im( V → X ) . Now, using that V → B JX → X is open, since cohomologically smooth for example, this is an opensubset of X . (cid:3) Let us begin with some concrete examples. In the following, B → ∗ is the v-sheaf O + on Perf k and A → ∗ is the v-sheaf O . Proposition

IV.3.3 . The morphisms B → ∗ , A → ∗ and Spd O E → ∗ are formally smooth. Proof.

Let S = Spa( R , R +0 ) (cid:44) → S = Spa( R, R + ) be a Zariski closed embedding of aﬃnoidperfectoid spaces. Then R → R is surjective, which immediately shows that A → ∗ is formallysmooth. The case of B → ∗ follows as B ⊂ A is open. For Spd O E , note that any untilt of S can be given by some element ξ ∈ W O E ( R +0 ) of the form ξ = π + (cid:80) ∞ n =0 π i [ r i, ] where all r i ∈ R ◦◦ .But R ◦◦ → R ◦◦ is surjective (cf. the discussion after [ Sch17a , Deﬁnition 5.7]), so one can lift all r i, ∈ R ◦◦ to r i ∈ R ◦◦ , and then ξ = π + (cid:80) ∞ n =0 π i [ r i ] deﬁnes an untilt of S over O E lifting the givenone on S . (cid:3) Corollary

IV.3.4 . Is f : Y → X is a smooth morphism of analytic adic spaces over Z p then f ♦ : Y ♦ → X ♦ is formally smooth. Proof.

Any smooth morphism is locally ´etale over a ﬁnite-dimensional ball. (cid:3)

Let us remark the following.

Proposition

IV.3.5 . If f : Y → X is a formally smooth and surjective map of v-stacks, then f is surjective as a map of ´etale stacks. Equivalently, in case X is a perfectoid space, the map f splits over an ´etale cover of X .

30 IV. GEOMETRY OF DIAMONDS

Proof.

We can suppose X is aﬃnoid. There exists a surjective morphism X (cid:48) → X with X (cid:48) aﬃnoid perfectoid and a section s : X (cid:48) → Y of Y → X over X (cid:48) . Let us choose a Zariski closedembedding X (cid:48) (cid:44) → B IX . Applying the formal smoothness property we deduce there is an ´etaleneighborhood U → B IX of X (cid:48) ⊂ B IX and a section over U of Y → X . It thus suﬃces to see that U → X admits a section over an ´etale cover of X . As in the proof of Proposition IV.3.2 there existsa ﬁnite subset J ⊂ I , and quasicompact ´etale map V → B JX such that U → B IX is the pullbackof V → B JX via the projection B IX → B JX . This reduces us to the case I is ﬁnite. We may alsoreplace V by its image in B JX . At geometric points, the splitting follows from [ Sch17a , Lemma9.5]. Approximating a section over a geometric point over an ´etale neighborhood then gives thedesired splitting on an ´etale cover. (cid:3)

According to [

Sch17a , Proposition 23.13] cohomological smoothness is cohomologically smoothlocal on the source. The same holds for formally smooth morphisms.

Corollary

IV.3.6 . Let f : Y → X be a morphism of v-stacks. Suppose there exists a v-surjective formally smooth morphism of v-stacks g : Y (cid:48) → Y such that f ◦ g is formally smooth.Then f is formally smooth. Proof.

Given a test diagram g : S → Y , h : S → X as in Deﬁnition IV.3.1, we can ﬁrstlift S → Y ´etale locally to Y (cid:48) by Proposition IV.3.5, and the required ´etale neighborhoods lift to S by [ Sch17a , Proposition 6.4] applied to S as the intersection of all open neighborhoods in S .Thus, the diagram can be lifted to a similar test diagram for Y (cid:48) → X , which admits a solution byassumption. (cid:3) Let us remark the following.

Proposition

IV.3.7 . The stack

Bun G → ∗ is formally smooth. Proof.

Let S = Spa( R , R +0 ) ⊂ S = Spa( R, R + ) be a Zariski closed immersion of aﬃnoidperfectoid spaces over Spd k , and ﬁx a pseudouniformizer (cid:36) ∈ R . Let E be a G -bundle on X S .Pick any geometric point Spa( C, C + ) → S ; we intend to ﬁnd an ´etale neighborhood U → S ofSpa( C, C + ) in S such that the G -bundle over U × S S extends to U .Note that the pullback of E to Y C, [1 ,q ] is a trivial G -bundle, by Theorem III.2.2. From [ GR03 ,Proposition 5.4.21] (applied with R = lim −→ V O + ( Y V, [1 ,q ] ), t = π and I = 0, where V → S runs over´etale neighborhoods of Spa( C, C + ) in S ; all of these lift to S ) it follows that after passing to an´etale neighborhood as above, we can assume that the pullback of E to Y S , [1 ,q ] is a trivial G -bundle.In that case, E is given by some matrix A ∈ G ( B R , [1 , ) encoding the descent. Applying [ GR03 ,Proposition 5.4.21] again, with R = lim −→ S ⊂ U ⊂ S O + ( Y U, [1 , ) , t = π, I = ker( R → lim −→ U O + ( Y S , [1 , )) , then shows that we may lift A into a neighborhood, as desired. (cid:3) The following is the analog of Proposition II.3.5 (iii).

V.4. A JACOBIAN CRITERION 131

Proposition

IV.3.8 . Let S be a perfectoid space and let [ E → E ] be a map of vector bundleson X S such that E is everywhere of positive Harder–Narasimhan slopes, and E is everywhere ofnegative Harder–Narasimhan slopes. Then BC ([ E → E ]) → S is formally smooth. Proof.

Using the exact sequence0 → BC ( E ) → BC ([ E → E ]) → BC ( E [1]) → BC ( E ) and BC ( E [1]). For E = E ,we can use Corollary II.3.3 to choose ´etale locally on S a short exact sequence0 → G → O X S ( r ) m → E → G is ﬁberwise on S semistable of positive slope. Moreover, by Proposition II.3.4 (iii), onecan also ensure that H ( X S (cid:48) , G| X S (cid:48) ) = 0 for all aﬃnoid perfectoid spaces S (cid:48) → S . In particular, if S ⊂ S is a Zariski closed immersion of aﬃnoid perfectoid spaces, the map O X S ( r ) m ( S ) → E ( S )is surjective, and we can replace E by O X S ( r ) m . But then Proposition II.2.5 (iv) shows that thisBanach–Colmez space is representable by a perfectoid open unit disc, which is formally smooth.For E = E , we can use Theorem II.2.6 to ﬁnd a short exact sequence0 → E → O X S ( d ) m → G → d, m >

0, and this induces an exact sequence0 → BC ( G ) → BC ( O X S ( d ) m ) → BC ( E [1]) → BC ( O X S ( d ) m ) → BC ( E [1])is formally smooth (as ´etale locally surjective and its ﬁbre BC ( G ) is formally smooth). We concludeby Corollary IV.3.6. (cid:3) IV.4. A Jacobian criterion

The goal of this section is to prove that certain geometrically deﬁned diamonds are cohomo-logically smooth when one expects them to be. We regard this result as the most profound in thetheory of diamonds so far: While we cannot control much of the geometry of these diamonds, inparticular we have no way to relate them to (perfectoid) balls in any reasonable way, we can stillprove relative Poincar´e duality for them. The spaces considered below also appear quite naturallyin a variety of contexts, so we expect the result to have many applications.The setup is the following. Let S be a perfectoid space and let Z → X S be a smooth map ofsous-perfectoid adic spaces — deﬁning this concept of smoothness will be done in a ﬁrst subsection,but it is essentially just a family of smooth rigid spaces over X S , in the usual sense. One can thenconsider the v-sheaf M Z of sections of Z → X S , sending any perfectoid space S (cid:48) → S to the setof maps X S (cid:48) → Z lifting X S (cid:48) → X S . In general, we cannot prove that M Z is a locally spatialdiamond, but this turns out to be true when Z is quasiprojective in the sense that it is a Zariskiclosed subspace of an open subset of (the adic space) P nX S for some n ≥ M Z → S is not (cohomologically) smooth: If tangent spaces of M Z → S would exist, one would expect their ﬁbre over S (cid:48) → M Z , given by some section s : X S (cid:48) → Z ,to be given by H ( X S (cid:48) , s ∗ T Z/X S ), where T Z/X S is the tangent bundle of Z → X S ; and then anobstruction space would be given by H ( X S (cid:48) , s ∗ T Z/X S ). Thus, one can expect smoothness to hold

32 IV. GEOMETRY OF DIAMONDS only when H ( X S (cid:48) , s ∗ T Z/X S ) vanishes. This holds true, locally on S (cid:48) , if all slopes of s ∗ T Z/X S arepositive (by Proposition II.3.4 (iii)), suggesting the following deﬁnition. Definition

IV.4.1 . Let M sm Z ⊂ M Z be the open subfunctor of all sections s : X S (cid:48) → Z suchthat s ∗ T Z/X S has everywhere positive Harder–Narasimhan slopes. Roughly speaking, one expects M sm Z to look inﬁnitesimally like the Banach–Colmez space BC ( s ∗ T Z/X S ); these indeed are cohomologically smooth when all slopes are positive, by Propo-sition II.3.5 (iii). Unfortunately, we are unable to prove a direct relation of this sort; however, wewill be able to relate these spaces via a “deformation to the normal cone”.Our goal is to prove the following theorem. Theorem

IV.4.2 . Let S be a perfectoid space and let Z → X S be a smooth map of sous-perfectoid spaces such that Z admits a Zariski closed immersion into an open subset of (the adicspace) P nX S for some n ≥ . Then M Z is a locally spatial diamond, the map M Z → S is compact-iﬁable, and M sm Z → S is cohomologically smooth.Moreover, for a geometric point x : Spa C → M sm Z given by a map Spa C → S and a section s : X C → Z , the map M sm Z → S is at x of (cid:96) -dimension equal to the degree of s ∗ T Z/X S . Remark

IV.4.3 . The map M sm Z → S is a natural example of a map that is only locally of ﬁnitedimension, but not globally so (as there are many connected components of increasing dimension). Remark

IV.4.4 . In the “classical context” of algebraic curves the preceding theorem is thefollowing (easy) result. Let

X/k be a proper smooth curve and Z → X be quasi-projective smooth.Consider M Z the functor on k -schemes that sends S to morphisms s : X × k S → Z over X . This isrepresentable by a quasi-projective scheme over Spec( k ). Let M smZ be the open sub-scheme deﬁnedby the condition that if s : X × k S → Z is an S -point of M Z then the vector bundle s ∗ T Z/X hasno H ﬁberwise on S . Then M Z → Spec( k ) is smooth. Remark

IV.4.5 . Suppose that W is a smooth quasi-projective E -scheme. The moduli space M Z with Z = W × E X S classiﬁes morphisms X S → W i.e. M Z is a moduli of morphisms fromfamilies of Fargues–Fontaine curves to W . This is some kind of “Gromov–Witten” situation. Remark

IV.4.6 . We could have made the a priori weaker assumption that Z admits a Zariskiclosed immersion inside an open subset of P ( E ) where E is a vector bundle on X S . Nevertheless,since the result is local on S and we can suppose it is aﬃnoid perfectoid, and since when S isaﬃnoid perfectoid O X S (1) “is ample” i.e. there is a surjection O X S ( − N ) n (cid:16) E for N, n (cid:29)

0, thisassumption is equivalent to the one we made i.e. we can suppose E is free. Example

IV.4.7 (The Quot diamond) . Let E be a vector bundle on X S . We denote byQuot E −→ S the moduli space over S of locally free quotients of E . Fixing the rank of such a quotient, one seesthat Quot E is a ﬁnite disjoint union of spaces M Z with Z → X S a Grassmannian of quotients of E . This is thus representable in locally spatial diamonds, compactiﬁable, of locally ﬁnite dim . trg.Let Quot sm E ⊂ Quot E be the open subset parametrizing quotients u : E → F such that ﬁberwise,the greatest slope of ker u is strictly less than the smallest slope of F . According to Theorem IV.4.2this is cohomologically smooth over S . V.4. A JACOBIAN CRITERION 133

Fix an integer n ≥

1. For some N ∈ Z and r ∈ N ≥ , let Quot n, sm , ◦O ( N ) r be the open subset ofQuot sm O ( N ) r where the quotient has rank n and its slopes are greater than N . When N and r vary one constructs, as in the “classical case”, cohomologically smooth charts on Bun GL n usingQuot n, sm , ◦O ( N ) r . In fact, the morphism Quot n, sm , ◦O ( N ) r −→ Bun GL n given by the quotient of O ( N ) r is separated cohomologically smooth. When pulled back by amorphism S → Bun GL n with S perfectoid, this is an open subset of a positive Banach–Colmezspace.We will not use the Quot diamond in the following. In section V.3, using the Jacobian criterion,we will construct charts on Bun G for any G that are better suited to our needs. IV.4.1. Smooth maps of sous-perfectoid adic spaces.

We need some background aboutsmooth morphisms of adic spaces in non-noetherian settings. We choose the setting of sous-perfectoid adic spaces as deﬁned by Hansen-Kedlaya, [

HK20 ], cf. [

SW20 , Section 6.3]. Recall thatan adic space X is sous-perfectoid if it is analytic and admits an open cover by U = Spa( R, R + )where each R is a sous-perfectoid Tate algebra, meaning that there is some perfectoid R -algebra (cid:101) R such that R → (cid:101) R is a split injection in the category of topological R -modules.The class of sous-perfectoid rings R is stable under passage to rational localizations, ﬁnite ´etalemaps, and R (cid:104) T , . . . , T n (cid:105) . As smooth maps should be built from these basic examples, we can hopefor a good theory of smooth maps of sous-perfectoid spaces.Recall that a map f : Y → X of sous-perfectoid adic spaces is ´etale if locally on the source andtarget it can be written as an open immersion followed by a ﬁnite ´etale map. Definition

IV.4.8 . Let f : Y → X be a map of sous-perfectoid adic spaces. Then f is smoothif one can cover Y by open subsets V ⊂ Y such that there are ´etale maps V → B dX for some integer d ≥ . It can immediately be checked that analytiﬁcations of smooth schemes satisfy this condition.

Proposition

IV.4.9 . Let X = Spa( A, A + ) be an aﬃnoid sous-perfectoid adic space, and let f : Y → Spec A be a smooth map of schemes. Let f : Y → X be the analytiﬁcation of f : Y → Spec A , representing the functor taking Spa(

B, B + ) → Spa(

A, A + ) to the Spec B -valued points of Y → Spec A . Then f : Y → X is smooth. Proof.

Locally, f is the composite of an ´etale and the projection from aﬃne space. Thismeans that its analytiﬁcation is locally ´etale over the projection from the analytiﬁcation of aﬃnespace, which is a union of balls, giving the result. (cid:3) Let us analyze some basic properties of smooth maps of sous-perfectoid adic spaces.

Proposition

IV.4.10 . Let f : Y → X and g : Z → Y be maps of sous-perfectoid adicspaces. (i) The property of f being smooth is local on Y .

34 IV. GEOMETRY OF DIAMONDS (ii) If f and g are smooth, then so is f ◦ g : Z → X . (iii) If h : X (cid:48) → X is any map of sous-perfectoid adic spaces and f is smooth, then the ﬁbre product Y (cid:48) = Y × X X (cid:48) in adic spaces exists, is sous-perfectoid, and f (cid:48) : Y (cid:48) → X (cid:48) is smooth. (iv) The map f is universally open. (v) If f is surjective, then there is some ´etale cover X (cid:48) → X with a lift X (cid:48) → Y . Regarding part (1), we note that we will see in Proposition IV.4.18 that the property of f beingsmooth is in fact ´etale local on Y (and thus smooth local on X and Y , using (5)). Proof.

Part (1) is clear from the deﬁnition. For part (2), the composite is locally a compositeof an ´etale map, a projection from a ball, an ´etale map, and another projection from a ball; but wecan swap the two middle maps, and use that composites of ´etale maps are ´etale. Part (3) is againclear, by the stability properties of sous-perfectoid rings mentioned above. For part (4), it is nowenough to see that f is open, and we can assume that f is a composite of an ´etale map and theprojection from a ball, both of which are open. For part (5), using that f is open, we can worklocally on Y and thus assume again that it is a composite of an ´etale map and the projection froma ball; we can then replace Y by its open image in the ball. By [ Sch17a , Lemma 9.5], for anygeometric point Spa(

C, C + ) → X of X , we can ﬁnd a lift to Y . Writing the geometric point asthe limit of aﬃnoid ´etale neighborhoods, the map to Y ⊂ B dX can be approximated at some ﬁnitestage, and then openness of Y ensures that it will still lie in Y . This gives the desired ´etale coverof X over which f splits. (cid:3) Of course, the most important structure of a smooth morphism is its module of K¨ahler diﬀer-entials. Recall that if Y is sous-perfectoid, then one can deﬁne a stack (for the ´etale topology) ofvector bundles on Y , such that for Y = Spa( R, R + ) aﬃnoid with R sous-perfectoid, the categoryof vector bundles is equivalent to the category of ﬁnite projective R -modules; see [ KL15 ], [

SW20 ,Theorem 5.2.8, Proposition 6.3.4]. By deﬁnition, a vector bundle on Y is an O Y -module that islocally free of ﬁnite rank. Definition

IV.4.11 . Let f : Y → X be a smooth map of sous-perfectoid adic spaces, withdiagonal ∆ f : Y → Y × X Y . Let I Y/X ⊂ O Y × X Y be the ideal sheaf. Then Ω Y/X := I Y/X / I Y/X considered as O Y × X Y / I Y/X = O Y -module. It follows from the deﬁnition that there is a canonical O X -linear derivation d : O Y → Ω Y/X ,given by g (cid:55)→ g ⊗ − ⊗ g . Proposition

IV.4.12 . Let f : Y → X be a smooth map of sous-perfectoid adic spaces. Then Ω Y/X is a vector bundle on Y . There is a unique open and closed decomposition Y = Y (cid:116) Y (cid:116) . . . (cid:116) Y n such that Ω Y/X | Y d is of rank d for all d = 0 , . . . , n . In that case, for any open subset V ⊂ Y d withan ´etale map V → B d (cid:48) X , necessarily d (cid:48) = d . We will say that f is smooth of dimension d if Ω Y/X is of rank d . By the proposition, this isequivalent to asking that Y can be covered by open subsets V that admit ´etale maps V → B dX . Inparticular, f is smooth of dimension 0 if and only if it is ´etale. V.4. A JACOBIAN CRITERION 135

Proof.

It is enough to show that if f is a composite of an ´etale map Y → B dX with theprojection to X , then Ω Y/X is isomorphic to O dY . Indeed, this implies that Ω Y/X is a vector bundlein general, of the expected rank; and the decomposition into open and closed pieces is then a generalproperty of vector bundles.Let Y (cid:48) = B dX . Then Y × X Y → Y (cid:48) × X Y (cid:48) is ´etale, and the map Y → Y (cid:48) × Y (cid:48) × X Y (cid:48) ( Y × X Y ) isan open immersion (as the diagonal of the ´etale map Y → Y (cid:48) ). It follows that I Y/X is the pullbackof I Y (cid:48) /X (cid:48) . But Y (cid:48) (cid:44) → Y (cid:48) × X Y (cid:48) is of the formSpa( R (cid:104) T , . . . , T n (cid:105) , R + (cid:104) T , . . . , T n (cid:105) ) (cid:44) → Spa( R (cid:104) T (1)1 , . . . , T (1) n , T (2)1 , . . . , T (2) n (cid:105) , R + (cid:104) T (1)1 , . . . , T (2) n (cid:105) )if X = Spa( R, R + ), and the ideal sheaf is given by ( T (1)1 − T (2)1 , . . . , T (1) n − T (2) n ). This deﬁnes aregular sequence after any ´etale localization, by the lemma below. This gives the claim. (cid:3) Lemma

IV.4.13 . Let X = Spa( R (cid:104) T , . . . , T n (cid:105) , R + (cid:104) T , . . . , T n (cid:105) ) where R is a sous-perfectoid Tatering, let Y = Spa( S, S + ) where S is a sous-perfectoid Tate ring, and let f : Y → X be a smoothmap. Then T , . . . , T n deﬁne a regular sequence on S and ( T , . . . , T n ) S ⊂ S is a closed ideal. Proof.

By induction, one can reduce to the case n = 1. The claim can be checked locally, sowe can assume that Y is ´etale over B dX for some d ; replacing X by B dX , we can then assume that f is´etale. Let Y ⊂ Y be the base change to X = Spa( R, R + ) = V ( T ) ⊂ X ; then Y and Y × X X areboth ´etale over X and become isomorphic over X ⊂ X . By spreading of ´etale maps, this impliesthat they are isomorphic after base change to X (cid:48) = Spa( R (cid:104) T (cid:48) (cid:105) , R + (cid:104) T (cid:48) (cid:105) ) where T (cid:48) = (cid:36) n T for some n (and (cid:36) is a pseudouniformizer of R ). This easily implies the result. (cid:3) Locally around a section, any smooth space is a ball:

Lemma

IV.4.14 . Let f : Y → X be a smooth map of sous-perfectoid spaces with a section s : Spa( K, K + ) → Y for some point Spa(

K, K + ) → X . Then there are open neighborhoods U ⊂ X of Spa(

K, K + ) and V ⊂ Y of s (Spa( K, K + )) such that U ∼ = B dV . Proof.

We can assume that X and Y are aﬃnoid. If f is ´etale, then any section extends to asmall neighborhood (e.g. by [ Sch17a , Lemma 15.6, Lemma 12.17]), and any section is necessarily´etale and thus open, giving the result in that case. In general, we may work locally around the givensection, so we can assume that f is the composite of an ´etale map Y → B dX and the projection to X .Using the ´etale case already handled, we can assume that Y is an open subset of B dX . Any sectionSpa( K, K + ) → B dX has a coﬁnal system of neighborhoods that are small balls over open subsetsof X : The section is given by d elements T , . . . , T d ∈ K + , and after picking a pseudouniformizer (cid:36) and shrinking X , one can ﬁnd global sections T (cid:48) , . . . , T (cid:48) d of O + X ( X ) such that T i ≡ T (cid:48) i mod (cid:36) n .Then {| T (cid:48) | , . . . , | T (cid:48) d | ≤ | (cid:36) | n } is a small ball over X , and the intersection of all these is Spa( K, K + ).Thus, one of these neighborhoods is contained in Y , as desired. (cid:3) Proposition

IV.4.15 . Let f i : Y i → X , i = 1 , , be smooth maps of sous-perfectoid adic spaces,and let g : Y → Y be a map over X . (i) If g is smooth, then the sequence → g ∗ Ω Y /X → Ω Y /X → Ω Y /Y → is exact.

36 IV. GEOMETRY OF DIAMONDS (ii)

Conversely, if g ∗ Ω Y /X → Ω Y /X is a locally split injection, then g is smooth.In particular, if g ∗ Ω Y /X → Ω Y /X is an isomorphism, then g is ´etale. Proof.

Part (1) follows from a routine reduction to the case of projections from balls, whereit is clear. For part (2), we may assume that Y → B d X and Y → B d X are ´etale. It suﬃcesto see that the composite Y → Y → B d X is smooth, as g is the composite of its base change Y × B d X Y → Y with the section Y → Y × B d X Y of the ´etale map Y × B d X Y → Y ; any suchsection is automatically itself ´etale. Thus, we may assume that Y = B d X . Locally on Y , we mayﬁnd a projection g (cid:48) : B d X → B d − d X so that( g (cid:48)∗ Ω B d − d X /X ) | Y is an orthogonal complement of g ∗ Ω Y /X . Thus, we can assume that d = d =: d , and g ∗ Ω Y /X → Ω Y /X is an isomorphism.Our aim is now to prove that g : Y → Y = B dX is ´etale. We may assume that all of X , Y and Y are aﬃnoid. Passing to the ﬁbre over a point S = Spa( K, K + ) → X , this follows from a result ofHuber, [ Hub96 , Proposition 1.6.9 (iii)]. The resulting ´etale map Y ,S → Y ,S deforms uniquely toa quasicompact separated ´etale map Y (cid:48) ,U → Y ,U for a small enough neighborhood U ⊂ X of S , by[ Sch17a , Lemma 12.17]. Moreover, the map Y ,U → Y ,U lifts uniquely to Y ,U → Y (cid:48) ,U for U smallenough, by the same result. Replacing X by U , Y by Y ,U and Y by Y (cid:48) ,U , we can now assumethat g : Y → Y is a map between sous-perfectoid spaces smooth over X that is an isomorphismon one ﬁbre. It is enough to see that it is then an isomorphism in a neighborhood. To see this, wemay in fact work locally on Y .For this, we study Y ⊂ Y × X Y → Y : Here Y × X Y → Y is smooth, and Y ⊂ Y × X Y is locally the vanishing locus of d functions (as Y ⊂ Y × X Y is). Moreover, over ﬁbres lying overthe given point of X , the map Y → Y becomes an isomorphism, and in particular gives a sectionof Y × X Y → Y . By Lemma IV.4.14, after shrinking Y , we can assume that there is an openneighborhood V ⊂ Y × X Y such that V ∼ = B dY . Inside there, Y is (locally) given by the vanishingof d functions, and is only a point in one ﬁbre. Now the result follows from the next lemma, using Y in place of X . (cid:3) Lemma

IV.4.16 . Let X = Spa( A, A + ) be a sous-perfectoid aﬃnoid adic space with a point X (cid:48) = Spa( K, K + ) → X . Let f , . . . , f n ∈ A + (cid:104) T , . . . , T n (cid:105) be functions such that K → K (cid:104) T , . . . , T n (cid:105) / ( f , . . . , f n ) is an isomorphism. Then, after replacing X by an open neighborhood of X (cid:48) , the map A → A (cid:104) T , . . . , T n (cid:105) / ( f , . . . , f n ) is an isomorphism. Proof.

For any ring B with elements g , . . . , g n ∈ B , consider the homological Koszul complexKos( B, ( g i ) ni =1 ) = [ B → B n → . . . → B n ( g ,...,g n ) −−−−−−→ B ] . V.4. A JACOBIAN CRITERION 137

We claim that, after shrinking X , we can in fact arrange that A → Kos( A (cid:104) T , . . . , T n (cid:105) , ( f i ) ni =1 )is a quasi-isomorphism.Note that all terms of these complexes are free Banach- A -modules, and thus the formation ofthis complex commutes with all base changes; and one can use descent to establish the statement.In particular, we can reduce ﬁrst to the case that X is perfectoid, and then to the case that X is strictly totally disconnected. In that case, the map A → K is automatically surjective,and so we can arrange that under the isomorphism K ∼ = K (cid:104) T , . . . , T n (cid:105) / ( f , . . . , f n ), all T i aremapped to 0. Moreover, applying another change of basis, we can arrange that the image of f i in K (cid:104) T , . . . , T n (cid:105) / ( T , . . . , T n ) is given by a i T i for some nonzero scalar a i ∈ K + . Note that weare in fact allowed to also localize on B nX around the origin, as away from the origin the functions f , . . . , f n locally generate the unit ideal (in the ﬁbre, but thus in a small neighborhood). Doingsuch a localization, we can now arrange that f i ≡ T i mod (cid:36) for some pseudouniformizer (cid:36) ∈ A + .But now in fact A + → Kos( A + (cid:104) T , . . . , T n (cid:105) , ( f i ) ni =1 )is a quasi-isomorphism, as can be checked modulo (cid:36) , where it is the quasi-isomorphism A + /(cid:36) → Kos( A + /(cid:36) [ T , . . . , T n ] , ( T i ) ni =1 ) . (cid:3) Let us draw some consequences. First, we have the following form of the Jacobian criterion inthis setting.

Proposition

IV.4.17 . Let f : Y → X be a smooth map of sous-perfectoid adic spaces, and let f , . . . , f r ∈ O Y ( Y ) be global functions such that df , . . . , df r ∈ Ω Y/X ( Y ) can locally be extended toa basis of Ω Y/X . Then Z = V ( f , . . . , f r ) ⊂ Y is a sous-perfectoid space smooth over X . Proof.

We can assume that all f i ∈ O + Y ( Y ) by rescaling, and we can locally ﬁnd f r +1 , . . . , f n ∈O + Y ( Y ) such that df , . . . , df n is a basis of Ω Y/X . This induces an ´etale map Y → B nX , and then V ( f , . . . , f r ) ⊂ Y is the pullback of B rX ⊂ B nX , giving the desired result. (cid:3) Moreover, we can prove that being smooth is ´etale local on the source.

Proposition

IV.4.18 . Let f : Y → X be a map of sous-perfectoid adic spaces. Assume thatthere is some ´etale cover j : V → Y such that f ◦ j is smooth. Then f is smooth. Proof.

By ´etale descent of vector bundles on sous-perfectoid adic spaces, Ω Y/X := I Y/X / I Y/X is a vector bundle, together with an O X -linear derivation d : O Y → Ω Y/X . We claim that locallywe can ﬁnd functions f , . . . , f n ∈ O Y such that df , . . . , df n ∈ Ω Y/X is a basis. To do this, itsuﬃces to ﬁnd such functions over all ﬁbres Spa(

K, K + ) → X , as any approximation will then stillbe a basis (small perturbations of a basis are still a basis). But over ﬁbres, the equivalence of theconstructions in [ Hub96 , 1.6.2] shows that the df for f ∈ O X form generators of Ω Y/X .Thus, assume that there are global sections f , . . . , f n such that df , . . . , df n ∈ Ω Y/X are abasis. Rescaling the f i if necessary, they deﬁne a map g : Y → B dX that induces an isomorphism

38 IV. GEOMETRY OF DIAMONDS g ∗ Ω B dX /X → Ω Y/X . By Proposition IV.4.15, the map Y → B dX is ´etale locally on Y ´etale. We mayassume that Y and X are aﬃnoid; in particular, all maps are separated. Then by [ Sch17a , Lemma15.6, Proposition 11.30] also Y → B dX is ´etale. (cid:3) Finally, we note that if Y and Y (cid:48) are both smooth over a sous-perfectoid space X , then theconcept of Zariski closed immersions Y (cid:44) → Y (cid:48) over X is well-behaved. Proposition

IV.4.19 . Let f : Y → X , f (cid:48) : Y (cid:48) → X be smooth maps of sous-perfectoid adicspaces, and let g : Y → Y (cid:48) be a map over X . The following conditions are equivalent. (i) There is a cover of Y (cid:48) by open aﬃnoid V (cid:48) = Spa( S (cid:48) , S (cid:48) + ) such that V = Y × Y (cid:48) V (cid:48) = Spa( S, S + ) is aﬃnoid and S (cid:48) → S is surjective, with S + ⊂ S the integral closure of the image of S (cid:48) + . (ii) For any open aﬃnoid V (cid:48) = Spa( S (cid:48) , S (cid:48) + ) ⊂ Y (cid:48) , the preimage V = Y × Y (cid:48) V (cid:48) = Spa( S, S + ) isaﬃnoid and S (cid:48) → S is surjective, with S + ⊂ S the integral closure of the image of S (cid:48) + .Moreover, in this case the ideal sheaf I Y ⊂ Y (cid:48) ⊂ O Y (cid:48) is pseudocoherent in the sense of [ KL16 ] ,and locally generated by sections f , . . . , f r ∈ O Y (cid:48) such that df , . . . , df r ∈ Ω Y (cid:48) /X can locally beextended to a basis. Proof.

We ﬁrst analyze the local structure under condition (1), so assume that Y (cid:48) = Spa( S (cid:48) , S (cid:48) + )and Y = Spa( S, S + ) are aﬃnoid, with S (cid:48) → S surjective and S + ⊂ S the integral closure of theimage of S (cid:48) + . It follows that g ∗ Ω Y (cid:48) /X → Ω Y/X is surjective, and letting d (cid:48) and d be the respectivedimensions of Y (cid:48) and Y (which we may assume to be constant), we see that r = d (cid:48) − d ≥ f , . . . , f r ∈ I Y ⊂ Y (cid:48) so that df , . . . , df r generate the kernel of g ∗ Ω Y (cid:48) /X → Ω Y/X (as the kernel is generated by the closure of the image of I Y ⊂ Y (cid:48) ). By Proposition IV.4.17, the van-ishing locus of the f i deﬁnes a sous-perfectoid space Z ⊂ Y (cid:48) that is smooth over X . The inducedmap Y → Z induces an isomorphism on diﬀerentials, hence is ´etale by Proposition IV.4.15; but itis also a closed immersion, hence locally an isomorphism.We see that the ideal sheaf I Y ⊂ Y (cid:48) is locally generated by sections f , . . . , f r as in the statementof the proposition. By the proof of Proposition IV.4.17 and Lemma IV.4.13, it follows that theideal sheaf I Y ⊂ Y (cid:48) is pseudocoherent in the sense of [ KL16 ].To ﬁnish the proof, it suﬃces to show that (1) implies (2). By the gluing result for pseudocoher-ent modules of [

KL16 ], the pseudocoherent sheaf I Y ⊂ Y (cid:48) over V (cid:48) corresponds to a pseudocoherentmodule I ⊂ S (cid:48) , and then necessarily V = Spa( S, S + ) where S = S (cid:48) /I with S + ⊂ S the integralclosure of the image of S (cid:48) + . (cid:3) Definition

IV.4.20 . In the setup of Proposition IV.4.19, the map g is a Zariski closed immer-sion if the equivalent conditions are satisﬁed. IV.4.2. Maps from X S into P n . Our arguments make critical use of the assumption that inTheorem IV.4.2, the space Z → X S is locally closed in P nX S . For this reason, we analyze the specialcase of P n in this section. Proposition

IV.4.21 . Let n ≥ and consider the small v-sheaf M P n taking any perfectoidspace S to the set of maps X S → P nE . Then M P n → ∗ is partially proper and representable in V.4. A JACOBIAN CRITERION 139 locally spatial diamonds, and admits a decomposition into open and closed subspaces M P n = (cid:71) m ≥ M m P n such that each M m P n → ∗ has ﬁnite dim . trg , and the degree of the pullback of O P n (1) to X M m P n is m . In fact, there is a canonical open immersion M m P n (cid:44) → ( BC ( O X S ( m ) n +1 ) \ { } ) /E × . Proof.

The degree of the pullback L /X S of O P n (1) to X S deﬁnes an open and closed decom-position according to all m ∈ Z . Fix some m . Then over the corresponding subspace M m P n , wecan ﬁx a trivialization L ∼ = O X S ( m ), which amounts to an E × -torsor. After this trivialization,one parametrizes n + 1 sections of L ∼ = O X S ( m ) without common zeroes. The condition of nocommon zeroes is an open condition on S : Indeed, the common zeroes form a closed subspace of | X S | , and the map | X S | → | S | is closed (see the proof of Lemma IV.1.20). This implies the desireddescription. (cid:3) Proposition

IV.4.22 . Let S be a perfectoid space and let Z → X S be a smooth map of sous-perfectoid adic spaces such that Z admits a Zariski closed embedding into an open subspace of P nX S .Then the induced functor M Z → M P nXS is locally closed. More precisely, for any perfectoid space T → M P nXS , the preimage of M Z isrepresentable by some perfectoid space T Z ⊂ T that is ´etale locally Zariski closed in T , i.e. thereis some ´etale cover of T by aﬃnoid perfectoid T (cid:48) = Spa( R, R + ) → T such that T Z × T T (cid:48) =Spa( R Z , R + Z ) is aﬃnoid perfectoid, with R → R Z surjective and R + Z ⊂ R Z the integral closure ofthe image of R + .In particular, the map M Z → S is representable in locally spatial diamonds and compactiﬁable,of locally ﬁnite dim . trg . Proof.

Choose an open subspace W ⊂ P nX S such that Z is Zariski closed in W . For anyperfectoid space T with a map T → M P nXS corresponding to a map X T → P nX S over X S , the locus T W ⊂ T where the section factors over W is open. Indeed, this locus is the complement of theimage in | T | of the preimage of | P nX S \ W | under | X T | → | P nX S | , and | X T | → | T | is closed.Replacing T by T W , we can assume that the section X T → P nX S factors over W . We may alsoassume that T = Spa( R, R + ) is aﬃnoid perfectoid and that S = T . Pick a pseudouniformizer (cid:36) ∈ R , in particular deﬁning the cover Y S, [1 ,q ] = {| π | q ≤ | [ (cid:36) ] | ≤ | π |} ⊂ Spa W O E ( R + )of X S . The pullback of the line bundle O P n (1) to X S along this section, and then to Y S, [1 ,q ] , is ´etalelocally trivial, as when S is a geometric point, Y S, [1 ,q ] is aﬃnoid with ring of functions a principalideal domain by Corollary II.1.12. Replacing W by a small ´etale neighborhood of this section andcorrespondingly shrinking S , we can assume that the pullback of O P n (1) to W [1 ,q ] = W × X S Y S, [1 ,q ] is trivial. In that case the pullback Z [1 ,q ] → Y S, [1 ,q ] of Z → X S is Zariski closed in an open subsetof A n +1 Y S, [1 ,q ] .

40 IV. GEOMETRY OF DIAMONDS

Inside A n +1 Y S, [1 ,q ] , the image of Y Spa(

K,K + ) , [1 ,q ] (via the given section) for a point Spa( K, K + ) → S isan intersection of small balls over Y S (cid:48) , [1 ,q ] for small neighborhoods S (cid:48) ⊂ S of Spa( K, K + ). Thus,one of these balls is contained in the open subset of which Z [1 ,q ] is a Zariski closed subset. Thus,after this further localization, we can assume that there is a Zariski closed immersion Z [1 ,q ] (cid:44) → B n +1 Y S, [1 ,q ] , and in particular Z [1 ,q ] is aﬃnoid and cut out by global functions on B n +1 Y S, [1 ,q ] by Proposition IV.4.19.Pulling back these functions along the given section Y S, [1 ,q ] → B n +1 Y S, [1 ,q ] , it suﬃces to see that if S = Spa( R, R + ) is an aﬃnoid perfectoid space of characteristic p with a choice of pseudouniformizer (cid:36) ∈ R and f ∈ B R, [1 ,q ] = O ( Y S, [1 ,q ] )is a function, then there is a universal perfectoid space S (cid:48) ⊂ S for which the pullback of f is zero,and S (cid:48) ⊂ S is Zariski closed. This is given by Lemma IV.4.23. (cid:3) Lemma

IV.4.23 . Let S = Spa( R, R + ) ∈ Perf F q be aﬃnoid perfectoid with a ﬁxed pseudo-uniformizer (cid:36) , I ⊂ (0 , ∞ ) a compact interval with rational ends, and Z ⊂ | Y S,I | a closed subsetdeﬁned by the vanishing locus of an ideal J ⊂ O ( Y S,I ) . Then, via the open projection υ : | Y S,I | → | S | ,the closed subset | S | \ υ ( | Y S,I | \ Z ) is Zariski closed. The corresponding Zariski closed perfectoidsubspace of S is universal for perfectoid spaces T → S such that J (cid:55)→ via O ( Y S,I ) → O ( Y T,I ) . Proof.

Since Y ♦ S,I → S is cohomologically smooth, υ is open. We can suppose J = ( f ) with f ∈ O ( Y S,I ). For any untilt of F q (( (cid:36) /p ∞ )) over E such that | π | b ≤ | [ (cid:36) (cid:93) ] | ≤ | π | a if I = [ a, b ], we geta corresponding untilt R (cid:93) of R over E , with a map B R,I → R (cid:93) . The locus where the image of f in R (cid:93) vanishes is Zariski closed by Proposition II.0.2. Intersecting these Zariski closed subsets overvarying such untilts gives the vanishing locus of f , as in any ﬁbre, f vanishes as soon at it vanishesat inﬁnitely many untilts (e.g., by Corollary II.1.12), and all rings are sous-perfectoid, in particularuniform, so vanishing at all points implies vanishing. (cid:3) IV.4.3. Formal smoothness of M smZ . The key result we need is the following.

Proposition

IV.4.24 . Let S = Spa( R, R + ) be an aﬃnoid perfectoid space over F q and let Z → X S be a smooth map of sous-perfectoid adic spaces that is Zariski closed in an open subspaceof P nX S . Then M sm Z → S is formally smooth. Proof.

Pick a test diagram as in Deﬁnition IV.3.1; we can and do assume that the S fromthere is the given S , replacing the S in this proposition if necessary. This means we have a diagram ZX S X Ss and, up to replacing S by an ´etale neighborhood of S we try to extend the section s to a sectionover X S (the dotted line in the diagram). Fix a geometric point Spa( C, C + ) → S ; we will always V.4. A JACOBIAN CRITERION 141 allow ourselves to pass to ´etale neighborhoods of this point. Fix a pseudouniformizer (cid:36) ∈ R andconsider the aﬃnoid cover Y S, [1 ,q ] → X S ; recall that Y S, [1 ,q ] = {| π | q ≤ | [ (cid:36) ] | ≤ | π |} ⊂ Spa W O E ( R + )and we also consider its boundary annuli Y S, [1 , = {| [ (cid:36) ] | = | π |} , Y S, [ q,q ] = {| π | q = | [ (cid:36) ] |} ⊂ Y S, [1 ,q ] . Let Z [1 ,q ] → Z be its pullback; with pullback Z [1 , , Z [ q,q ] ⊂ Z [1 ,q ] of Y S, [1 , , Y S, [ q,q ] ⊂ Y S, [1 ,q ] . Inparticular, Z is obtained from Z [1 ,q ] via identiﬁcation of its open subsets Z [1 , , Z [ q,q ] along theisomorphism ϕ : Z [1 , → Z [ q,q ] .Arguing as in the proof of Proposition IV.4.22, we can after ´etale localization on S embed Z [1 ,q ] (cid:44) → B n +1 Y S, [1 ,q ] as a Zariski closed subset. We thus have a diagram Z [1 ,q ] B n +1 Y S, [1 ,q ] Y S , [1 ,q ] Y S, [1 ,q ] . Zariski closed

In particular, Z [1 ,q ] is aﬃnoid.Next, consider the K¨ahler diﬀerentials Ω Z [1 ,q ] /Y S, [1 ,q ] . Again, as B C, [1 ,q ] is a principal idealdomain, its restriction to the section Y Spa(

C,C + ) , [1 ,q ] ⊂ Z [1 ,q ] is trivial, and thus it is trivial in a smallneighborhood. It follows that after a further ´etale localization we can assume that Ω Z [1 ,q ] /Y S, [1 ,q ] ∼ = O rZ [1 ,q ] is trivial. On the Zariski closed subset Z , [1 ,q ] ⊂ Z [1 ,q ] , this implies that we may ﬁnd functions f , . . . , f r ∈ O ( Z , [1 ,q ] ) vanishing on the section Y S , [1 ,q ] → Z , [1 ,q ] and locally generating the idealof this closed immersion (use Proposition IV.4.19). In particular, df , . . . , df r ∈ Ω Z , [1 ,q ] /Y S , [1 ,q ] are generators at the image of the section Y S , [1 ,q ] → Z , [1 ,q ] , and thus in an open neighborhood.Picking lifts of the f i to O ( Z [1 ,q ] ) and shrinking Z [1 ,q ] , Proposition IV.4.15 implies that they deﬁnean ´etale map Z [1 ,q ] → B rY S, [1 ,q ] . Moreover, over { } Y S , [1 ,q ] ⊂ B rY S, [1 ,q ] , this map admits a section. Shrinking further around thissection, we can thus arrange that there are open immersions( π N B ) rY S, [1 ,q ] ⊂ Z [1 ,q ] ⊂ B rY S, [1 ,q ] , and that the section over Y S , [1 ,q ] is given by the zero section.The isomorphism ϕ : Z [1 , → Z [ q,q ] induces a map ϕ (cid:48) : ( π N B ) rY S, [1 , → B rY S, [ q,q ] . Recall that for any compact interval I ⊂ (0 , ∞ ), the space Y S,I = Spa( B R,I , B +( R,R + ) ,I )

42 IV. GEOMETRY OF DIAMONDS is aﬃnoid. The map ϕ (cid:48) is then given by a map α : B R, [ q,q ] (cid:104) T , . . . , T r (cid:105) → B R, [1 , (cid:104) π − N T , . . . , π − N T r (cid:105) linear over the isomorphism ϕ : B R, [ q,q ] → B R, [1 , . The map α is determined by the images of T , . . . , T n which are elements α i ∈ B +( R,R + ) , [1 , (cid:104) π − N T , . . . , π − N T r (cid:105) . These have the property that on the quotient B R , [1 , they vanish at T = . . . = T r = 0 (as over S , the zero section is ϕ -invariant). Moreover, over the geometric point Spa( C, C + ) → S ﬁxed atthe beginning, we can apply a linear change of coordinates in order to ensure that the derivativeat the origin is given by a standard matrix for an isocrystal of negative slopes; i.e., there are cycles1 , . . . , r ; r + 1 , . . . , r ; . . . ; r a − + 1 , . . . , r a = r and positive integers d , . . . , d a such that α i ≡ T i +1 in B C, [1 , [ T , . . . , T r ] / ( T , . . . , T r ) if i (cid:54) = r j for some j = 1 , . . . , a , and α r j ≡ π − d j T r j − +1 in B C, [1 , [ T , . . . , T r ] / ( T , . . . , T r ) . (Here, we set r = 0.) Approximating this linear change of basis over an ´etale neighborhood, werespect the condition that the α i ’s vanish at T = . . . = T r = 0 over S , while we can for any large M arrange α i ≡ T i +1 in B +( R,R + ) , [1 , /π M [ π − N T , . . . , π − N T r ] / ( π − N T , . . . , π − N T r ) if i (cid:54) = r j and α r j ≡ π − d j T r j − +1 in B +( R,R + ) , [1 , /π M [ π − N T , . . . , π − N T r ] / ( π − N T , . . . , π − N T r ) . Moreover, rescaling all T i by powers of π , and passing to a smaller neighborhood around S , wecan then even ensure that α i ∈ T i +1 + π M B +( R,R + ) , [1 , (cid:104) T , . . . , T r (cid:105) for i (cid:54) = r j and α r j ∈ π − d j T r j − +1 + π M B +( R,R + ) , [1 , (cid:104) T , . . . , T r (cid:105) . At this point, the integers d , . . . , d a are ﬁxed, while we allow ourselves to choose M later, dependingonly on these.From this point on, we will no longer change S and S , and instead will merely change coordi-nates in the balls (by automorphisms). More precisely, we study the eﬀect of replacing T i by T i + (cid:15) i for some (cid:15) i ∈ π d ker( B +( R,R + ) , [1 ,q ] → B R , [1 ,q ] )where we take d to be at least the maximum of all d j . This replaces α i by a new power series α (cid:48) i ,given by α (cid:48) i ( T , . . . , T r ) = α i ( T , . . . , T i + (cid:15) i , . . . , T r ) − ϕ ( (cid:15) i )and the α (cid:48) i ’s still vanish at T = . . . = T r = 0 over S . Their nonconstant coeﬃcients will still havethe same properties as for α i (the linear coeﬃcients are unchanged, while all other coeﬃcients aredivisible by π M ), and the constant coeﬃcient satisﬁes α (cid:48) i (0 , . . . , ≡ α i (0 , . . . ,

0) + (cid:15) i +1 − ϕ ( (cid:15) i ) in B +( R,R + ) , [1 , /π M + d V.4. A JACOBIAN CRITERION 143 if i (cid:54) = r j and α (cid:48) r j (0 , . . . , ≡ α r j (0 , . . . ,

0) + π − d j (cid:15) r j − +1 − ϕ ( (cid:15) r j ) in B +( R,R + ) , [1 , /π M + d . Assume that by some inductive procedure we already achieved α i (0 , . . . , ∈ π N (cid:48) B +( R,R + ) , [1 , forsome N (cid:48) ≥ M . By Lemma IV.4.25 below, there is some constant c depending only on d , . . . , d a such that we can then ﬁnd (cid:15) i ∈ π N (cid:48) − c B +( R,R + ) , [1 , , vanishing over R , with α i (0 , . . . ,

0) = ϕ ( (cid:15) i ) − (cid:15) i +1 for i (cid:54) = r j and α r j (0 , . . . ,

0) = ϕ ( (cid:15) r j ) − π − d j (cid:15) r j − +1 . This means that α (cid:48) i (0 , . . . , ∈ π M + N (cid:48) − c B +( R,R + ) , [1 , , so if we choose M > c in the beginning (whichwe can), then this inductive procedure converges, and in the limit we get a change of basis afterwhich the zero section deﬁnes a ϕ -invariant section of Z [1 ,q ] , thus a section s : X S → Z , as desired.Note that we arranged that this section agrees with s over S , as all coordinate changes did notaﬀect the situation over S . (cid:3) We used the following quantitative version of vanishing of H ( X S , E ) for E of positive slopes. Lemma

IV.4.25 . Fix a standard Dieudonn´e module of negative slopes, given explicitly on abasis e , . . . , e r by ﬁxing cycles , . . . , r ; r + 1 . . . , r ; . . . ; r a − + 1 , . . . , r a = r and positive integers d , . . . , d a > , via ϕ ( e i ) = e i +1 for i (cid:54) = r j , ϕ ( e r j ) = π − d j e r j − +1 . Then there is an integer c ≥ with the following property.Let S = Spa( R, R + ) be an aﬃnoid perfectoid space over F q with Zariski closed subspace S =Spa( R , R +0 ) , and a pseudouniformizer (cid:36) ∈ R . Let I +[1 ,q ] = ker( B +( R,R + ) , [1 ,q ] → B +( R ,R +0 ) , [1 ,q ] ) , I +[1 , = ker( B +( R,R + ) , [1 , → B +( R ,R +0 ) , [1 , ) . Then for all f , . . . , f r ∈ I +[1 , one can ﬁnd g , . . . , g r ∈ π − c I +[1 ,q ] such that f i = ϕ ( g i ) − g i +1 for i (cid:54) = r j , f r j = ϕ ( g r j ) − π − d j g r j − +1 . Proof.

We may evidently assume that a = 1; set d = d . By linearity, we can assume that allbut one of the f i ’s is equal to zero. Thus, it suﬃces to see that for all positive integers r and N there is c ≥ f ∈ I +[1 , one can ﬁnd some g ∈ π − c I +[1 ,q r ] (for the evident deﬁnitionof I +[1 ,q r ] ) such that f = ϕ r ( g ) − π − d g. Replacing E by its unramiﬁed extension of degree r , we can then assume that r = 1. At this point,we want to reduce to the qualitative version given by Lemma IV.4.26 below, saying that the map ϕ − π − d : I [1 ,q ] → I [1 , is surjective. Indeed, assume a constant c as desired would not exist. Then for any integer i ≥ S ,i = Spa( R ,i , R +0 ,i ) ⊂ S i = Spa( R i , R + i ), with choices ofpseudouniformizers (cid:36) i ∈ R i , as well as elements f i ∈ I +[1 , ,i such that there is no g ∈ π − i I +[1 ,q ] ,i with

44 IV. GEOMETRY OF DIAMONDS f i = ϕ ( g i ) − π − d g i . Then we can deﬁne R + = (cid:81) i R + i with (cid:36) = ( (cid:36) i ) i ∈ R + , and R = R + [ (cid:36) ], whichdeﬁnes an aﬃnoid perfectoid space S = Spa( R, R + ), containing a Zariski closed subspace S ⊂ S deﬁned similarly. Moreover, the sequence ( π i f i ) i deﬁnes an element of f ∈ I +[1 , . As ϕ − π − d : I [1 ,q ] → I [1 , is surjective by Lemma IV.4.26, we can ﬁnd some g ∈ I [1 ,q ] with f = ϕ ( g ) − π − d g . Then π c g ∈ I +[1 ,q ] for some c , and restricting g to S ,i ⊂ S i with i > c gives the desired contradiction. (cid:3) We reduced to the following qualitative version.

Lemma

IV.4.26 . Let d be a positive integer, let S = Spa( R , R +0 ) ⊂ S = Spa( R, R + ) be aZariski closed immersion of aﬃnoid perfectoid spaces over F q , and let (cid:36) ∈ R be a pseudouni-formizer. Let I [1 ,q ] = ker( B R, [1 ,q ] → B R , [1 ,q ] ) , I [1 , = ker( B R, [1 , → B R , [1 , ) . Then the map ϕ − π − d : I [1 ,q ] → I [1 , is surjective. Proof.

By the snake lemma and the vanishing H ( X S , O X S ( d )) = 0 (Proposition II.2.5 (iii)),the lemma is equivalent to the surjectivity of H ( X S , O X S ( d )) → H ( X S , O X S ( d )) . For d ≤ [ E : Q p ] (or if E is of equal characteristic), this follows directly from Proposition II.2.5 (iv)and the surjectivity of R ◦◦ → R ◦◦ . In general, we can either note that the proof of Proposi-tion II.2.5 (iii) also proves the lemma, or argue by induction by choosing an exact sequence0 → O X S ( d − → O X S ( d − → O X S ( d ) → H ( X S , O X S (1))), and use the van-ishing of H ( X S , O X S ( d − d >

2, Proposition II.2.5 (iii). This induction gets startedas long as E (cid:54) = Q p . For E = Q p , we can write O X S ( d ) as a direct summand of π ∗ π ∗ O X S ( d ) for anyextension π : X S,E → X S with E (cid:54) = Q p . (cid:3) IV.4.4. Universal local acyclicity of M smZ → S . The next step in the proof of Theo-rem IV.4.2 is to show that F (cid:96) is universally locally acyclic. Proposition

IV.4.27 . Let S be a perfectoid space and let Z → X S be a smooth map of sous-perfectoid spaces such that Z is Zariski closed inside an open subset of P nX S for some n ≥ . Then,for any (cid:96) (cid:54) = p , the sheaf F (cid:96) is universally locally acyclic for the map M sm Z → S. Proof.

Recall from Proposition IV.4.22 that M Z → M P nXS is a locally closed immersion, andthe open embedding M P nXS (cid:44) → (cid:71) m ≥ ( BC ( O X S ( m ) n +1 ) \ { } ) /E × from Proposition IV.4.21. In the following, we ﬁx some m and work on the preimage of( BC ( O X S ( m ) n +1 ) \ { } ) /E × . V.4. A JACOBIAN CRITERION 145

We choose a surjection T → BC ( O X S ( m ) n +1 \ { } ) /E × from a perfectoid space T as inLemma IV.4.28; in particular, g is separated, representable in locally spatial diamonds, coho-mologically smooth, and formally smooth. Moreover, locally T admits a Zariski closed immersioninto the perfectoid ball (cid:101) B nS over S . Taking the pullback of T to M Z , we get a surjection T Z → M Z for some perfectoid space T Z such that ´etale locally T Z admits a Zariski closed immersion into aspace ´etale over (cid:101) B nS .It follows that one can cover M Z via maps h : T → M Z that are separated, representable inlocally spatial diamonds, cohomologically smooth, and formally smooth, and such that T admitsa Zariski closed immersion into some space ´etale over (cid:101) B nS . By Proposition IV.4.24, we can, up tofurther replacement of T by an ´etale cover, assume that the map h extends to a map h : T → M Z for some perfectoid space T ´etale over (cid:101) B nS . Moreover, as T → M Z is formally smooth, we can,after a further ´etale localization, lift the map T → M Z to a retraction T → T ; thus, T is a retractof a space that is ´etale over a perfectoid ball. Now the result follows from Corollary IV.2.27. (cid:3) We used the following presentation of certain projectivized Banach–Colmez spaces.

Lemma

IV.4.28 . Let S be a perfectoid space over F q and let E be a vector bundle on X S that iseverywhere of nonnegative Harder–Narasimhan slopes. There is a perfectoid space T → S that islocally Zariski closed in a perfectoid ball (cid:101) B nS over S and that admits a surjective map T → ( BC ( E ) \ { } ) /E × over S that is separated, representable in locally spatial diamonds, cohomologically smooth, andformally smooth. Proof.

The target parametrizes line bundles L on X S of slope zero together with a section of E ⊗ L that is nonzero ﬁbrewise on S . Parametrizing in addition an injection L (cid:44) → O X S (1) deﬁnesa map that is separated, representable in locally spatial diamonds, cohomologically smooth, andformally smooth (by Proposition II.3.5 and Proposition IV.3.8). Over this cover, one has locallyon S an untilt S (cid:93) over E corresponding to the support of the cokernel of L → O X S (1), and oneparametrizes nonzero sections of E (1) that vanish at S (cid:93) (cid:44) → X S . This is Zariski closed (by [ BS19 ,Theorem 7.4, Remark 7.5]) inside the space of all sections of E (1). We see that it suﬃces to provethe similar result with ( BC ( E ) \ { } ) /E × replaced by BC ( E (1)) × Spd E , and this reduces to theindividual factors. For BC ( E (1)), the result follows from the argument in Proposition IV.3.8. ForSpd E , there is nothing to do in equal characteristic, so assume that E is p -adic. Then we reduceto [ ∗ / O × E ] as the ﬁbres of Spd E → [ ∗ / O × E ] over perfectoid spaces are given by BC ( L ) \ { } forsome line bundle L of slope 1, and this in turn admits covers of the desired form. Finally, for[ ∗ / O × E ], we can pass to the ´etale cover [ ∗ / p O E ] ∼ = [ ∗ / O E ], or to [ ∗ /E ]. This, ﬁnally, admits asurjection from a perfectoid open unit disc BC ( O X S (1)) with the desired properties by passing toBanach–Colmez spaces in an exact sequence0 → O X S → O X S ( ) → O X S (1) → (cid:3)

46 IV. GEOMETRY OF DIAMONDS

IV.4.5. Deformation to the normal cone.

The ﬁnal step in the proof of Theorem IV.4.2is a deformation to the normal cone.By Proposition IV.4.22 and Proposition IV.4.27 (and Proposition IV.2.33), in order to proveTheorem IV.4.2 it only remains to prove that Rf ! F (cid:96) is invertible and sitting in the expected co-homological degree. Picking a v-cover T → M sm Z by some perfectoid space T and using that theformation of Rf ! F (cid:96) commutes with any base change by Proposition IV.2.15, it suﬃces to prove thefollowing result. Proposition

IV.4.29 . Let S be a perfectoid space and let Z → X S be a smooth map of sous-perfectoid spaces such that Z admits a Zariski closed immersion into an open subset of (the adicspace) P nX S for some n ≥ . Let f : M Z → S be the moduli space of sections of Z → X S . Moreover,let s : X S → Z be a section such that s ∗ T Z/X S is everywhere of positive Harder–Narasimhan slopes,and of degree d .Let t : S → M Z be the section of f corresponding to s . Then t ∗ Rf ! F (cid:96) is ´etale locally on S isomorphic to F (cid:96) [2 d ] . Proof.

We will prove this by deformation to the normal cone. In order to avoid a generaldiscussion of blow-ups etc., we will instead take an approach based on the local structure of Z neara section as exhibited in the proof of Proposition IV.4.24.We are free to make v-localizations on S (as being cohomologically smooth can be checked aftera v-cover), and replace Z by an open neighborhood of s ( X S ). With this freedom, we can follow theproof of Proposition IV.4.24 and ensure that S = Spa( R, R + ) is strictly totally disconnected withpseudouniformizer (cid:36) , the pullback Z [1 ,q ] → Y S, [1 ,q ] of Z → X S to Y S, [1 ,q ] = {| π | q ≤ | [ (cid:36) ] | ≤ | π |} ⊂ Spa W O E ( R + )satisﬁes π N B rY S, [1 ,q ] ⊂ Z [1 ,q ] ⊂ B rY S, [1 ,q ] and the gluing isomorphism is given by power series α i ∈ T i +1 + π M B +( R,R + ) , [1 , (cid:104) T , . . . , T r (cid:105) resp. α r j ∈ π − d j T r j − +1 + π M B +( R,R + ) , [1 , (cid:104) T , . . . , T r (cid:105) with notation following the proof of Proposition IV.4.24. Moreover, the constant coeﬃcients of all α i vanish. These in fact deﬁne a map ϕ : π d B rY S, [1 , → B rY S, [ q,q ] preserving the origin, where d is the maximum of the d j .For any n ≥ N, d , we can look at the subset Z ( n )[1 ,q ] = π n B rY S, [1 ,q ] ∪ ϕ ( π n B rY S, [1 , ) ⊂ Z [1 ,q ] , which descends to an open subset Z ( n ) ⊂ Z . Letting T ( n ) i = π − n T i , the gluing is then given bypower series α ( n ) i given by α ( n ) i = π − n α i ( π n T , . . . , π n T n ) V.4. A JACOBIAN CRITERION 147 which satisfy the same conditions, but the nonlinear coeﬃcients of α ( n ) i become more divisible by π . The limit α ( ∞ ) i = lim −→ n →∞ α ( n ) i ∈ B R, [1 , (cid:104) T , . . . , T r (cid:105) exists, and is linear in the T i .Let S (cid:48) = S × N ≥ N ∪ {∞} , using the proﬁnite set N ≥ N ∪ {∞} . Let Z (cid:48) → X S (cid:48) be the smoothmap of sous-perfectoid spaces obtained by descending Z (cid:48) [1 ,q ] = B rY S (cid:48) , [1 ,q ] ∪ ϕ (cid:48) ( B rY S (cid:48) , [ q,q ] )along the isomorphism ϕ (cid:48) given by the power series α (cid:48) i = ( α ( N ) i , α ( N +1) i , . . . , α ( ∞ ) i ) ∈ B R (cid:48) , [1 , (cid:104) T , . . . , T r (cid:105) . Then the ﬁbre of Z (cid:48) → X S (cid:48) over S × { n } is given by Z ( n ) , while its ﬁbre over S × {∞} is givenby an open subset Z ( ∞ ) of the Banach–Colmez space BC ( s ∗ T Z/X S ). Moreover, letting S (cid:48) ( >N ) ⊂ S (cid:48) be the complement of S × { N } , there is natural isomorphism γ : S (cid:48) ( >N ) → S (cid:48) given by the shift S × { n + 1 } ∼ = S × { n } , and this lifts to an open immersion γ : Z (cid:48) ( >N ) = Z (cid:48) × X S (cid:48) X S (cid:48) ( >N ) (cid:44) → Z (cid:48) . We need to check that Z (cid:48) → X S (cid:48) still satisﬁes the relevant quasiprojectivity assumption. Lemma

IV.4.30 . The space Z (cid:48) → X S (cid:48) admits a Zariski closed immersion into an open subsetof P mX S (cid:48) for some m ≥ . Proof.

One may perform a parallel construction with Z replaced by an open subset of P mX S ,reducing us to the case that Z is open in P mX S . In that case, the key observation is that the blow-upof P mX S at the section s : X S → P mX S is still projective, which is an easy consequence of X S admittingenough line bundles. (cid:3) Let f (cid:48) : M Z (cid:48) → S (cid:48) be the projection, with ﬁbres f ( n ) and f ( ∞ ) . By Proposition IV.4.27, both F (cid:96) and Rf (cid:48) ! F (cid:96) are f (cid:48) -universally locally acyclic. In particular, the formation of Rf (cid:48) ! F (cid:96) commuteswith base change, and we see that the restriction of Rf (cid:48) ! F (cid:96) to the ﬁbre over ∞ is ´etale locallyisomorphic to F (cid:96) [2 d ], as an open subset of BC ( s ∗ T Z/X S ). As S is strictly totally disonnected, onecan choose a global isomorphism with F (cid:96) [2 d ].The map from F (cid:96) [2 d ] to the ﬁbre of Rf (cid:48) ! F (cid:96) over ∞ extends to a small neighborhood; passing tothis small neighborhood, we can assume that there is a map β : F (cid:96) [2 d ] → Rf (cid:48) ! F (cid:96) that is an isomorphism in the ﬁbre over ∞ . We can assume that this map is γ -equivariant (passingto a smaller neighborhood). Let Q be the cone of β . Then Q is still f (cid:48) -universally locally acyclic,as is its Verdier dual D M Z (cid:48) /S (cid:48) ( Q ) = R H om M Z (cid:48) ( Q, Rf (cid:48) ! F (cid:96) ) . In particular, Rf (cid:48) ! D M Z (cid:48) /S (cid:48) ( Q ) ∈ D ´et ( S (cid:48) , F (cid:96) ) is constructible, and its restriction to S ×{∞} is trivial.This implies (e.g. by [ Sch17a , Proposition 20.7]) that its restriction to S × { n, n + 1 , . . . , ∞} is

48 IV. GEOMETRY OF DIAMONDS trivial for some n (cid:29)

0. Passing to this subset, we can assume that Rf (cid:48) ! D M Z (cid:48) /S (cid:48) ( Q ) = 0. TakingVerdier duals and using Corollary IV.2.25, this implies that Rf (cid:48)∗ Q = 0.In particular, for all n ≥ n , we have Rf ( n ) ∗ Q | M Z ( n ) = 0. Using the γ -equivariance, this isequivalent to Rf ( n ) ∗ ( Q | M ( n Z ) | M Z ( n ) = 0 , regarding M Z ( n ) ⊂ M Z ( n as an open subset. Taking the colimit over all n and using that thesystem M Z ( n ) ⊂ M Z ( n has intersection s ( S ) ⊂ M Z and is coﬁnal with a system of spatialdiamonds of ﬁnite cohomological dimension (as can be checked in the case of projective space),[ Sch17a , Proposition 14.9] implies that s ∗ Q | M Z ( n = lim −→ n Rf ( n ) ∗ ( Q | M Z ( n ) | M Z ( n ) = 0(by applying it to the global sections on any quasicompact separated ´etale (cid:101) S → S ), and thus themap s ∗ β | M Z : F (cid:96) [2 d ] → s ∗ Rf ! F (cid:96) is an isomorphism, as desired. This ﬁnishes the proof of Proposition IV.4.29 and thus of Theo-rem IV.4.2. (cid:3) The idea of the preceding proof is the following. Let C → X S × A be the open subset ofthe deformation to the normal cone of s : X S (cid:44) → Z (we did not develop the necessary formalismto give a precise meaning to this in the context of smooth sous-perfectoid spaces, but it could bedone) whose ﬁber at 0 ∈ A is the normal cone of the immersion s (the divisor over 0 ∈ A of thedeformation to the normal cone is the union of two divisors: the projective completion of the normalcone and the blow-up of Z along X S , both meeting at inﬁnity inside the projective completion).One has a diagram X S × A (cid:31) (cid:127) (cid:47) (cid:47) (cid:36) (cid:36) C (cid:15) (cid:15) A where outside t = 0 ∈ A this is given by the section s : X S (cid:44) → Z , i.e. the pullback over G m of thepreceding diagram gives the inclusion X S × G m (cid:44) → Z × G m , and at t = 0 this is the inclusion of X S inside the normal cone of the section s . Let us note moreover that C is equipped with a G m -actioncompatible with the one on A .This gives rise to an E × -equivariant morphism with an equivariant section M Cg (cid:15) (cid:15) S × E s (cid:67) (cid:67) whose ﬁber at 0 ∈ E is the zero section of BC ( s ∗ T Z/X S ) → S , and is isomorphic to M Z × E × equipped with the section s outside of 0. Now, the complex s (cid:48)∗ Rg ! F (cid:96) is E × -equivariant on S × E .Its ﬁber outside 0 ∈ E , i.e. its restriction to S × E × , is s ∗ Rf ! F (cid:96) , and its ﬁber at 0 is F (cid:96) ( d )[2 d ], V.5. PARTIAL COMPACTLY SUPPORTED COHOMOLOGY 149 d = deg( T Z/X S ) (since g is universally locally acyclic the dualizing complexe commutes with basechange).Now one checks that one can replace the preceding diagram by a quasicompact O E \ { } -invariant open subset U ⊂ M C together with an equivariant diagram U h (cid:15) (cid:15) (cid:31) (cid:127) (cid:47) (cid:47) M C g (cid:15) (cid:15) S × O Et (cid:65) (cid:65) (cid:31) (cid:127) (cid:47) (cid:47) S × E. s (cid:67) (cid:67) In the preceding proof one replaces O E by π N ∪{∞} ⊂ O E , which does not change anything forthe argument. One concludes using that × π “contracts everything to 0” and some constructibilityargument using the fact that U is spatial and some complexes are h -universally locally acyclic (seethe argument “ Rf (cid:48) ! D M Z (cid:48) /S (cid:48) ( Q ) ∈ D ´et ( S (cid:48) , F (cid:96) ) is constructible” at the end of the proof of PropositionIV.4.29). IV.5. Partial compactly supported cohomology

Let us start by recalling the following basic vanishing result. Let C be a complete algebraicallyclosed nonarchimedean ﬁeld C with pseudouniformizer (cid:36) ∈ C . Let Spa Z (( t )) × Z Spa C = D ∗ C bethe punctured open unit disc over C , and consider the subsets j : U = {| t | ≤ | (cid:36) |} (cid:44) → D ∗ C , j (cid:48) : U (cid:48) = {| t | ≥ | (cid:36) |} (cid:44) → D ∗ C Note that the punctured open unit disc has two ends: Towards the origin, and towards the boundary.The open subsets U and U (cid:48) contain one end each. Lemma

IV.5.1 . The partially compactly supported cohomology groups R Γ( D ∗ C , j ! Λ) = 0 = R Γ( D ∗ C , j (cid:48) ! Λ) vanish. As usual Λ is any coeﬃcient ring killed by an integer n prime to p . Proof.

We treat the vanishing R Γ( D ∗ C , j ! Λ) = 0, the other one being similar. Let k : D ∗ C (cid:44) → D C be the inclusion. One has an exact triangle( kj ) ! Λ −→ Rk ∗ j ! Λ −→ i ∗ A +1 −−−→ where i : { } (cid:44) → D C . One has H ( A ) = Λ, H ( A ) = Λ(1), H i ( A ) = 0 for i (cid:54) = 0 ,

1, since A =lim −→ n R Γ( U n , Λ) with U n = {| t | ≤ | (cid:36) n |} ⊂ D ∗ C being a punctured disc. We thus have to prove thatthe preceding triangle induces an isomorphism A ∼ −→ R Γ c ( U, Λ)[1]. Let (cid:101) j : P \ { , ∞} (cid:44) → P \ { } .There is a commutative diagram R Γ c ( U, Λ) R Γ( D ∗ C , j ! Λ) AR Γ c ( P \ { , ∞} , Λ) R Γ( P \ { } , (cid:101) j ! Λ) A . (cid:39) +1+150 IV. GEOMETRY OF DIAMONDS

It thus suﬃces to check that R Γ( P \ { } , (cid:101) j ! Λ) = 0 , for example in the algebraic setting using comparison theorems, which is an easy exercise. (cid:3) Our goal now is to prove a very general version of such a result. Fix an algebraically closedﬁeld k | F q and work on Perf k . Let X be a spatial diamond such that f : X → ∗ = Spd k is partiallyproper with dim . trg f < ∞ . Then the base change X × k S of X to any spatial diamond S is notitself quasicompact. Rather, it has two ends, and we will in this section study the cohomology withcompact support towards one of the ends.To analyze the situation, pick quasi-pro-´etale and universally open surjections (cid:101) X → X and (cid:101) S → S from aﬃnoid perfectoid spaces (using [ Sch17a , Proposition 11.24]), and pick maps (cid:101) X → Spa k (( t )) and (cid:101) S → Spa k (( u )) by choosing pseudouniformizers. We get a correspondence (cid:101) X × (cid:101) SX × S Spa k (( t )) × Spa k (( u )) = D ∗ k (( u )) where all maps are qcqs, and the left map is (universally) open. Now Spa k (( t )) × Spa k (( u )) is apunctured open unit disc over Spa k (( u )), and one can write it as the increasing union of the aﬃnoidsubspaces {| u | b ≤ | t | ≤ | u | a } ⊂ Spa k (( t )) × Spa k (( u ))for varying rational 0 < a ≤ b < ∞ . For any two choices of pseudouniformizers, a power of onedivides the other, so it follows that if (cid:101) U a,b ⊂ (cid:101) X × (cid:101) S denotes the preimage of {| t | b ≤ | u | ≤ | t | a } ,then the doubly-indexed ind-system { (cid:101) U a,b } U a,b and let j a,b : U a,b → X × S , j a : U a → X × S , j b : U b → X × S be the open immersions. We can now deﬁne the cohomology groups of interest, or rather the versionof pushforward along β : X × S → S . As usual, Λ is a coeﬃcient ring killed by some integer n prime to p . Definition

IV.5.2 . The functors Rβ !+ , Rβ ! − : D ´et ( X × S, Λ) → D ´et ( S, Λ) are deﬁned by Rβ !+ C := lim −→ a Rβ ∗ ( j a ! C | U a ) ,Rβ ! − C := lim −→ b Rβ ∗ ( j b ! C | U b ) for C ∈ D ´et ( X × S, Λ) . V.5. PARTIAL COMPACTLY SUPPORTED COHOMOLOGY 151

As the ind-systems of U a and U b are independent of all choices, these functors are canonical.The main result is the following. Here α : X × S → X and β : X × S → S are the twoprojections. Theorem

IV.5.3 . Assume that C = α ∗ A ⊗ L Λ β ∗ B for A ∈ D ´et ( X, Λ) and B ∈ D ´et ( S, Λ) . Then Rβ !+ C = 0 = Rβ ! − C. Remark

IV.5.4 . The essential case for applications is C = α ∗ A , i.e. B = Λ, and S = Spa k (( t )).In other words, we take any coeﬃcient system A on X , pull it back to X × Spa k (( t )), and then takethe partially compactly supported cohomology (relative to S ). However, it is sometimes useful toknow the result in the relative case, i.e. for general S , and then it is also natural to allow twists by B ∈ D ´et ( S, Λ).

Proof.

We write the proof for Rβ !+ ; the other case is exactly the same. Let X • → X be asimplicial hypercover by aﬃnoid perfectoid spaces X i = Spa( R i , R + i ) which are partially properover Spa k (i.e., R + i is minimal, i.e. the integral closure of k + R ◦◦ ). As X is a spatial diamond,we can arrange that the X i are the compactiﬁcations of quasi-pro-´etale maps to X (since X isspatial it admits an hypercover X • → X with X i aﬃnoid perfectoid and X i → X quasi-pro-´etale,since X → Spd k is partially proper this extends to a hypercover X c • → X where X ci is Huber’scanonical compactiﬁcation over Spa( k )); in particular, g i : X i → X satisﬁes dim . trg g i = 0 < ∞ .Let β • : X • × S → S be the corresponding projection. We claim that Rβ !+ C is the limit of Rβ • , !+ ( C | X • × S ), for any C ∈ D ´et ( X × S, Λ). Writing C as a limit of its Postnikovtruncations ([ Sch17a , Proposition 14.15]), we can assume C ∈ D +´et ( X × S, Λ). Now g i : X i → X is a qcqs map between spaces partially proper over ∗ , so g i is proper, and hence so its base change h i : X i × S → X × S . This implies that Rβ i, !+ ( h ∗ i C ) = Rβ !+ ( Rh i ∗ h ∗ i C ) , as j U ! commutes with Rh i ∗ by [ Sch17a , Theorem 19.2]. Now by [

Sch17a , Proposition 17.3], onesees that C is the limit of Rh i ∗ h ∗ i C . But Rβ !+ commutes with this limit, using that the ﬁlteredcolimit does as everything lies in D + (with a uniform bound).By the preceding reduction, we may assume that X = Spa( R, R + ) is an aﬃnoid perfectoidspace. We can even assume that X has no nonsplit ﬁnite ´etale covers (by taking the X i above tobe compactiﬁcations of strictly totally disconnected spaces). In that case, there is a map g : X → Y = Spa K , where K is the completed algebraic closure of k (( t )), which is necessarily proper (as X and Y are partially proper over ∗ ), and as above one has Rβ !+ C = Rβ Y, !+ ( Rh ∗ C )where β Y : Y × S → S is the projection and h : X × S → Y × S is the base change of g . Let α Y : Y × S → Y be the other projection. Then the projection formula (and properness of h )[ Sch17a , Proposition 22.11] show that Rh ∗ C = Rh ∗ ( α ∗ A ⊗ L Λ β ∗ B ) ∼ = Rh ∗ α ∗ A ⊗ L Λ β ∗ Y B and Rh ∗ α ∗ A ∼ = α ∗ Y Rg ∗ A by proper base change.

52 IV. GEOMETRY OF DIAMONDS

In other words, we can reduce to the case X = Spa K ; in particular A ∈ D ´et (Spa K, Λ) = D (Λ)is just a complex of Λ-modules. In that case, deﬁne U a,b and U a as above but taking X = (cid:101) X → Spa k (( t )) the natural map. We claim that in this case for all a > Rβ ∗ ( j a ! C | U a ) = 0 . To prove this, it suﬃces to see that for all a (cid:48) > a >

0, the cone of Rβ ∗ ( j a (cid:48) ! C | U a (cid:48) ) → Rβ ∗ ( j a ! C | U a )vanishes, as Rβ ∗ ( j a ! C | U a ) is the limit of these cones as a (cid:48) → ∞ . Now these cones depend on onlya quasicompact part of X × S , and hence their formation commutes with any base change in S ,cf. [ Sch17a , Proposition 17.6]. Therefore, we can reduce to the case S = Spa( L, L + ) for somecomplete algebraically closed nonarchimedean ﬁeld L with open and bounded valuation subring L + ⊂ L , and check on global sections R Γ( S, − ). Moreover, the cone commutes with all direct sumsin C , so one can assume that A ∈ D ´et (Spa K, Λ) = D ´et (Λ) is simply given by A = Λ.It remains to prove the following statement: For all B ∈ D ´et (Spa( L, L + ) , Λ) one has R Γ(Spa K × Spa(

L, L + ) , j a ! B | U a ) = 0 . If the stalk of B at the closed point vanishes, this follows from proper base change (writing Spa K × Spa(

L, L + ) as the union of its subspaces proper over Spa( L, L + )), [ Sch17a , Theorem 19.2]. Thuswe may assume that B is concentrated at the closed point of S . Analyzing the structure of U a ,one checks that in fact there is some map k (( u )) → L such that the ﬁbres over the closed pointof Spa( L, L + ) of U a and {| t | ≤ | u | a } agree. (A priori, it is a union of such subsets for a proﬁniteset of maps k (( u )) → L , but for any two such choices one is contained in the other, by comparingvaluations of the pseudouniformizers.) Thus, we can now assume that U a = {| t | ≤ | u | a } , and wecan also reduce to the case that B is constant. Now using as above that the cones for a (cid:48) > a > S and commute with direct sums in B , we can reduce to B = Λand the rank-1-geometric point S = Spa L where L is the completed algebraic closure of k (( u )).At this point, we can further replace Spa K by Spa k (( t )): One can write Spa K as the inverselimit over ﬁnite extensions of Spa k (( t )), each of which is isomorphic to Spa k (( t (cid:48) )), and although apriori R Γ(Spa K × Spa

L, j a ! Λ) does not take this inverse limit to a colimit, this does happen afterpassing to cones for maps for a (cid:48) > a >

0, which suﬃces as above. Finally, we have reduced toLemma IV.5.1. (cid:3)

IV.6. Hyperbolic localization

In this section we extend some results of Braden, [

Bra03 ], to the world of diamonds. Ourpresentation is also inspired by the work of Richarz, [

Ric19 ]. We will use these results throughoutour discussion of geometric Satake, starting in Section VI.3.Let S be a small v-stack, and let f : X → S be proper and representable in spatial diamondswith dim . trg f < ∞ , and assume that there is a G m -action on X/S , where G m is the v-sheafsending Spa( R, R + ) to R × . The ﬁxed point space X := X G m ⊂ X deﬁnes a closed subfunctor.We make the following assumption about the G m -action. Here, ( A ) + (resp. ( A ) − ) denotesthe aﬃne line Spa( R, R + ) (cid:55)→ R with the natural G m -action (resp. its inverse). V.6. HYPERBOLIC LOCALIZATION 153

Hypothesis

IV.6.1 . There is a decomposition of X into open and closed subsets X , . . . , X n such that for each i = 1 , . . . , n , there are locally closed G m -stable subfunctors X + i , X − i ⊂ X with X ∩ X + i = X i (resp. X ∩ X − i = X i ) such that the G m -action on X + i (resp. X − i ) extends to a G m -equivariant map ( A ) + × X + i → X + i (resp. ( A ) − × X − i → X − i ), and such that X = n (cid:91) i =1 X + i = n (cid:91) i =1 X − i . We let X + = n (cid:71) i =1 X + i X − = n (cid:71) i =1 X − i , so that there are natural maps q + : X + → Xq − : X − → X, as well as closed immersions i + : X → X + i − : X → X − and projections p + : X + → X p − : X − → X ;here p + is given by the restriction of ( A ) + × X + i → X + i to { }× X + i , and p − is deﬁned analogously.Although the decomposition of X into X i for i = 1 , . . . , n is a choice, ultimately the functors X + and X − are independent of any choice. Indeed, we have the following functorial description. Proposition

IV.6.2 . Consider the functor ( X + ) (cid:48) sending any perfectoid space T over S to theset of G m -equivariant maps from ( A ) + to X . There is a natural map X + → ( X + ) (cid:48) , as there is anatural G m -equivariant map ( A ) + × X + → X + → X . The map X + → ( X + ) (cid:48) is an isomorphism.Analogously, X − classiﬁes the set of G m -equivariant maps from ( A ) − to X . Proof.

It is enough to handle the case of X + . There is a natural map ( X + ) (cid:48) → X given byevaluating the G m -equivariant map on ( A ) + × ( X + ) (cid:48) → X on { }× X + . Let ( X + i ) (cid:48) = ( X + ) (cid:48) × X X i ;it is enough to prove that X + i → ( X + i ) (cid:48) is an isomorphism. For this, it is enough to prove that themap ( X + i ) (cid:48) → X given by evaluation at 1 is an injection whose image is contained in the locallyclosed subspace X + i ⊂ X . This can be checked after pullback to an aﬃnoid perfectoid base space S = Spa( R, R + ). As X/S is proper (in particular, separated) and G m × ( X + i ) (cid:48) ⊂ ( A ) + × ( X + i ) (cid:48) is dense, it follows that the map ( X + i ) (cid:48) → X is an injection. To bound its image, we can argue ongeometric points. If x ∈ | X | is any point in the image of | ( X + i ) (cid:48) | , and (cid:36) ∈ R is a pseudouniformizer

54 IV. GEOMETRY OF DIAMONDS with induced action γ on X , then the sequence γ n ( x ) converges to a point of | X i | for n → ∞ . Onthe other hand, if x (cid:54)∈ | X + i | , then x ∈ | X + j | for some j (cid:54) = i , which implies that γ n ( x ) converges toa point of | X j | for n → ∞ ; this is a contradiction.Thus, ( X + i ) (cid:48) embeds into X + i ⊂ X , but it also contains X + i , so indeed X + i = ( X + i ) (cid:48) . (cid:3) Lemma

IV.6.3 . The map j : X → X + × X X − is an open and closed immersion. Moreprecisely, for any i = 1 , . . . , n , the map j i : X i → X + i × X X − i is an isomorphism. Proof.

It is enough to prove that for any i = 1 , . . . , n , the map j i : X i → X + i × X X − i isan isomorphism. As it is a closed immersion (as X i → X is a closed immersion and the targetembeds into X ), it is enough to prove that it is bijective on geometric rank 1 points. Thus, we canassume S = Spa C , and let x : Spa C = S → X be a section that factors over X + i × X X − i . Thenthe G m -action on x extends to a G m -equivariant map g : P C → X . Consider the preimage of X + i under g ; this is a locally closed subfunctor, and it contains all geometric points. Indeed, on ( A ) + C ,the map g factors over X + i by hypothesis, and at ∞ , it maps into X i ⊂ X + i . This implies that thepreimage of X + i under g is all of P C . In particular, we get a map( A ) + × P C → ( A ) + × X + i → X + i which, when restricted to the copy of G m embedded via t (cid:55)→ ( t, t − ), is constant with value x .By continuity (and separatedness of X + i ), this implies that it is also constant with value x whenrestricted to A embedded via t (cid:55)→ ( t, t − ), i.e. the point (0 , ∞ ) ∈ ( A ) + × P C maps to x . On theother hand, when restricted to G m × {∞} , the map is constant with values in X i , and thus bycontinuity also on ( A ) + × {∞} . This implies that x ∈ X i , as desired. (cid:3) In this setup, we can deﬁne two functors D ´et ( X, Λ) → D ´et ( X , Λ). We use the diagrams X ± XX . p ± q ± Definition

IV.6.4 . Deﬁne the functors L + X/S = R ( p + ) ! ( q + ) ∗ : D ´et ( X, Λ) → D ´et ( X , Λ) ,L − X/S = R ( p − ) ∗ R ( q − ) ! : D ´et ( X, Λ) → D ´et ( X , Λ) , and a natural transformation L − X/S → L + X/S as follows. First, there are natural transformations R ( i + ) ! = R ( p + ) ! R ( i + ) ! R ( i + ) ! → R ( p + ) ! , R ( p − ) ∗ = ( i − ) ∗ ( p − ) ∗ R ( p − ) ∗ → ( i − ) ∗ , and the desired transformation L − X/S → L + X/S arises as a composite L − X/S = R ( p − ) ∗ R ( q − ) ! → ( i − ) ∗ R ( q − ) ! → ( Ri + ) ! ( q + ) ∗ → R ( p + ) ! ( q + ) ∗ = L + X/S , V.6. HYPERBOLIC LOCALIZATION 155 where the middle map ( i − ) ∗ R ( q − ) ! → ( Ri + ) ! ( q + ) ∗ of functors D ´et ( X, Λ) → D ´et ( X , Λ) is deﬁnedas the following composite ( i − ) ∗ R ( q − ) ! → ( i − ) ∗ R ( q − ) ! R ( q + ) ∗ ( q + ) ∗ = ( i − ) ∗ R ( (cid:101) q − ) ∗ R ( (cid:101) q + ) ! ( q + ) ∗ → ( i − ) ∗ R ( (cid:101) q − ) ∗ j ∗ j ∗ R ( (cid:101) q + ) ! ( q + ) ∗ = ( i − ) ∗ ( i − ) ∗ Rj ! R ( (cid:101) q + ) ! ( q + ) ∗ = R ( i + ) ! ( q + ) ∗ , using base change in the cartesian diagram X (cid:31) (cid:127) j (cid:47) (cid:47) X + × X X − (cid:101) q + (cid:15) (cid:15) (cid:101) q − (cid:47) (cid:47) X + q + (cid:15) (cid:15) X − q − (cid:47) (cid:47) X. Equivalently, it is enough to deﬁne for each i = 1 , . . . , n a natural transformation ( i − i ) ∗ R ( q − i ) ! → ( Ri + i ) ! ( q + i ) ∗ of functors D ´et ( X, Λ) → D ´et ( X i , Λ) . As X i i + i (cid:47) (cid:47) i − i (cid:15) (cid:15) X + iq + i (cid:15) (cid:15) X − i q − i (cid:47) (cid:47) X is cartesian, this arises as the composite ( i − i ) ∗ R ( q − i ) ! → ( i − i ) ∗ R ( q − i ) ! R ( q + i ) ∗ ( q + i ) ∗ = ( i − i ) ∗ ( i − i ) ∗ R ( i + ) ! ( q + i ) ∗ = R ( i + ) ! ( q + i ) ∗ . The following is our version of Braden’s theorem, [

Bra03 ], cf. [

Ric19 , Theorem B].

Theorem

IV.6.5 . For any A ∈ D ´et ( X/ G m , Λ) whose restriction to X we continue to denoteby A , the map L − X/S A → L + X/S A is an isomorphism. In fact, moreover for any A + ∈ D ´et ( X + / G m , Λ) , the map R ( i + ) ! A + → R ( p + ) ! A + is an isomorphism, and for any A − ∈ D ´et ( X − / G m , Λ) , the map R ( p − ) ∗ A − → ( i − ) ∗ A − is an isomorphism, so that L − X/S A = R ( p − ) ∗ R ( q − ) ! A ∼ = ( i − ) ∗ R ( q − ) ! A ∼ = R ( i + ) ! ( q + ) ∗ A ∼ = R ( p + ) ! ( q + ) ∗ A = L + X/S A is a series of isomorphisms. Before we start with the proof, we prove a certain general result about cohomology groups onspaces with “two ends”, a ﬂow connecting the two ends, and cohomology of sheaves, equivariantfor the ﬂow, that are compactly supported at only one end.

56 IV. GEOMETRY OF DIAMONDS

Proposition

IV.6.6 . Let S = Spa( R, R + ) be an aﬃnoid perfectoid space, (cid:36) ∈ R a pseudouni-formizer, let f : Y → S be a partially proper map of locally spatial diamonds, and assume that Y isequipped with a G m -action over S . Assume that the quotient v-stack Y / G m is qcqs. In that case, wecan ﬁnd a quasicompact open subset V ⊂ Y such that G m × V → Y is surjective and quasicompact.Write G m,S = lim −→ n ≥ U n , U n = { x ∈ G m,S | | x | ≤ | (cid:36) | − n } , and let j n : V n ⊂ Y be the open image of the cohomologically smooth map U n × S V ⊂ G m,S × S Y → Y .In this situation, we deﬁne for any A ∈ D ´et ( Y, Λ) the relative cohomology with partial supports lim −→ n Rf ∗ ( j n ! A | V n ) ∈ D ´et ( S, Λ) ; this functor is canonically independent of the choices made in its deﬁnition.For any A ∈ D +´et ( Y / G m , Λ) (resp. any A ∈ D ´et ( Y / G m , Λ) if dim . trg f < ∞ ), lim −→ n Rf ∗ ( j n ! A | V n ) = 0 . Remark

IV.6.7 . Assume that S = Spa C is a geometric rank 1 point. Then we set R Γ c + ( Y, A ) = lim −→ n R Γ( Y, j n ! A V n ) , which is exactly the above functor lim −→ n Rf ∗ ( j n ! A | V n ) under the identiﬁcation D ´et ( S, Λ) = D (Λ).Roughly speaking, the space Y has two ends, one given by (cid:83) n< γ n ( V ) for V large enough, where γ is the automorphism of Y induced by (cid:36) ∈ G m ( S ), and the other given by (cid:83) n> γ n ( V ). We areconsidering the cohomology groups of Y that have compact support in one of these directions, butnot in the other. If one replaces the G m -action by its inverse, this implies a similar result for thedirection of compact support interchanged. Proof.

The proof is analogous to the proof of Theorem IV.5.3, reducing to the case Y = G m × S , where it is a simple computation. Indeed, one can assume that A ∈ D +´et ( Y / G m , Λ) by aPostnikov limit argument (in case dim . trg f < ∞ ). Finding a v-hypercover of Y by spaces with G m -action of the form G m × X i , where each X i is a proper spatial diamond over S , and using v-hyperdescent, one reduces to the case Y = G m × X i . Then there is a projection G m × X i → G m,S ,and one reduces to Y = G m,S . In that case, the same arguments as in the previous section apply. (cid:3) Proof of Theorem IV.6.5.

We can assume that A ∈ D +´et ( X/ G m , Λ) by pulling through thePostnikov limit lim ←− n τ ≥− n A , noting that L + X/S commutes with limits while ( q + ) ∗ commutes withPostnikov limits and R ( p + ) ! as well by ﬁnite cohomological dimension.By choosing a v-hypercover of S by disjoint unions of strictly totally disconnected spaces S • ,and using v-hyperdescent, we can assume that S is a strictly totally disconnected space; indeed, L + X/S commutes with all limits, while ( q + ) ∗ and R ( p + ) ! commute with any base change and sopreserve cartesian objects, and thus also commute with the hyperdescent.We start by proving that for any A + ∈ D ´et ( X + / G m , Λ), the map R ( i + ) ! A + → R ( p + ) ! A +V.6. HYPERBOLIC LOCALIZATION 157 is an isomorphism, and similarly for any A − ∈ D ´et ( X − / G m , Λ), the map R ( p − ) ∗ A − → ( i − ) ∗ A − is an isomorphism. Let j + : X + \ X (cid:44) → X + , j − : X − \ X (cid:44) → X − denote the open embeddings.Then there are exact triangles( i + ) ∗ R ( i + ) ! A + → A + → R ( j + ) ∗ ( j + ) ∗ A + , ( j − ) ! ( j − ) ∗ A − → A − → ( i − ) ∗ ( i − ) ∗ A − . Using these triangles, we see that it is enough to see that for any B + ∈ D ´et (( X + \ X ) / G m , Λ), B − ∈ D ´et (( X − \ X ) / G m , Λ), one has R ( p + ) ! R ( j + ) ∗ B + = 0 , R ( p − ) ∗ R ( j − ) ! B − = 0as objects in D ´et ( X , Λ). This follows from Proposition IV.6.6 applied to S = X and Y = X + \ X (resp. Y = X − \ X ), and the following lemma. Lemma

IV.6.8 . The G m -action on X + \ X (resp. X − \ X ) has the property that the quotientv-stack ( X + \ X ) / G m (resp. ( X − \ X ) / G m ) is qcqs over S (thus, over X ). Proof.

It is enough to do the case of X + \ X , and we may restrict to X + i \ X i . We canassume that S = Spa( R, R + ) is an aﬃnoid perfectoid space, and ﬁx a pseudouniformizer (cid:36) ∈ R . As G m,S /(cid:36) Z is qcqs (in fact proper — a Tate elliptic curve), it is equivalent to prove that ( X + \ X ) /γ Z is qcqs, where γ is the automorphism given by the action of (cid:36) ∈ G m ( S ).Now we use the criterion of Lemma II.2.17 for the action of γ on | X + i | . As a locally closedpartially proper subspace of the proper spatial diamond X over S , the locally spectral space | X + i | istaut, and the condition on generizations is always fulﬁlled for locally spatial diamonds. The spectralclosed subspace | X i | ⊂ | X + i | is ﬁxed by γ , and by assumption for all x ∈ | X + i | , the sequence γ n ( x )for n → ∞ converges to a point of | X i | (as the G m -action extends to ( A ) + ). It remains to see thatfor all x ∈ | X + i \ X i | , the sequence γ n ( x ) for n → −∞ diverges in | X + i | . But x ∈ | X − j | for some j , and j (cid:54) = i by Lemma IV.6.3. Thus, γ n ( x ), for n → −∞ , converges to a point of | X j | , which isoutside of | X + i | , so the sequence diverges in | X + i | . (cid:3) Now it remains to see that for any A ∈ D +´et ( X/ G m , Λ), the map( i − ) ∗ R ( q − ) ! A → R ( i + ) ! ( q + ) ∗ A in D +´et ( X , Λ) = D + (( X ) ´et , Λ) is an isomorphism. This can be done locally on X , so ﬁx some i ∈ { , . . . , n } , and choose a quasicompact open neighborhood U ⊂ X of X i that does not meetany X j for j (cid:54) = i and such that X + i ∩ U , X − i ∩ U ⊂ U are closed. The G m -orbit Y = G m · U ⊂ X is still open, and contains X + i and X − i , necessarily as closed subsets.We are now in the situation of the next proposition. To check conditions (ii) and (iii) of thatproposition, note that we may ﬁnd a quasicompact open subspace V ⊂ Y such that Y = γ Z · V byaveraging U over {| (cid:36) | ≤ | t | ≤ } ⊂ G m . Let W be the closure of (cid:83) n ≥ γ n ( V ) ⊂ X . To check (iii),it suﬃces (by symmetry) to see that (cid:92) m ≥ γ m ( W ) = X − i

58 IV. GEOMETRY OF DIAMONDS in X . Note that X − i ⊂ (cid:83) n ≥ γ n ( V ) (as for all x ∈ X − i , the sequence γ − n ( x ) converges into X i ⊂ V ), so X − i is contained in W , and thus in (cid:84) m ≥ γ m ( W ). To prove the converse inclusion, let W (cid:48) = (cid:84) m ≥ γ m ( W ). If X − i (cid:40) W (cid:48) , then there is some j (cid:54) = i such that X − j contains a quasicompactopen subset A ⊂ W (cid:48) . Then (cid:101) A = (cid:83) n ≥ γ − n ( A ) is a γ − -invariant open subset of W (cid:48) whose closureis γ − ( N ∪{∞} ) · A ; in particular, replacing A by γ − n ( A ) if necessary, we can arrange that this closureis contained in any given small neighborhood of X j , and in particular intersects V trivially. Then γ n ( V ) ∩ A = γ n ( V ∩ γ − n ( A )) = ∅ for all n ≥

0, and hence A intersects (cid:83) n ≥ γ n ( V ) trivially, andthen also its closure W . But we assumed that A ⊂ W (cid:48) ⊂ W , giving a contradiction. (cid:3) Proposition

IV.6.9 . Let S = Spa( R, R + ) be a strictly totally disconnected perfectoid space, let f : Y → S be a compactiﬁable map of locally spatial diamonds, and assume that Y /S is equippedwith a G m -action, with ﬁxed points Y ⊂ Y , and the following properties. (i) There are G m -invariant closed subspaces q + : Y + ⊂ Y , q − : Y − ⊂ Y , containing Y (via i + : Y → Y + , i − : Y → Y − ) such that the action maps extend to maps ( A ) + × Y + → Y + resp. ( A ) − × Y − → Y − . (ii) The quotient v-stack

Y / G m is quasicompact. In particular, picking a pseudouniformizer (cid:36) ∈ R with induced action γ on Y , we can ﬁnd some quasicompact open V ⊂ Y such that Y = γ Z · V . (iii) With V as in (ii), let W − be the closure of γ N · V and W + the closure of γ − N · V . Then (cid:84) n ≥ γ n ( W − ) = Y − and (cid:84) n ≥ γ − n ( W + ) = Y + .Then Y is a spatial diamond, the diagram Y i + (cid:47) (cid:47) i − (cid:15) (cid:15) Y + q + (cid:15) (cid:15) Y − q − (cid:47) (cid:47) Y is cartesian, the quotient v-stacks ( Y \ Y + ) / G m and ( Y \ Y − ) / G m are qcqs, and for all A ∈ D +´et ( Y / G m , Λ) (resp. all A ∈ D ´et ( Y / G m , Λ) if dim . trg f < ∞ ) whose pullback to Y we continue todenote by A , the map ( i − ) ∗ R ( q − ) ! A → ( i − ) ∗ R ( q − ) ! ( q + ) ∗ ( q + ) ∗ A = ( i − ) ∗ ( i − ) ∗ R ( i + ) ! ( q + ) ∗ A = R ( i + ) ! ( q + ) ∗ A in D ´et ( Y , Λ) is an isomorphism. Proof.

Note that Y / G m ⊂ Y / G m is closed and thus Y / G m is quasicompact. As the G m -action is trivial on Y , this implies that Y is quasicompact. As Y ⊂ Y is closed and Y → S iscompaciﬁable and in particular quasiseparated, we see that Y → S is qcqs. That the diagram iscartesian follows from the proof of Lemma IV.6.3.Next, we check that ( Y \ Y + ) / G m and ( Y \ Y − ) / G m are qcqs. By symmetry and as G m,S /γ Z is qcqs, it suﬃces to see that ( Y \ Y − ) /γ Z is qcqs. First, we check that it is quasiseparated. Takeany quasicompact open subspace V − ⊂ Y \ Y − i ; we need to see there are only ﬁnitely many n with V − ∩ γ n ( V − ) (cid:54) = ∅ . We can assume that V − ⊂ (cid:83) n ≥ ( V ) (translating by a power of γ ifnecessary), and then V − is covered by the open subsets V − \ γ m ( W − ) ⊂ V − by the claim above. V.6. HYPERBOLIC LOCALIZATION 159

By quasicompacity, the intersection of V − with γ m ( W − ) is empty for some large enough m , butthen also the intersection of V − with γ m (cid:48) ( V − ) ⊂ γ m ( W − ) for m (cid:48) ≥ m is empty.To see that ( Y \ Y − ) /γ Z is quasicompact, note that V \ (cid:83) n> γ n ( V ) is a spectral space (as itis closed in V ) that maps bijectively to ( Y \ Y − ) /γ Z .Now, for the cohomological statement, we can as usual assume that A ∈ D +´et ( Y / G m , Λ) bya Postnikov limit argument. Then we are interested in checking that a map in D +´et ( Y , Λ) = D + ( Y , ´et , Λ) (cf. [

Sch17a , Remark 14.14]) is an isomorphism, so we need to check that the sectionsover all quasicompact separated ´etale Y (cid:48) → Y agree. Now we claim that any such quasicompactseparated ´etale Y (cid:48) → Y lifts to a G m -equivariant quasicompact separated ´etale map Y (cid:48) → Y ; thiswill then allow us to assume Y (cid:48) = Y via passing to the pullback of everything to Y (cid:48) .To see that one may lift Y (cid:48) → Y to Y (cid:48) → Y , consider the open subspace V ( n ) ⊂ Y givenas the intersection of (cid:83) m ≥ n γ m ( V ) with (cid:83) m ≤− n γ m ( V ). It follows from the topological situationthat this is still quasicompact, and that the intersection of all V ( n ) is equal to Y (using condition(iii)). Let Y ( n ) = γ Z · V ( n ) ⊂ Y . Then γ -equivariant quasicompact separated ´etale maps to Y ( n ) are equivalent to quasicompact separated ´etale maps to V ( n ) together with isomorphisms betweenthe two pullbacks to V ( n ) ∩ γ ( V ( n ) ). The latter data extends uniquely from Y to V ( n ) for smallenough n by [ Sch17a , Proposition 11.23]. Repeating a similar argument after taking a productwith G m,S /γ Z (which is qcqs), and observing that the Y ( n ) are coﬁnal with their G m -orbits, onecan then attain G m -equivariance.We have now reduced to checking the statement on global sections. Now consider the compact-iﬁcation j : Y (cid:44) → Y = Y /S → S . Note that Y satisﬁes all the same conditions of the proposition.Restricted to Y , this gives a quasicompact open immersion j : Y (cid:44) → Y . By the above argument,this quasicompact open immersion spreads to a quasicompact open immersion into Y , and by tak-ing it small enough in the argument above, we can assume that it is contained in Y . This allowsus to assume that j is quasicompact. In that case the functor Rj ∗ commutes with all operations inquestion by [ Sch17a , Proposition 17.6, Proposition 23.16 (i)]. Thus, we can now moreover assumethat Y is partially proper.Our goal now is to prove that when Y is partially proper and A ∈ D +´et ( Y / G m , Λ), the map( i − ) ∗ R ( q − ) ! A → R ( i + ) ! ( q + ) ∗ A becomes an isomorphism after applying Rf ∗ where f : Y → S is the proper map. For this, wedeﬁne another functor D ´et ( Y, Λ) → D ´et ( S, Λ), as follows. Let j n : V n = (cid:83) m ≥− n γ n ( V ) (cid:44) → Y for n ≥

0. Then we consider A (cid:55)→ F ( A ) = lim −→ n Rf ∗ ( j n ! A | V n ) : D ´et ( Y, Λ) → D ´et ( S, Λ) . Lemma

IV.6.10 . Let j − : Y \ Y − → Y , j + : Y \ Y + → Y denote the open immersions. (i) If A = Rj −∗ A − for A − ∈ D +´et (( Y \ Y − ) / G m , Λ) , then F ( A ) = 0 . (ii) If A = j +! A + for A + ∈ D +´et (( Y \ Y + ) / G m , Λ) , then F ( A ) = 0 . Proof.

This follows Proposition IV.6.6 and condition (iii). (cid:3)

60 IV. GEOMETRY OF DIAMONDS

There are natural transformations Rf ∗ R ( p − ) ∗ R ( q − ) ! → F → Rf ∗ R ( p + ) ! ( q + ) ∗ , and the lemmaimplies that these are equivalences when evaluated on A ∈ D +´et ( Y / G m , Λ). Using that also ( Y + \ Y ) / G m and ( Y − \ Y ) / G m are qcqs (as closed subspaces of ( Y \ Y − ) / G m resp. ( Y \ Y + ) / G m ) sothat we can apply Proposition IV.6.6 again as in the beginning of the proof of Theorem IV.6.5, weget an isomorphism Rf ∗ ( i − ) ∗ R ( q − ) ! A ∼ = Rf ∗ R ( p − ) ∗ R ( q − ) ! A ∼ = F ( A ) ∼ = Rf ∗ R ( p + ) ! ( q + ) ∗ A ∼ = Rf ∗ R ( i + ) ! ( q + ) ∗ A. We need to see that this implies that also the map Rf ∗ ( i − ) ∗ R ( q − ) ! A → Rf ∗ R ( i + ) ! ( q + ) ∗ A deﬁned in the statement of the proposition is an isomorphism. For this, observe that this map isan isomorphism if and only if for A = j +! A + with A + ∈ D ´et (( Y \ Y + ) / G m , Λ), one has Rf ∗ ( i − ) ∗ R ( q − ) ! A = 0 . But this follows from the existence of some isomorphism Rf ∗ ( i − ) ∗ R ( q − ) ! A ∼ = Rf ∗ R ( i + ) ! ( q + ) ∗ A = 0 , using ( q + ) ∗ A = ( q + ) ∗ j +! A + = 0. (cid:3) Using Theorem IV.6.5, we give the following deﬁnition.

Definition

IV.6.11 . Let f : X → S with G m -action be as above, satisfying Hypothesis IV.6.1.Let D ´et ( X, Λ) G m -mon ⊂ D ´et ( X, Λ) be the full subcategory generated under ﬁnite colimits and retractsby the image of D ´et ( X/ G m , Λ) → D ´et ( X, Λ) . The hyperbolic localization functor is the functor L X/S : D ´et ( X, Λ) G m -mon → D ´et ( X , Λ) given by L − X/S ∼ = L + X/S . We observe that Theorem IV.6.5 implies the following further results.

Proposition

IV.6.12 . In the situation of Deﬁnition IV.6.11, let g : S (cid:48) → S be a map of smallv-stacks, with pullback f (cid:48) : X (cid:48) = X × S S (cid:48) → S (cid:48) , g X : X (cid:48) → X , g : X (cid:48) → X . Then there arenatural equivalences g ∗ L X/S ∼ = L X (cid:48) /S (cid:48) g ∗ X , L X/S Rg X ∗ ∼ = Rg ∗ L X (cid:48) /S (cid:48) , L X/S Rg X ! ∼ = Rg L X (cid:48) /S (cid:48) , Rg L X/S ∼ = L X (cid:48) /S (cid:48) Rg ! X , the latter two in case g is compactiﬁable and representable in locally spatial diamonds with dim . trg g < ∞ (so that the relevant functors are deﬁned). Proof.

The ﬁrst and third assertions are clear for L + X/S , while the second and fourth assertionsare clear for L − X/S . (cid:3) Proposition

IV.6.13 . In the situation of Deﬁnition IV.6.11, let A ∈ D ´et ( X, Λ) G m -mon and B ∈ D ´et ( S, Λ) . Let L (cid:48) X/S denote the hyperbolic localization functor for the inverse G m -action.Then there is natural isomorphism R H om( L X/S ( A ) , Rf B ) ∼ = L (cid:48) X/S R H om( A, Rf ! B ) . In particular, taking B = Λ , hyperbolic localization commutes with Verdier duality, up to changingthe G m -action. V.7. DRINFELD’S LEMMA 161

Proof.

More generally, for all A ∈ D ´et ( X, Λ) and B ∈ D ´et ( S, Λ), we have a natural isomor-phism R H om( L + X/S ( A ) , Rf B ) ∼ = L (cid:48)− X/S R H om( A, Rf ! B ). Indeed, R H om( L + X/S ( A ) , Rf B ) = R H om( R ( p + ) ! ( q + ) ∗ A, Rf B ) ∼ = R ( p + ) ∗ R H om(( q + ) ∗ A, R ( p + ) ! Rf B ) ∼ = R ( p + ) ∗ R H om(( q + ) ∗ A, R ( q + ) ! Rf ! B ) ∼ = R ( p + ) ∗ R ( q + ) ! R H om( A, Rf ! B ) . (cid:3) Proposition

IV.6.14 . In the situation of Deﬁnition IV.6.11, assume that A ∈ D ´et ( X, Λ) G m -mon is f -universally locally acyclic. Then L X/S ( A ) ∈ D ´et ( X , Λ) is universally locally acyclic withrespect to f : X ⊂ X → S . Proof.

As the assumption is stable under base change, we may assume that S is strictly totallydisconnected, and it suﬃces to see that L X/S ( A ) is f -locally acyclic. For condition (a), we can infact assume that S = Spa( C, C + ) is strictly local; let j : S = Spa( C, O C ) ⊂ S be the generic openpoint. Then we have to see that L X/S ( A ) = Rj ∗ ( L X/S ( A ) | X × S S ), where j : X × S S → X isthe pullback of j . But this follows from Proposition IV.6.12 and the corresponding property of A .For condition (b), it suﬃces to see that the functor R H om Λ ( L X/S ( A ) , Rf − ) commutes withall direct sums, as then its left adjoint Rf ( A ⊗ L Λ − ) preserves perfect-constructible complexes. Forthis, we compute this functor: R H om Λ ( L X/S ( A ) , Rf − ) ∼ = L (cid:48) X/S R H om Λ ( A, Rf ! − ) ∼ = L (cid:48) X/S ( D X/S ( A ) ⊗ L Λ f ∗ − ) . Here, we used Proposition IV.6.13 and Proposition IV.2.19. The ﬁnal functor clearly commuteswith all direct sums, giving the desired result. (cid:3)

IV.7. Drinfeld’s lemma

As a ﬁnal topic of this chapter, we prove the version of Drinfeld’s lemma that we will need inthis paper. Contrary to the classical formulation [

Dri80 , Theorem 2.1], cf. also [

Lau04 , Theorem8.1.4], this version actually makes the Weil group of E , not the absolute Galois group of E , appear.(Also, it is worth remarking that usually, a global Galois group appears, not a local Galois group.)In this section, we work on Perf k where k = F q . In that case, we can write the moduli spaceof degree 1 Cartier divisors on the Fargues–Fontaine curve as Div = Spd ˘ E/ϕ Z . This admits anatural map ψ : Div → [ ∗ /W E ]to the classifying space of the Weil group of E . Indeed, if C = (cid:98) E is a completed algebraic closureof E , then there is an action of W E on Spd C , with the inertia subgroup I E ⊂ W E acting via itsusual action, while Frobenius elements act via the composite of the usual action and the Frobeniusof Spd C . More precisely, τ ∈ W E acts as τ ◦ Frob − deg τ where deg : W E → Z is the projection;note that this as a map over Spd k as on Spd k the two Frobenii cancel. The natural map[Spd C/W E ] → [Spd ˘ E/ϕ Z ]

62 IV. GEOMETRY OF DIAMONDS is an isomorphism, thus yielding the natural map ψ : [Spd ˘ E/ϕ Z ] ∼ = [Spd C/W E ] → [ ∗ /W E ] . One could equivalently compute W E × Spd C ∼ = Spd C × Div Spd C for the natural map Spd C → Div to arrive at the result.In particular, for any small v-stack X , we get a natural map ψ X : X × Div → X × [ ∗ /W E ] . Proposition

IV.7.1 . The functor ψ ∗ X : D ´et ( X × [ ∗ /W E ] , Λ) → D ´et ( X × Div , Λ) is fully faithful. If the natural pullback functor D ´et ( X, Λ) → D ´et ( X × Spd C, Λ) is an equivalence, then ψ ∗ X is also an equivalence. Proof.

We apply descent along ∗ → [ ∗ /W E ]. This describes D ´et ( X × [ ∗ /W E ] , Λ) in terms ofcartesian objects in D ´et ( X × W E • , Λ), and D ´et ( X × Div , Λ) in terms of cartesian objects in D ´et ( X × Spd C × W E • , Λ). By [

Sch17a , Theorem 1.13], all functors D ´et ( X × W E • , Λ) → D ´et ( X × Spd C × W E • , Λ) are fully faithful; this implies the fully faithfulness. Moreover, for essential surjectivity oncartesian objects it is enough to know essential surjectivity on the degree 0 part of the simplicialresolution, i.e. for D ´et ( X, Λ) → D ´et ( X × Spd C, Λ), giving the desired result. (cid:3)

We note the following immediate corollary.

Corollary

IV.7.2 . For any ﬁnite set I , pullback along X × (Div ) I → X × [ ∗ /W IE ] induces afully faithful functor D ´et ( X × [ ∗ /W IE ] , Λ) → D ´et ( X × (Div ) I , Λ) . Proof.

This follows inductively from Proposition IV.7.1. (cid:3)

We need the following reﬁnement, see Proposition VI.9.2. For any small v-stack Y , let D lc ( Y, Λ) ⊂ D ´et ( Y, Λ)be the full subcategory of all objects that are locally constant with perfect ﬁbres.

Proposition

IV.7.3 . For any ﬁnite set I and any small v-stack X , the functor D lc ( X × [ ∗ /W IE ] , Λ) → D lc ( X × (Div ) I , Λ) is an equivalence of categories. We will mostly be using this in case X is a point. V.7. DRINFELD’S LEMMA 163

Proof.

By Corollary IV.7.2, the functor is fully faithful. By induction, we can reduce to thecase that I has one element. By descent, we can assume that X is strictly totally disconnected.Note that X × Div is a spatial diamond, and using [ Sch17a , Proposition 20.15] we can reduce tothe case that X = Spa( C, C + ) is strictly local (by writing any connected component as a coﬁlteredinverse limit of its open and closed neighborhoods to see that then any object is locally in the imageof the functor). Moreover, the category D lc is unchanged if we replace Spa( C, C + ) by Spa( C, O C ),so we can assume that X is even a geometric rank 1 point.At this point, we need to simplify the coeﬃcient ring Λ. The algebra Λ is a Z /n Z -algebrafor some n prime to p ; we can then assume n is a power of some prime (cid:96) (cid:54) = p , and in fact even n = (cid:96) by an induction argument. By [ Sch17a , Proposition 20.15], we can also assume that Λ is aﬁnitely generated F (cid:96) -algebra. Taking a surjection from a polynomial algebra, one can then assumethat Λ = F (cid:96) [ T , . . . , T d ]. Applying [ Sch17a , Proposition 20.15] again, we can assume that Λ is thelocalization of F (cid:96) [ T , . . . , T d ] at a closed point, or applying faithfully ﬂat descent in the coeﬃcients,that Λ is the completion of F (cid:96) [ T , . . . , T d ] at a closed point, but equipped with the discrete topology.Also note that this ring is regular, so all truncations of perfect complexes are perfect, and we canassume that the complex is concentrated in degree 0.We are now in the following situation. We have an ´etale sheaf A of Λ ∼ = F (cid:96) r [[ T , . . . , T d ]]-modules on S = Spa C × Div , such that for some ﬁnitely generated Λ-module M , there are ´etalelocal isomorphisms between A and the constant Λ-module associated to M . Our goal is to see thatafter pullback along the W E -torsor (cid:101) S = Spa C × Spd (cid:98) E → S = Spa C × Div , there is an isomorphism between A and M . To see this, we will also need to analyze the behaviourat a carefully chosen geometric point. In fact, by Lemma II.1.14 we can ﬁnd a point Spa K → Y C of the curve Y C associated with C such that the induced map Gal( K | K ) → I E is surjective. Thisinduces a point y : Spd K → S , and we can lift it to a geometric point (cid:101) y : Spd (cid:98) K → (cid:101) S . Deﬁne M as the stalk of A at (cid:101) y ; our goal is then to prove the existence of a unique isomorphism between A | (cid:101) Y and M that is the identity at (cid:101) y .To prove this, we ﬁrst reduce modulo ( T , . . . , T d ) n . Then Λ n := Λ / ( T , . . . , T d ) n is a ﬁnite ring,and the space of isomorphisms between A and M is parametrized by a scheme ﬁnite ´etale over Y .By [ SW20 , Lemma 16.3.2], all such ﬁnite ´etale covers come via pullback from ﬁnite ´etale covers ofDiv , and are thus trivialized after pullback to (cid:101) Y ; this implies that there is a unique isomorphism A/ ( T , . . . , T d ) n ∼ = M/ ( T , . . . , T d ) n reducing to the identity at (cid:101) y .Taking the limit over n , we get an isomorphism (cid:98) A | (cid:101) Y ∼ = (cid:99) M | (cid:101) Y between the pro-´etale sheaves (cid:98) A =lim ←− n A/ ( T , . . . , T d ) n and (cid:99) M = lim ←− n M/ ( T , . . . , T d ) n after pullback to (cid:101) Y . This gives in particularan automorphism of (cid:99) M over (cid:101) Y × Y (cid:101) Y ∼ = W E × (cid:101) Y , and thus by connectedness of (cid:101) Y a continuous map W E → Aut Λ ( (cid:99) M ) (in fact, it extends continuouslyto the absolute Galois group of E ). We claim that this map is trivial on an open subgroup of I E (but not necessarily on an open subgroup of the absolute Galois group of E — here it is necessaryto pass to the Weil group). Indeed, restricting the map W E → Aut Λ ( (cid:99) M ) to Gal( K | K ) gives a

64 IV. GEOMETRY OF DIAMONDS map Gal( K | K ) → Aut Λ ( (cid:99) M ) that is in fact continuous for the discrete topology on the target,as a local system of A -modules on Spd K is given a continuous representation of Gal( K | K ). AsGal( K | K ) → I E is surjective, we get the claim.By equivariance under an open subgroup of I E , we ﬁnd that the isomorphism (cid:98) A | (cid:101) Y ∼ = (cid:99) M | (cid:101) Y descends, necessarily uniquely, to an isomorphism overSpa C × Spd E (cid:48) for some ﬁnite extension E (cid:48) | ˘ E . Now we take the pushforward of the isomorphism (cid:98) A | Spa C × Spd E (cid:48) ∼ = (cid:99) M | Spa C × Spd E (cid:48) to the ´etale site of Spa C × Spd E (cid:48) . As any ´etale U → Spa C × Spd E (cid:48) is locallyconnected, we have H ( U, M ) = H ( U, (cid:99) M ) and then also H ( U, A ) = H ( U, (cid:98) A ) (as A is ´etale locallyisomorphic to M ) for all such U , so we get the desired isomorphism A | Spa C × Spd E (cid:48) ∼ = M | Spa C × Spd E (cid:48) . (cid:3) HAPTER V D ´et (Bun G ) In this chapter, we want to understand the basic structure of D ´et (Bun G , Λ), building it up fromall D ´et (Bun bG , Λ), where we continue to work in the setting where Λ is killed by some integer n prime to p .Throughout this chapter, we ﬁx an algebraically closed ﬁeld k | F q and work on Perf k . Our goalis to prove the following theorem. Theorem

V.0.1 (Theorem V.3.7, Proposition V.3.6; Proposition V.2.2, Theorem V.1.1; Theo-rem V.4.1; Theorem V.5.1; Theorem V.7.1) . Let Λ be any ring killed by some integer n prime to p . (o) For any b ∈ B ( G ) , there is a map π b : M b → Bun G that is representable in locally spatial diamonds, partially proper and cohomologically smooth, where M b parametrizes G -bundles E together with an increasing Q -ﬁltration whose associated graded is,at all geometric points, isomorphic to E b with its slope grading. The v-stack M b is representable inlocally spatial diamonds, partially proper and cohomologically smooth over [ ∗ /G b ( E )] . (i) Via excision triangles, there is an inﬁnite semiorthogonal decomposition of D (Bun G , Λ) into thevarious D (Bun bG , Λ) for b ∈ B ( G ) . (ii) For each b ∈ B ( G ) , pullback along Bun bG ∼ = [ ∗ / (cid:101) G b ] → [ ∗ /G b ( E )] gives an equivalence D ([ ∗ /G b ( E )] , Λ) ∼ = D (Bun bG , Λ) , and D ([ ∗ /G b ( E )] , Λ) ∼ = D ( G b ( E ) , Λ) is equivalent to the derived category of the category of smoothrepresentations of G b ( E ) on Λ -modules. (iii) The category D (Bun G , Λ) is compactly generated, and a complex A ∈ D (Bun G , Λ) is compactif and only if for all b ∈ B ( G ) , the restriction i b ∗ A ∈ D (Bun bG , Λ) ∼ = D ( G b ( E ) , Λ) is compact, and zero for almost all b . Here, compactness in D ( G b ( E ) , Λ) is equivalent to lying inthe thick triangulated subcategory generated by c - Ind G b ( E ) K Λ as K runs over open pro- p -subgroupsof G b ( E ) . D ´et (Bun G ) (iv) On the subcategory D (Bun G , Λ) ω ⊂ D (Bun G , Λ) of compact objects, there is a Bernstein–Zelevinsky duality functor D BZ : ( D (Bun G , Λ) ω ) op → D (Bun G , Λ) ω with a functorial identiﬁcation R Hom(

First, we want to understand D ´et ([ ∗ /G ] , Λ) for a locally pro- p -group G . Fix a coeﬃcient ringΛ such that n Λ = 0 for some n prime to p , and assume that G is locally pro- p . Our aim is to provethe following theorem. Theorem

V.1.1 . Let D ( G, Λ) be the derived category of the category of smooth representationsof G on Λ -modules. There is a natural symmetric monoidal equivalence D ( G, Λ) (cid:39) D ´et ([ ∗ /G ] , Λ) under which the functor D ( G, Λ) → D (Λ) forgetting the G -action gets identiﬁed with the pullbackfunctor D ´et ([ ∗ /G ] , Λ) → D ´et ( ∗ , Λ) = D (Λ) under the projection ∗ → [ ∗ /G ] .The same result holds true for the base change [Spa C/G ] = [ ∗ /G ] × Spa C for any completealgebraically closed nonarchimedean ﬁeld C/k ; more precisely, the base change functor D ´et ([ ∗ /G ] , Λ) → D ´et ([Spa C/G ] , Λ) is an equivalence. Note that indeed D ´et ( ∗ , Λ) = D (Λ) . This follows from applying [

Sch17a , Theorem 1.13 (ii)] to the small v-stack X = ∗ . In fact, forany complete algebraically closed ﬁeld C , one has D ´et (Spa C, Λ) = D (Λ) and there is a sequence D (Λ) −→ D ´et ( ∗ , Λ) fully faithful −−−−−−−−−→ D ´et (Spa C, Λ) = D (Λ)and D (Λ) → D ´et ( ∗ , Λ) is thus an equivalence. .1. CLASSIFYING STACKS 167

Proof.

We start by constructing a functor D ( G, Λ) → D ´et ([ ∗ /G ] , Λ)compatible with the derived tensor product and the forgetful functors. For this, one ﬁrst con-structs a functor from the category of smooth representations of G on Λ-modules to the heart of D ´et ([ ∗ /G ] , Λ); note that this heart is a full subcategory of the heart of D ([ ∗ /G ] v , Λ), which is thecategory of v-sheaves on [ ∗ /G ]. Now one can send a smooth G -representation V to the v-sheaf F V on [ ∗ /G ] that takes a perfectoid space X with a G -torsor (cid:101) X → X to the set of all continuous G -equivariant maps from | (cid:101) X | to V . In fact one checks that for any perfectoid space S and anylocally proﬁnite set A , | S × A | = | S | × A, and thus | (cid:101) X | has a continuous G -action. As v-covers induce quotient maps by [ Sch17a , Proposition12.9], this is indeed a v-sheaf. Moreover, after pullback along ∗ → [ ∗ /G ], it is given by the functorwhich sends X to the set of continuous G -equivariant maps from | X | × G = | X × G | to V . These arecanonically the same (via restriction to X × { } ) as continuous maps | X | → V , so that F V | ∗ = V isthe v-sheaf corresponding to V . As V is discrete, this is a disjoint union of points, and in particular(after pullback to any Spa C ) an ´etale sheaf. According to [ Sch17a , Deﬁnition 14.13], this impliesthat F V ∈ D ´et ([ ∗ /G ] , Λ), as desired.From now on, we will simply write V for F V . Given any complex of smooth G -representations V • , one can form the corresponding complex V • of v-sheaves on [ ∗ /G ], which deﬁnes an objectof D ´et ([ ∗ /G ] , Λ) ⊂ D ([ ∗ /G ] v , Λ) (using [

Sch17a , Proposition 14.16]), giving the desired functor D ( G, Λ) → D ´et ([ ∗ /G ] , Λ) compatible with the forgetful functors. One checks that this functor iscompatible with derived tensor products by unraveling the deﬁnitions.To check whether the functor is an equivalence, we may by [

Sch17a , Theorem 1.13 (ii)] replace[ ∗ /G ] by its base change [Spa C/G ] = [ ∗ /G ] × Spa C , where C is some complete algebraically closednonarchimedean ﬁeld.For the v-stack X = [Spa C/G ], we can also consider its ´etale site X ´et ⊂ X v consisting of all Y ∈ X v which are ´etale (and locally separated) over X . This recovers a classical site. Lemma

V.1.2 . The ´etale site X ´et is equivalent to the category G - Set of discrete G -sets, viasending a discrete set S with continuous G -action to [ S × Spa

C/G ] . Proof.

It is clear that the functor S (cid:55)→ [ S × Spa

C/G ] maps to X ´et ⊂ X v (as the pullback toSpa C is given by S × Spa C ), and is fully faithful. Conversely, if Y → X = [Spa C/G ] is ´etale,then the pullback of Y to Spa C is a discrete set, on which G acts continuously, giving the descentdatum deﬁning Y . (cid:3) Lemma

V.1.3 . There is a natural equivalence D ( G, Λ) (cid:39) D ( G - Set , Λ) , such that the followingdiagram commutes D ( G, Λ) ∼ = (cid:47) (cid:47) (cid:15) (cid:15) D ( G - Set , Λ) ∼ = (cid:47) (cid:47) D ([Spa C/G ] ´et , Λ) (cid:15) (cid:15) D ´et ([ ∗ /G ] , Λ) (cid:31) (cid:127) (cid:47) (cid:47) D ´et ([Spa C/G ] , Λ) .

68 V. D ´et (Bun G ) Proof.

It is enough to give an equivalence of abelian categories between smooth G -representationson Λ-modules, and sheaves of Λ-modules on discrete G -sets. The construction of the functor is asbefore, and it is clearly fully faithful. But any sheaf of Λ-modules F on G - Set comes from thesmooth G -representation V = lim −→ H ⊂ G F ( G/H ), where H runs over all open subgroups of G . Onedirectly veriﬁes that the diagram commutes. (cid:3) It remains to see that the natural functor D ([Spa C/G ] ´et , Λ) → D ´et ([Spa C/G ] , Λ)is an equivalence. We claim that this reduces to the case that G is pro- p : We ﬁrst reduce fullyfaithfulness to this case. For this, we have to see that, if λ : X v → X ´et denotes the map of sites,then for any A ∈ D ([Spa C/G ] ´et , Λ), the natural map A → Rλ ∗ λ ∗ A is an equivalence. This can be checked locally on [Spa C/G ] ´et , meaning that we can replace G by an open pro- p -subgroup. Similarly, for essential surjectivity, one needs to see that for all B ∈ D ´et ([Spa C/G ] , Λ), the map λ ∗ Rλ ∗ B → B is an equivalence, which can again be checked locally.Thus, we can assume that G is pro- p . Note that ([Spa C/G ] ´et , Λ) is locally of cohomologicaldimension 0, as there is no continuous group cohomology of pro- p -groups on Λ-modules if n Λ = 0for n prime to p . This implies (cf. [ Sta , Tag 0719]) that D ([Spa C/G ] ´et , Λ) is left-complete. As D ´et ([Spa C/G ] , Λ) is also left-complete by [

Sch17a , Proposition 14.11], it is enough to see that thefunctor D + ([Spa C/G ] ´et , Λ) → D +´et ([Spa C/G ] , Λ)is an equivalence. First, we check fully faithfulness, i.e. that the unit id → Rλ ∗ λ ∗ of the adjunctionis an equivalence. For this, it is enough to see that for any ´etale sheaf of Λ-modules, i.e. any smooth G -representation V , one has R Γ([Spa

C/G ] v , V ) = V G , i.e. its H is V G and there are no higher H i . However, one can compute v-cohomology using theCech nerve for the cover Spa C → [Spa C/G ], which produces the complex of continuous cochains,giving the desired result.Finally, for essential surjectivity, it is now enough to check on the heart. But if F is a v-sheafon [Spa C/G ] whose pullback to Spa C is an ´etale sheaf, then this pullback is a disjoint union ofpoints, thus separated and ´etale, and therefore F is itself a v-stack which is ´etale over [Spa C/G ],and so deﬁnes an object in the topos [Spa

C/G ] ´et . (cid:3) Corollary

V.1.4 . The operation R H om Λ ( − , Λ) : D ´et ([ ∗ /G ] , Λ) op → D ´et ([ ∗ /G ] , Λ) corresponds to the derived smooth duality functor A (cid:55)→ ( A ∗ ) sm : D ( G, Λ) op → D ( G, Λ) induced on derived categories by the left-exact smooth duality functor V (cid:55)→ ( V ∗ ) sm = { f : V → Λ | ∃ H ⊂ G open ∀ h ∈ H, v ∈ V : f ( hv ) = f ( v ) } . .2. ´ETALE SHEAVES ON STRATA 169 Proof.

The operation A (cid:55)→ ( A ∗ ) sm on D ( G, Λ) satisﬁes the adjunctionHom D ( G, Λ) ( B, ( A ∗ ) sm ) = Hom D ( G, Λ) ( B ⊗ L Λ A, Λ)for all B ∈ D ( G, Λ). As R H om Λ ( − , Λ) is characterized by the similar adjunction in D ´et ([ ∗ /G ] , Λ),we get the result. (cid:3)

V.2. ´Etale sheaves on strata

We want to describe D ´et (Bun G , Λ) via its strata Bun bG . For this, we need the following resultsaying roughly that connected Banach–Colmez spaces are “contractible”. Proposition

V.2.1 . Let f : S (cid:48) → S be a map of small v-stacks that is a torsor under BC ( E ) resp. BC ( E [1]) , where E is a vector bundle on X S that is everywhere of positive (resp. negative)slopes. Then the pullback functor f ∗ : D ´et ( S, Λ) → D ´et ( S (cid:48) , Λ) is fully faithful. Proof.

By descent [

Sch17a , Proposition 17.3, Remark 17.4], the problem is v-local on S ,and in particular one can assume that the torsor is split. In the positive case, we can use Corol-lary II.3.3 (iv) to ﬁnd pro-´etale locally on S a short exact sequence0 → O X S ( r ) m (cid:48) → O X S ( r ) m → E → , inducing a similar sequence on Banach–Colmez spaces. This reduces us to the case E = O X S ( n )for some n (as then pullback under BC ( O X S ( r ) m ) → S is fully faithful, as is pullback under BC ( O X S ( r ) m ) → S (cid:48) = BC ( E )). In that case, BC ( E ) is a 1-dimensional perfectoid open unit ballover S by Proposition II.2.5 (iv), in particular cohomologically smooth. It suﬃces to see that Rf ! is fully faithful, for which it suﬃces that for all A ∈ D ´et ( S, Λ), the adjunction map Rf ! Rf ! A → A is an equivalence. But note that both Rf ! and Rf ! commute with any base change by [ Sch17a ,Proposition 22.19, Proposition 23.12]. Thus, we may by passage to stalks reduce to the case S = Spa( C, C + ) where C is a complete algebraically closed nonarchimedean ﬁeld and C + ⊂ C anopen and bounded valuation subring, and in fact we only need to check the statement on globalsections. If the stalk of A at the closed point s ∈ S is zero, then the same holds true for Rf ! Rf ! A as Rf ! A agrees with f ∗ A up to twist, so this follows from proper base change, [ Sch17a , Theorem19.2]. This allows us to reduce to the case that A is constant, and then as both Rf ! and Rf ! commute with all direct sums, even to the case A = Λ. Thus, we are reduced to the computationof the cohomology of the perfectoid open unit disc.The case of negative Banach–Colmez spaces follows by taking an exact sequence0 → E → O X S ( d ) m → G → → BC ( O X S ( d ) m ) → BC ( G ) → BC ( E [1]) → . (cid:3) Now we can formulate the desired result.

70 V. D ´et (Bun G ) Proposition

V.2.2 . For any b ∈ B ( G ) , the map Bun bG = [ ∗ / (cid:101) G b ] → [ ∗ /G b ( E )] induces via pullback an equivalence D ´et ( G b ( E ) , Λ) ∼ = D ´et ([ ∗ /G b ( E ) , Λ]) ∼ = D ´et ([ ∗ / (cid:101) G b ] , Λ) . Moreover, for any complete algebraically closed nonarchimedean ﬁeld

C/k , the map D ´et ([ ∗ / (cid:101) G b ] , Λ) → D ´et ([Spa C/ (cid:101) G b ] , Λ) is an equivalence. Proof.

Using [

Sch17a , Theorem 1.13] and Theorem V.1.1, it is enough to prove that thefunctor D ´et ([Spa C/G b ( E )] , Λ) → D ´et ([Spa C/ (cid:101) G b ] , Λ)is an equivalence. For this, it is enough to prove that pullback under the section [Spa

C/G b ( E )] → [Spa C/ (cid:101) G b ] induces a fully faithful functor D ´et ([Spa C/ (cid:101) G b ] , Λ) → D ´et ([Spa C/G b ( E )] , Λ) , which follows from Proposition V.2.1. (cid:3) We see that D ´et (Bun G , Λ) is glued from the categories D ´et (Bun bG , Λ) ∼ = D ( G b ( E ) , Λ), which areentirely representation-theoretic. In particular, this implies that the base ﬁeld plays no role:

Corollary

V.2.3 . For any complete algebraically closed nonarchimedean ﬁeld C and any lo-cally closed substack U ⊂ Bun G , the functor D ´et ( U, Λ) → D ´et ( U × Spa C, Λ) is an equivalence of categories. Although this seems like a purely technical result, it will actually play a key role when we studyHecke operators.

Proof.

Fully faithfulness holds true by [

Sch17a , Theorem 1.13 (ii)]. To see that it is anequivalence of categories, it is enough to check on all quasicompact locally closed substacks U ⊂ Bun G . These are stratiﬁed into ﬁnitely many locally closed substacks Bun bG ⊂ Bun G , and we havethe corresponding excision exact sequences, so to get essential surjectivity, it is enough to check oneach stratum Bun bG . Now it follows from Proposition V.2.2. (cid:3) It would be very interesting to understand the gluing of these categories in terms of pure representation theory. .3. LOCAL CHARTS 171

V.3. Local charts

For any b ∈ B ( G ) we wish to construct a chart π b : M b → Bun G whose image contains Bun bG , such that π b is separated, representable in locally spatial diamondsand cohomologically smooth , and whose geometry can be understood explicitly. Example

V.3.1 . Before we discuss the general case, let us brieﬂy discuss the ﬁrst interestingcase, namely G = GL and the non-basic element b corresponding to O (1) ⊕ O . In that case, welet M b be the moduli space of extensions0 → L → E → L (cid:48) → L is of degree 0 and L (cid:48) is of degree 1. Mapping such an extension to E deﬁnes the map M b → Bun G .Note that there is a natural E × × E × -torsor (cid:102) M b → M b , parametrizing isomorphisms L ∼ = O and L (cid:48) ∼ = O (1). On the other hand, it is clear that (cid:102) M b = BC ( O ( − M b is very explicit.Any extension E parametrized by M b is either isomorphic to O ⊕O (1), or to O ( ). The ﬁbres of π b : M b → Bun GL over a rank 2 bundle E are given by an open subset of the projectivized Banach–Colmez space ( BC ( E ) \{ } ) /E × . Thus, these ﬁbres interpolate between ( BC ( O ( )) \{ } ) /E × , whichis cohomologically smooth by Proposition II.3.7, and( BC ( O ⊕ O (1)) \ { } ) /E × = ( E × D \ { } ) /E × . The latter is still cohomologically smooth, although E × D is not — the quotient by E × gets ridof the disconnected nature of the space. In this case, and in fact in complete generality for all b ∈ B (GL n ), one can actually check cohomological smoothness of π b by hand. To handle thegeneral case, we had to prove the Jacobian criterion, Theorem IV.4.2.Coming back to the general case, we can in fact construct all M b together, as follows. Definition

V.3.2 . The v-stack M is the moduli stack taking S ∈ Perf k to the groupoid of G -bundles E on X S together with an increasing separated and exhaustive Q -ﬁltration ( ρ ∗ E ) ≤ λ ⊂ ρ ∗ E (ranging over algebraic representations ρ : G → GL n , and compatible with exact sequences andtensor products) on the corresponding ﬁbre functor such that (letting ( ρ ∗ E ) <λ = (cid:83) λ (cid:48) <λ ( ρ ∗ E ) ≤ λ (cid:48) )the quotient ( ρ ∗ E ) λ = ( ρ ∗ E ) ≤ λ / ( ρ ∗ E ) <λ is a semistable vector bundle of slope λ , for all λ ∈ Q and representations ρ : G → GL n . Note that by passing to the associated graded, M maps to the moduli stack of G -bundles in thecategory of Q -graded vector bundles on X S where the weight λ piece is semistable of slope λ . ByProposition III.4.7, this is isomorphic to (cid:70) b ∈ B ( G ) [ ∗ /G b ( E )]. In particular M decomposes naturallyinto a disjoint union M = (cid:71) b ∈ B ( G ) M b ,

72 V. D ´et (Bun G ) and for each b ∈ B ( G ), we have natural maps q b : M b → [ ∗ /G b ( E )] . Example

V.3.3 . When G = GL n , M sends S to the groupoid of ﬁltered vector bundles0 = Fil E (cid:40) Fil E (cid:40) · · · (cid:40) Fil r E = E for some r , such that Fil i +1 E / Fil i E is semistable andthe slopes ( µ (Fil i +1 E / Fil i E )) ≤ i

V.3.4 . Suppose G is quasisplit. Let M b be the centralizer of the slope morphism, asa Levi subgroup. Let P b be the parabolic subgroup with Levi M b such that the weight of ν b onLie P b are positive. There is a diagram Bun P b Bun G Bun M b induced by the inclusion P b ⊂ G and the quotient map P b → M b . There is a cartesian diagram M b Bun P b Bun bM b Bun M b . Proposition

V.3.5 . For any b ∈ B ( G ) , the map q b : M b → [ ∗ /G b ( E )] . is partially proper, representable in locally spatial diamonds, and cohomologically smooth, of dimen-sion (cid:104) ρ, ν b (cid:105) . In fact, after pullback along ∗ → [ ∗ /G b ( E )] , it is a successive torsor under negativeBanach–Colmez spaces.In particular, M b is a cohomologically smooth Artin v-stack, of dimension (cid:104) ρ, ν b (cid:105) . Proof.

It suﬃces to check everything after pullback after pullback by the v -cover ∗ → [ ∗ /G b ( E )], (cid:102) M b → M b . Let (cid:101) G → X S be the automorphism group of E b → X S , see Proposition III.5.2, the pureinner twisting of G × X S by E b . This is equipped with a parabolic subgroup (cid:101) G ≤ , and moreovera ﬁltration ( (cid:101) G ≤ λ ) λ ≤ with unipotent radical (cid:101) G < . This is the opposite parabolic subgroup to theone used in the proof of Proposition III.5.1. Then (cid:102) M b ( S ) is identiﬁed with the set of (cid:101) G < -torsors.The result is deduced using the description of the graded pieces of ( (cid:101) G ≤ λ ) λ< as vector bundles ofnegative slopes. (cid:3) We ﬁrst prove some structural results about M b and its universal G b ( E )-torsor (cid:102) M b → M b . Ageneral theme here is the subtle distinction between the absolute property of being a (locally spatial)diamond (which (cid:102) M b is not, but it has a large open part (cid:102) M ◦ b ⊂ (cid:102) M b that is) and the relative notionof (cid:102) M b → ∗ being representable in (locally spatial) diamonds (which (cid:102) M b is), and some related subtledistinctions on absolute and relative quasicompactness. .3. LOCAL CHARTS 173 Proposition

V.3.6 . The map M b → [ ∗ /G b ( E )] has a section [ ∗ /G b ( E )] → M b given by theclosed substack where E is (at every geometric point) isomorphic to E b , in which case ( ρ ∗ E ) ≤ λ ⊂ ρ ∗ E is a splitting of the Harder–Narasimhan ﬁltration of ρ ∗ E for all representations ρ : G → GL n .Consider the open complement M ◦ b = M b \ [ ∗ /G b ( E )] , with preimage (cid:102) M ◦ b = (cid:102) M b \ {∗} . Then (cid:102) M ◦ b is a spatial diamond.Moreover, if U π = ν Nb ( π ) ∈ G b ( E ) for any large enough N (so that ν Nb : G m → G b is awell-deﬁned cocharacter), then (cid:102) M ◦ b /U Z π → ∗ is proper. Proof.

To check that the substack where E is at every geometric point isomorphic to E b isclosed, note that by semicontinuity it suﬃces to see that everywhere on M b , the Newton point of E is bounded by b . By [ RR96 , Lemma 2.2 (iv)], this reduces to the case of G = GL n , where it isa simple consequence of the Harder–Narasimhan formalism (the Newton polygon of an extensionis always bounded by the Newton polygon of the split extension). On this closed substack, E hastwo transverse ﬁltrations, given by ( ρ ∗ E ) ≤ λ and the Harder–Narasimhan ﬁltration; it follows that E upgrades to a G -bundle in Q -graded vector bundles on the Fargues–Fontaine curve, with the weight λ -piece semistable of slope λ . We see that this gives the desired section of M b → [ ∗ /G b ( E )], usingagain Proposition III.4.7.We claim that the action of U π on | (cid:102) M b × k Spa k (( t )) | satisﬁes the hypotheses of Lemma II.2.17,with ﬁxed point locus given by the closed subspace ∗ considered in the previous paragraph. Writing (cid:102) M b as a successive extension of Banach–Colmez spaces, it is clear that for all x ∈ | (cid:102) M b × k Spa k (( t )) | which are not in the closed substack, the sequence U − nπ ( x ) leaves any quasicompact open subspacefor large n : Look at the ﬁrst step in the successive extensions where x does not project to the origin.Then x gives an element in the ﬁber over the origin, which is a negative Banach–Colmez space,on which U π = ν Nb ( π ) acts via a positive power of π ; thus U − nπ ( x ) leaves any quasicompact opensubspace of this Banach–Colmez space. In particular, it follows that the ﬁxed points locus of U π isprecisely the origin. To apply Lemma II.2.17, it remains to see that for all x ∈ | (cid:102) M b × k Spa k (( t )) | and quasicompact open neighborhoods U of the origin, one has γ m ( x ) ∈ U for all suﬃciently large m . This can be reduced to the case of GL n by the Tannakian formalism, so assume G = GL n forthis argument. Now ﬁx a map f : Spa( C, C + ) → (cid:102) M b × k Spa k (( t )) having x in its image; it suﬃcesto construct a map Spa( C, C + ) × N ∪ {∞} → (cid:102) M b × k Spa k (( t ))whose restriction to Spa( C, C + ) × { } is f and which is equivariant for the γ -action, with γ actingon the left via shift on the proﬁnite set N ∪ {∞} . The map f classiﬁes some Q -ﬁltered vector bundle E ≤ λ ⊂ E of rank n on X C with (cid:76) λ E λ ∼ = E b as Q -graded vector bundles. After pullback to Y C, [1 ,q ] ,the ﬁltration is split, so we can ﬁnd an isomorphism α : E| Y C, [1 ,q ] ∼ = E b | Y C, [1 ,q ] of Q -ﬁltered vector bundles on Y C, [1 ,q ] , such that α reduces on graded pieces to the given identiﬁ-cation. The descent datum is now given by some isomorphism of Q -ﬁltered vector bundles β : ϕ ∗ ( E b | Y C, [ q,q ] ) ∼ = E b | Y C, [1 ,

74 V. D ´et (Bun G ) that reduces to the standard Frobenius on graded pieces. In other words, β is the standard Frobeniuson E b multiplied by some β (cid:48) : E b | Y C, [1 , ∼ = E b | Y C, [1 , and with respect to the Q -grading on E b , the map β (cid:48) is the identity plus a lower triangular matrix.The action of γ replaces β (cid:48) by its U π -conjugate. This multiplies all lower triangular entries bypowers of π , so ( β (cid:48) , γ ( β (cid:48) ) , γ ( β (cid:48) ) , . . . , ϕ ∗ ( E b | Y S, [ q,q ] ) ∼ = E b | Y S, [1 , where S = Spa( C, C + ) × N ∪ {∞} . Using this as a descent datum deﬁnes a Q -ﬁltered vector bundleon X S deﬁning the required map S = Spa( C, C + ) × N ∪ {∞} → (cid:102) M b × k Spa k (( t )) . This ﬁnishes the veriﬁcation of the hypotheses of Lemma II.2.17.It is clear from the deﬁnition that (cid:102) M ◦ b → ∗ is partially proper (as the theory of vector bundleson the Fargues–Fontaine curve does not depend on R + ). Thus, to show that (cid:102) M ◦ b /U Z π → ∗ is proper,it suﬃces to see that the map is quasicompact, which can be checked after base change to Spa k (( t ));then it follows from the previous discussion and Lemma II.2.17.It remains to see that (cid:102) M ◦ b is a spatial diamond. For this, we pick a representative b ∈ G ( ˘ E ) ofthe σ -conjugacy class that is decent in the sense of [ RZ96 , Deﬁnition 1.8]. In particular, b ∈ G ( E s )for some unramiﬁed extension E s | E of degree s , and (cid:102) M ◦ b is already deﬁned over Perf F qs . Let Frob s be the Frobenius x (cid:55)→ x q s . As b is decent, the action of U π = ν Nb ( π ) on (cid:102) M ◦ b agrees with the actionof a power of Frob s for N large enough. We know that (cid:102) M ◦ b /U Z π × k Spa k (( t ))is a spatial diamond (as it is proper over Spa k (( t ))). Replacing U π by Frob s , and moving thequotient by Frobenius to the other factor (which is allowed as the absolute Frobenius acts triviallyon topological spaces) one sees that also (cid:102) M ◦ b × k Spa k (( t )) / Frob Z s is a spatial diamond. But Spa k (( t )) / Frob Z s → ∗ is proper and cohomologically smooth. Thus,Lemma II.3.8 shows that it is a spatial v-sheaf. By [ Sch17a , Theorem 12.18], to see that (cid:102) M ◦ b is aspatial diamond, it suﬃces to check on points. Writing (cid:102) M b as a successive extension of Banach–Colmez spaces, any point in (cid:102) M ◦ b has a minimal step where it does not map to the origin. Thenits image is a nontrivial point of an absolute Banach–Colmez space, and a punctured absoluteBanach–Colmez space is a diamond by Proposition II.3.7; the result follows. (cid:3) The following theorem gives the desired local charts for Bun G ; its proof is based on the Jacobiancriterion for (cohomological) smoothness, Theorem IV.4.2. Theorem

V.3.7 . The map π b : M b → Bun G forgetting the ﬁltration is partially proper, repre-sentable in locally spatial diamonds, and cohomologically smooth of (cid:96) -dimension (cid:104) ρ, ν b (cid:105) . .4. COMPACT GENERATION 175 Proof.

Let S → Bun G be some map for a perfectoid space S , given by some G -bundle E on X S . Then M b × Bun G S parametrizes Q -ﬁltrations on E whose associated graded Q -bundlecorresponds to b . Such Q -ﬁltrations are parametrized by sections of a smooth projective ﬂag variety Z = E / P → X S for some parabolic P ⊂ G × E X . Moreover, the condition on the associated gradedbundle and Proposition III.5.2 imply that M b is an open subspace of M sm Z , so the result followsfrom Theorem IV.4.2. (cid:3) V.4. Compact generation

We are now revisiting the notion of ﬁnite type smooth representation in terms of D ´et (Bun G , Λ) . The goal of this section is to prove the following theorem. As above, we ﬁx some coeﬃcientring Λ such that n Λ = 0 for some n prime to p . Theorem

V.4.1 . For any locally closed substack U ⊂ Bun G , the triangulated category D ´et ( U, Λ) is compactly generated. An object A ∈ D ´et ( U, Λ) is compact if and only if for all b ∈ B ( G ) containedin U , the restriction i b ∗ A ∈ D ´et (Bun bG , Λ) ∼ = D ( G b ( E ) , Λ) along i b : Bun bG ⊂ Bun G is compact, and zero for almost all b . Here, compactness in D ( G b ( E ) , Λ) is equivalent to lying in the thick triangulated subcategory generated by c - Ind G b ( E ) K Λ as K runs overopen pro- p -subgroups of G b ( E ) . To prove the theorem, we exhibit a class of compact projective generators. The key result isthat (cid:102) M b behaves like a strictly local scheme; in some vague sense, (cid:102) M b is the strict henselization of Bun G at b . Proposition

V.4.2 . Let b ∈ B ( G ) . For any A ∈ D ´et ( (cid:102) M b , Λ) with stalk A = i ∗ A ∈ D ´et ( ∗ , Λ) ∼ = D (Λ) at the closed point i : ∗ ⊂ (cid:102) M b , the map R Γ( (cid:102) M b , A ) → A is an isomorphism. In particular, R Γ( (cid:102) M b , − ) commutes with all direct sums. Proof.

Replacing A by the cone of A → i ∗ A , we can assume that A = j ! A (cid:48) for some A (cid:48) ∈ D ´et ( (cid:102) M ◦ b , Λ). We have to see that R Γ( (cid:102) M b , j ! A (cid:48) ) = 0 . But this follows from Theorem IV.5.3 (applied with X = (cid:102) M ◦ b and S = Spa k (( t )), noting that basechange along S → ∗ follows from smooth base change), using that the partial compactiﬁcation (cid:102) M ◦ b ⊂ (cid:102) M b is precisely a compactiﬁcation towards one of the two ends of (cid:102) M ◦ b , as follows from thebehaviour of the Frobenius action exhibited in the proof of Proposition V.3.6. (cid:3) Remark

V.4.3 . (i) Consider the v -sheaf X = Spd( k [[ x , . . . , x d ]]) and the quasicompact open subset U = Spd( k [[ x , . . . , x d ]]) \ V ( x , . . . , x d ) that is representable by a perfectoid space. When base changed to S = Spa( k (( t ))), X becomes isomorphic to an open unit disk, U becomes the punctured unit disk that has two ends:the origin and the exterior of the disk. The picture is thus analogous to the preceding one with (cid:102) M b , and for any A ∈ D ´et ( X, Λ) one has R Γ( X, A ) ∼ = i ∗ A where i : Spd( k ) (cid:44) → X is V ( x , . . . , x d ).

76 V. D ´et (Bun G ) (ii) This applies more generally to the v -sheaf associated to any W ( k )-aﬃne formal scheme Spf( R )with R an I -adic ring. Then, for any A ∈ D ´et (Spd( R ) , Λ) one has R Γ(Spd( R ) , A ) = R Γ(Spd(

R/I ) , i ∗ A )with i : Spd( R/I ) (cid:44) → Spd( R ). In particular, for A ∈ D ´et (Spd( R ) \ V ( I ) , Λ), where here Spd( R ) \ V ( I ) is representable by a spatial diamond and even a perfectoid space if R is a k -algebra, one has R Γ(Spd( R ) \ V ( I ) , A ) ∼ = R Γ(Spd(

R/I ) , i ∗ Rj ∗ A ) . Thus, Proposition V.4.2 can be seen as a result about “nearby cycles on the strict henselianizationof Bun G at b ”. Corollary

V.4.4 . Let b ∈ B ( G ) and let K ⊂ G b ( E ) be an open pro- p -subgroup. Then for any A ∈ D ´et ( (cid:102) M b /K, Λ) with pullback A = i ∗ A ∈ D ´et ([ ∗ /K ] , Λ) ∼ = D ( K, Λ) corresponding to a complex V of smooth K -representations, the map R Γ( (cid:102) M b /K, A ) → R Γ([ ∗ /K ] , A ) ∼ = V K is an isomorphism. In particular, R Γ( (cid:102) M b /K, − ) commutes with all direct sums. Proof.

This follows formally from Proposition V.4.2 by descent along ψ : (cid:102) M b → (cid:102) M b /K ; moreprecisely, by writing any A as a direct summand of ψ ∗ ψ ∗ A . (cid:3) Remark

V.4.5 . Let i : [ ∗ /G b ( E )] (cid:44) → M b ← (cid:45) M ◦ b : j be the usual diagram. From the corollary, one deduces that if one regards i ∗ Rj ∗ A ∈ D ´et ([ ∗ /G b ( E )] , Λ) ∼ = D ( G b ( E ) , Λ)as a complex of G b ( E )-representations, then this is given by R Γ( (cid:102) M ◦ b , A ) . As (cid:102) M ◦ b is qcqs (and of ﬁnite cohomological dimension) by Proposition V.3.6, this commutes withall direct sums in A . For example, in the case of GL and E b = O (1) ⊕ O , one has (cid:102) M ◦ b = BC ( O ( − \ { } = Spa k (( t /p ∞ )) /SL ( D )by Example II.3.12 and Example V.3.1. Thus, in this case one compute i ∗ Rj ∗ A = R Γ(Spa k (( t /p ∞ )) /SL ( D ) , A )which is a very explicit formula. If one would use the presentation BC ( O ( − × Spa C = ( A C (cid:93) ) ♦ /E instead, it would be considerably more diﬃcult to compute the answer. In particular, we criticallyused quasicompacity of the absolute (cid:102) M ◦ b , its base change (cid:102) M ◦ b × k Spa C is no longer quasicompact. This highlights the importance of working with absolute objects, and of using the right local charts.

In fact, Theorem V.3.7, smooth base change, and this formula for i ∗ Rj ∗ show that the gluingof the representation-theoretic strata D ´et (Bun bG , Λ) ∼ = D ( G b ( E ) , Λ) into D ´et (Bun G , Λ) is encodedin the spaces (cid:102) M ◦ b , showing that the local charts M b are of fundamental and not just technicalimportance. .5. BERNSTEIN–ZELEVINSKY DUALITY 177 Now we can prove Theorem V.4.1.

Proof of Theorem V.4.1.

As for closed immersions i , the functor i ∗ preserves compact ob-jects, it is enough to handle the case that U is an open substack. Let b ∈ | U | ⊂ Bun G ∼ = B ( G ) beany point of U , and let K ⊂ G b ( E ) be an open pro- p -subgroup, giving rise to the map f K : (cid:102) M b /K → Bun G . By Corollary V.4.4 and Theorem V.3.7, we see that A bK := Rf K ! Rf ! K Λ ∈ D ´et (Bun G , Λ) is compact;in fact, R Hom( A bK , B ) ∼ = R Hom( Rf ! K Λ , Rf ! K B ) ∼ = R Hom( Rf ! K Λ , f ∗ K B ⊗ L Λ Rf ! K Λ) ∼ = R Γ( (cid:102) M b /K, f ∗ K B ) ∼ = ( i b ∗ B ) K . From this computation, we see that the collection of objects A bK for varying b ∈ | U | and K ⊂ G b ( E )open pro- p form a class of compact generators: Indeed, if B is nonzero, then ( i b ∗ B ) K must benonzero for some b and K .To prove the characterization of compact objects, we argue by induction on the number ofpoints of | U | , noting that any compact object must be concentrated on a quasicompact substack,and thus on ﬁnitely many points. So assume that | U | is ﬁnite, b ∈ | U | is a closed point and j : V = U \ { b } ⊂ U is the open complement, so we know the result for V . It suﬃces to prove that j ∗ preserves compact objects. Indeed, then A ∈ D ´et ( U, Λ) is compact if and only if j ∗ A and i b ∗ A are compact, and this gives by induction the desired characterization.To see that j ∗ preserves compact objects, we can check on the given generators. For generators A b (cid:48) K corresponding to b (cid:48) (cid:54) = b we get j ∗ A b (cid:48) K = A b (cid:48) K , so there is nothing to prove. On the other hand, j ∗ A bK = Rf ◦ K ! f ◦ ! K Λ for f ◦ K : (cid:102) M ◦ b /K → V ⊂ Bun G . The compactness of j ∗ A bK then follows from R Γ( (cid:102) M ◦ b /K, − ) commuting with all direct sums. Butthis is true as (cid:102) M ◦ b is a spatial diamond of ﬁnite dim . trg, by Proposition V.3.6 and Proposition V.3.5,and taking cohomology under K is exact. (cid:3) V.5. Bernstein–Zelevinsky duality

We note that one can deﬁne a Bernstein–Zelevinsky involution on (the compact objects of) D ´et (Bun G , Λ). More precisely, we have the following result. Here, in anticipation of some functorintroduced later, we write π (cid:92) : D ´et (Bun G , Λ) → D ( ∗ , Λ) = D (Λ) : A (cid:55)→ Rπ ! ( A ⊗ L Λ Rπ ! Λ)for the left adjoint of π ∗ , π : Bun G → ∗ . Theorem

V.5.1 . For any compact object A ∈ D ´et (Bun G , Λ) , there is a unique compact object D BZ ( A ) ∈ D ´et (Bun G , Λ) with a functorial identiﬁcation R Hom( D BZ ( A ) , B ) ∼ = π (cid:92) ( A ⊗ L Λ B ) for B ∈ D ´et (Bun G , Λ) . Moreover, the functor D BZ is a contravariant autoequivalence of D ´et (Bun G , Λ) ω ,and D is naturally isomorphic to the identity.

78 V. D ´et (Bun G ) If U ⊂ Bun G is an open substack and A is concentrated on U , then so is D BZ ( A ) . In particular, D BZ restricts to an autoequivalence of the compact objects in D ´et (Bun bG , Λ) ∼ = D ( G b ( E ) , Λ) for b ∈ B ( G ) basic, and in that setting it is the usual Bernstein–Zelevinsky involution. Proof.

By the Yoneda lemma, the uniqueness of D BZ ( A ) is clear. For the existence, it suﬃcesto check on a system of generators, like A = i b ! c -Ind G b ( E ) K Λ for varying b ∈ B ( G ) and K ⊂ G b ( E )open pro- p . In that case, D BZ ( A ) = A bK by Corollary V.4.4. This also shows that if A is concentratedon U , then so is D BZ ( A ).Now note that R Hom( D BZ ( A ) , B ) ∼ = π (cid:92) ( A ⊗ L Λ B ) ∼ = π (cid:92) ( B ⊗ L Λ A ) ∼ = R Hom( D BZ ( B ) , A ) . In particular, taking B = D BZ ( A ), we see that there is a natural functorial map D ( A ) → A .We claim that this is an equivalence. It suﬃces to check on generators. We have seen that theBernstein–Zelevinsky dual of i b ! c -Ind G b ( E ) K Λ is A bK . Its restriction to Bun bG is again i b ! c -Ind G b ( E ) K Λ,so one easily checks that the map D ( A ) → A is an isomorphism over Bun bG . To see that it isan isomorphism everywhere, one needs to see that if B = Rj ∗ B (cid:48) , B (cid:48) ∈ D ´et ( U, Λ) for some opensubstack j : U ⊂ Bun G not containing Bun bG , then π (cid:92) ( A bK ⊗ L Λ B ) = 0 . Twisting a few things away and using the deﬁnition of A bK = Rf K ! Rf ! K Λ, this follows from theassertion that for all A (cid:48) ∈ D ´et ( (cid:102) M ◦ b /K, Λ), with j K : (cid:102) M ◦ b /K (cid:44) → (cid:102) M b /K the open immersion, onehas R Γ c ( (cid:102) M b /K, Rj K ∗ A (cid:48) ) = 0 . Using the trace map for (cid:102) M b → (cid:102) M b /K , this follows from Theorem IV.5.3, applied as before with X = (cid:102) M ◦ b and S = Spa k (( t )), noting that base change along S → ∗ holds by smooth base changeand is conservative.The comparison to Bernstein–Zelevinsky duality follows formally by taking B correspondingto the regular representation of G b ( E ), in which case π (cid:92) ( A ⊗ L Λ B ) is isomorphic to the underlyingchain complex of A . Moreover, as the regular representation has two commuting G b ( E )-actions,there is a residual G b ( E )-action, which is the usual action on A . This gives the usual deﬁnition ofthe Bernstein–Zelevinsky involution as R Hom into the regular representation. (cid:3)

V.6. Verdier duality

It turns out that one can also understand how Verdier duality acts on D ´et (Bun G , Λ). The keyresult is the following.

Theorem

V.6.1 . Let j : V (cid:44) → U be an open immersion of open substacks of Bun G . For any A ∈ D ´et ( V, Λ) , the natural map j ! R H om Λ ( A, Λ) → R H om Λ ( Rj ∗ A, Λ) is an isomorphism in D ´et ( U, Λ) . .6. VERDIER DUALITY 179 Note that one always has Rj ∗ R H om Λ ( A, Λ) = R H om Λ ( j ! A, Λ) ;the theorem asserts that this is also true with j ! and Rj ∗ exchanged, which is related to a localbiduality statement: If A ∈ D ´et ( U, Λ) is reﬂexive, the theorem implies formally that j ! A ∈ D ´et ( U, Λ)is reﬂexive.

Proof.

We can assume that U and V are quasicompact, and then by induction, we can assumethat V = U \ { b } for some closed b ∈ | U | . The map is clearly an isomorphism over V , so it suﬃcesto see that for the compact objects A bK = Rf K ! f ! K Λ , f K : (cid:102) M b /K → U ⊂ Bun G , one gets an isomorphism after applying R Hom( A bK , − ). As R Hom( A bK , B ) = ( i b ∗ B ) K , we see thatthe left-hand side R Hom( A bK , j ! R H om Λ ( A, Λ)) = 0vanishes. On the other hand, using the left adjoint π (cid:92) : D ´et (Bun G , Λ) → D ( ∗ , Λ) ∼ = D (Λ) ofpullback, we have R Hom( A bK , R H om Λ ( Rj ∗ A, Λ)) ∼ = R Hom( A bK ⊗ L Λ Rj ∗ A, Λ) ∼ = R Hom( π (cid:92) ( A bK ⊗ L Λ Rj ∗ A ) , Λ) . But by Theorem V.5.1, and the identiﬁcation of the Bernstein–Zelevinsky dual of A bK as i b ! c -Ind G b ( E ) K Λ,one has π (cid:92) ( A bK ⊗ L Λ Rj ∗ A ) ∼ = R Hom( i b ! c -Ind G b ( E ) K Λ , Rj ∗ A ) ∼ = R Hom( c -Ind G b ( E ) K Λ , Ri b ! Rj ∗ A ) = 0 . (cid:3) Recall that as a cohomologically smooth Artin stack of dimension 0, Bun G admits a dualizingcomplex D Bun G ∈ D ´et (Bun G , Λ) that is locally isomorphic to Λ[0].

Theorem

V.6.2 . For any open substack U ⊂ Bun G , an object A ∈ D ´et ( U, Λ) is reﬂexive,i.e. the natural map A → R H om Λ ( R H om Λ ( A, D U ) , D U ) is an equivalence, if and only if for all b ∈ B ( G ) lying in U with corresponding locally closed stratum i b : Bun bG → U , the restriction i b ∗ A ∈ D ´et (Bun bG , Λ) = D ´et ([ ∗ / (cid:101) G b ] , Λ) ∼ = D ´et ([ ∗ /G b ( E )] , Λ) = D ( G b ( E ) , Λ) is reﬂexive as a complex of admissible G b ( E ) -representations; this means that the complex of K -invariants is reﬂexive in D (Λ) for all open pro- p -subgroups K ⊂ G b ( E ) . In the deﬁnition of reﬂexivity, we can replace D U by Λ (as this changes the dual by a twist,and then the bidual stays the same). The theorem follows immediately from the following result. Lemma

V.6.3 . Let U ⊂ Bun G be an open substack and A ∈ D ´et ( U, Λ) . For any b ∈ B ( G ) lyingin U , there is a natural isomorphism i b ∗ R H om Λ ( R H om Λ ( A, Λ) , Λ) ∼ = R H om Λ ( R H om Λ ( i b ∗ A, Λ) , Λ) .

80 V. D ´et (Bun G ) Proof.

We may assume that U ⊂ Bun G is the set of generalizations of b , and let j : V = U \ { b } (cid:44) → U . Let B = j ∗ A . Using the exact triangle j ! B → A → i b ∗ i b ∗ A → , and the invertibility of i b ! Λ (as Bun bG is also cohomologically smooth), it is enough to prove that i b ∗ R H om Λ ( R H om Λ ( j ! B, Λ) , Λ) = 0 . But R H om Λ ( j ! B, Λ) = Rj ∗ R H om Λ ( B, Λ), and by Theorem V.6.1, R H om Λ ( Rj ∗ R H om Λ ( B, Λ) , Λ) = j ! R H om Λ ( R H om Λ ( B, Λ) , Λ) . (cid:3) V.7. ULA sheaves

Finally, we want to classify the objects A ∈ D ´et (Bun G , Λ) that are universally locally acyclicwith respect to Bun G → ∗ . Our goal is to prove the following theorem. This gives a geometricinterpretation of the classical notion of admissible representation in terms of D ´et (Bun G , Λ) . Theorem

V.7.1 . Let A ∈ D ´et (Bun G , Λ) . Then A is universally locally acyclic with respect to Bun G → ∗ if and only if for all b ∈ B ( G ) , the pullback i b ∗ A to i b : Bun bG (cid:44) → Bun G correspondsunder D ´et (Bun bG , Λ) ∼ = D ( G b ( E ) , Λ) to a complex M b of smooth G b ( E ) -representations for which M Kb is a perfect complex of Λ -modules for all open pro- p subgroups K ⊂ G b ( E ) . We want to use Proposition IV.2.32. As preparation, we need to understand D ´et (Bun G × Bun G , Λ).More generally, we have the following result.

Proposition

V.7.2 . Let G and G be two reductive groups over E , and let G = G × G .Then Bun G ∼ = Bun G × Bun G , giving rise to an exterior tensor product − (cid:2) − : D ´et (Bun G , Λ) × D ´et (Bun G , Λ) → D ´et (Bun G , Λ) . For all compact objects A i ∈ D ´et (Bun G i , Λ) , i = 1 , , the exterior tensor product A (cid:2) A ∈ D ´et (Bun G , Λ) is compact, these objects form a class of compact generators, and for all furtherobjects B i ∈ D ´et (Bun G i , Λ) , i = 1 , , the natural map R Hom( A , B ) ⊗ L Λ R Hom( A , B ) → R Hom( A (cid:2) A , B (cid:2) B ) is an isomorphism. Remark

V.7.3 . The proposition says that as Λ-linear presentable stable ∞ -categories, theexterior tensor product functor D ´et (Bun G , Λ) ⊗ D (Λ) D ´et (Bun G , Λ) → D ´et (Bun G , Λ)is an equivalence.

Proof.

We use the compact generators A i = A b i K i for certain b i ∈ B ( G i ), K i ⊂ G i,b i ( E ) openpro- p . These give rise to b = ( b , b ) ∈ B ( G ) and K = K × K ⊂ G b ( E ) = G ,b ( E ) × G ,b ( E ),and using M b ∼ = M b × M b .7. ULA SHEAVES 181 and the K¨unneth formula, one concludes that A (cid:2) A ∼ = A bK , which is again compact. As B ( G ) = B ( G ) × B ( G ) and open pro- p subgroups of the form K × K ⊂ G b ( E ) are coﬁnal, we see thatthese objects form a set of compact generators.Moreover, as R Hom( A bK , B ) = ( i b ∗ B ) K for all B ∈ D ´et (Bun G , Λ) and similarly for A b i K i , we alsosee that the map R Hom( A b K , B ) ⊗ L Λ R Hom( A b K , B ) → R Hom( A bK , B (cid:2) B )is an isomorphism. As these objects generate, the same follows for all compact A , A . (cid:3) Now we can prove Theorem V.7.1.

Proof of Theorem V.7.1.

By Proposition IV.2.32, we see that A being universally locallyacyclic is equivalent to the map p ∗ R H om( A, Λ) ⊗ L Λ p ∗ A → R H om( p ∗ A, p ∗ A )in D ´et (Bun G × Bun G , Λ) ∼ = D ´et (Bun G × G , Λ) being an isomorphism.By Proposition V.7.2, this is equivalent to being an isomorphism after applying R Hom( A (cid:2) A , − ) for varying compact A i ∈ D ´et (Bun G i , Λ). Using Proposition V.7.2, the left-hand side isgiven by R Hom( A , R H om( A, Λ)) ⊗ L Λ R Hom( A , A ) ∼ = R Hom( π (cid:92) ( A ⊗ L Λ A ) , Λ) ⊗ L Λ R Hom( A , A ) . The right-hand side is, using π : Bun G → ∗ for the projection, R Hom( A (cid:2) A , R H om( p ∗ A, p ∗ A )) ∼ = R Hom(( A ⊗ L Λ A ) (cid:2) A , p ∗ A ) ∼ = R Hom( p ∗ ( A ⊗ L Λ A ) , p ∗ R H om Λ ( A , A )) ∼ = R Hom( A ⊗ L Λ A, Rp ∗ p ∗ R H om Λ ( A , A )) ∼ = R Hom( A ⊗ L Λ A, π ∗ R Hom( A , A )) ∼ = R Hom( π (cid:92) ( A ⊗ L Λ A ) , R Hom( A , A )) , using usual adjunctions and smooth base change for p and π several times. Under these isomor-phisms, the map R Hom( π (cid:92) ( A ⊗ L Λ A ) , Λ) ⊗ L Λ R Hom( A , A ) → R Hom( π (cid:92) ( A ⊗ L Λ A ) , R Hom( A , A ))is the natural map. This is an isomorphism as soon as π (cid:92) ( A ⊗ L Λ A ) ∈ D (Λ) is perfect for all compact A . In fact, the converse is also true: If one takes A = D BZ ( A ), then R Hom( A , A ) = π (cid:92) ( A ⊗ L Λ A ),and hence it follows that for M = π (cid:92) ( A ⊗ L Λ A ) ∈ D (Λ), the map R Hom( M, Λ) ⊗ L Λ M → R Hom(

M, M )is an isomorphism, which means that M is dualizable in D (Λ), i.e. perfect.Now we use the system of compact generators given by i b ! c -Ind G b ( E ) K Λ for varying b ∈ B ( G ),with locally closed immersion i b : Bun bG → Bun G , and K ⊂ G b ( E ) open pro- p . This translates thecondition on perfectness of Rπ ! ( A ⊗ L Λ Rπ ! Λ ⊗ L Λ A ) into the desired condition on stalks. (cid:3) HAPTER VI

Geometric Satake

As before, we ﬁx a nonarchimedean local ﬁeld E with residue ﬁeld F q of characteristic p and auniformizer π ∈ E . Recall that for any perfectoid space S over F q , we deﬁned the “curve” Y S over O E , as well as Y S = Y S \ V ( π ) and the quotient X S = Y S /ϕ Z .In this chapter, we are interested in studying modiﬁcations of G -torsors on these spaces, andperverse sheaves on such. Our discussion will mirror this three-step procedure of the constructionof X S : If one has understood the basic theory over Y S , the basic results carry over easily to Y S and then to X S . While as in previous chapters our main focus is on X S , in this chapter we willactually make critical use of Y S in order to degenerate to the Witt vector aﬃne Grassmannian, andhence to apply some results from the setting of schemes (notably the decomposition theorem). Asthe discussion here is very much of a local sort, one can usually reduce easily to the case that G issplit, and hence admits a (split) reductive model over O E , and we will often ﬁx such a split modelof G .For any d ≥

0, we consider the moduli space Div d Y parametrizing degree d Cartier divisors D ⊂ Y S . For aﬃnoid S , one can form the completion B + of O X S along D . Inverting D deﬁnes alocalization B of B + . One can then deﬁne a positive loop group L +Div d Y G and loop group L Div d Y G ,with values given by G ( B + ) resp. G ( B ); for brevity, we will simply write L + G and LG here. Onecan then deﬁne the local Hecke stack H ck G, Div d Y = [ L + G \ LG/L + G ] → Div d Y We will often break symmetry, and ﬁrst take the quotient on the right to deﬁne the Beilinson–Drinfeld Grassmannian Gr G, Div d Y = LG/L + G → Div d Y so that H ck IG = L + G \ Gr IG . The Beilinson–Drinfeld Grassmannian Gr G, Div d Y → Div d Y is a small v-sheaf that can be writtenas an increasing union of closed subsheaves that are proper and representable in spatial diamonds,by bounding the relative position; this is one main result of [ SW20 ]. On the other hand, L + G canbe written as an inverse limit of truncated positive loop groups, which are representable in locallyspatial diamonds and cohomologically smooth; moreover, on each bounded subset, it acts throughsuch a ﬁnite-dimensional quotient. This essentially reduces the study of all bounded subsets of H ck G, Div d Y to Artin stacks. For any small v-stack S → Div d Y , we let H ck G,S/

Div d Y = H ck G,S/

Div d Y be the pullback. Let D ´et ( H ck G,S/

Div d Y , Λ) bd ⊂ D ´et ( H ck G,S/

Div d Y , Λ)be the full subcategory of all objects with quasicompact support over Div d Y . This is a monoidalcategory under convolution (cid:63) . Here, we use the convolution diagram H ck G,S/

Div d Y × S H ck G,S/

Div d Y ( p ,p ) ←−−−− L + G \ LG × L + G LG/L + G m −→ H ck G,S/

Div d Y and deﬁne A (cid:63) B = Rm ∗ ( p ∗ A ⊗ L Λ p ∗ B ) . On D ´et ( H ck G,S/

Div d Y , Λ) bd , one can deﬁne a relative perverse t -structure (where an object isperverse if and only if it is perverse over any geometric ﬁbre of S ), see Section VI.7. In particular,this t -structure is compatible with any base change in S . For this t -structure, the convolution (cid:63) isleft t -exact (and t -exactness only fails for issues related to non-ﬂatness over Λ). To prove this, wereinterpret convolution as fusion, and use some results on hyperbolic localization.Moreover, one can restrict to the complexes A ∈ D ´et ( H ck G,S/

Div d Y , Λ) bd that are universallylocally acyclic over S . This condition is also preserved under convolution. For d = 1, or in generalwhen S maps to the locus of distinct untilts (Div d Y ) (cid:54) = ⊂ Div d Y , one can describe the categoryof universally locally acyclic by the condition that the restriction to any Schubert cell is locallyconstant with perfect ﬁbres. To prove that all such sheaves are universally locally acyclic, we alsointroduce (for d = 1) the aﬃne ﬂag variety, in Section VI.5, and use their Demazure resolutions. Definition

VI.0.1 . The Satake category

Sat( H ck G,S/

Div d Y , Λ) ⊂ D ´et ( H ck G,S/

Div d Y , Λ) bd is the category of all A ∈ D ´et ( H ck G,S/

Div d Y , Λ) bd that are perverse, ﬂat over Λ (i.e., for all Λ -modules M , also A ⊗ L Λ M is perverse), and universally locally acyclic over (Div ) I . Intuitively, Sat( H ck G,S/

Div d Y , Λ) are the “ﬂat families of perverse sheaves on H ck G,S/

Div d Y → S ”, where ﬂatness refers both to the geometric aspect of ﬂatness over S (encoded in universallocal acyclicity) and the algebraic aspect of ﬂatness in the coeﬃcients Λ. The Satake categorySat( H ck G,S/

Div d Y , Λ) is a monoidal category under convolution. The forgetful functorSat( H ck G,S/

Div d Y , Λ) → D ´et (Gr G,S/

Div d Y , Λ)is fully faithful. If d = 1 and S = Spd F q , then one can compare it to the category considered byZhu [ Zhu17 ] and Yu [

Yu19 ], deﬁned in terms of the Witt vector aﬃne Grassmannian. Moreover,the categories for S = Spd O C and S = Spd C are naturally equivalent to the category for S =Spd F q , via the base change functors; here C = (cid:98) E . Thus, the Satake category is, after picking areductive model of G , naturally the same for the Witt vector aﬃne Grassmannian and the B +dR -aﬃneGrassmannian. At this point, we could in principle use Zhu’s results [ Zhu17 ] (reﬁned integrally byYu [

Yu19 ]) to identify the Satake category with the category of representations of (cid:98) G , at least when I. GEOMETRIC SATAKE 185 G is unramiﬁed. However, for the applications we actually need ﬁner knowledge of the functorialityof the Satake equivalence including the case for d >

1; we thus prove everything we need directly.More precisely, we now switch to (Div X ) d in place of Div d Y , replacing also the use of Y S by X S ;the two local Hecke stacks are locally isomorphic, so this poses no problems. For any ﬁnite set I ,let H ck IG = H ck G, (Div X ) I and consider the monoidal category Sat IG (Λ) = Sat( H ck IG , Λ) . In fact, the monoidal structure naturally upgrades to a symmetric monoidal structure. This relieson the fusion product, for which it is critical to allow general ﬁnite sets I . Namely, given ﬁnite sets I , . . . , I k , letting I = I (cid:116) . . . (cid:116) I k , one has an isomorphism H ck IG × (Div ) I (Div ) I ; I ,...,I k ∼ = k (cid:89) j =1 H ck I j G × (Div ) I (Div ) I ; I ,...,I k where (Div ) I ; I ,...,I k ⊂ (Div ) I is the open subset where x i (cid:54) = x i (cid:48) whenever i, i (cid:48) ∈ I lie in diﬀerent I j ’s. The exterior tensor product then deﬁnes a functor (cid:2) kj =1 : k (cid:89) j =1 Sat I j G (Λ) → Sat I ; I ,...,I k G (Λ)where Sat I ; I ,...,I k G (Λ) is the variant of Sat IG (Λ) for H ck IG × (Div ) I (Div ) I ; I ,...,I k . However, the re-striction functor Sat IG (Λ) → Sat I ; I ,...,I k G (Λ)is fully faithful, and the essential image of the exterior product lands in its essential image. Thus,we get a natural functor ∗ kj =1 : k (cid:89) j =1 Sat I j G (Λ) → Sat IG (Λ) , independent of the ordering of the I j . In particular, for any I , we get a functorSat IG (Λ) × Sat IG (Λ) → Sat I (cid:116) IG (Λ) → Sat IG (Λ) , using functoriality of Sat JG (Λ) in J , which deﬁnes a symmetric monoidal structure ∗ on Sat IG (Λ),commuting with (cid:63) . This is called the fusion product. In general, for any symmetric monoidalcategory ( C , ∗ ) with a commuting monoidal structure (cid:63) , the monoidal structure (cid:63) necessarily agreeswith ∗ ; thus, the fusion product reﬁnes the convolution product. (As usual in geometric Satake, weactually need to change ∗ slightly by introducing certain signs into the commutativity constraint,depending on the parity of the support of the perverse sheaves.)Moreover, restricting A ∈ Sat IG (Λ) to Gr IG and taking the pushforward to (Div ) I , all cohomol-ogy sheaves are local systems of Λ-modules on (Div ) I . By a version of Drinfeld’s lemma, theseare equivalent to representations of W IE on Λ-modules. This deﬁnes a symmetric monoidal ﬁbrefunctor F I : Sat IG (Λ) → Rep W IE (Λ) ,

86 VI. GEOMETRIC SATAKE where Rep W IE (Λ) is the category of continuous representations of W IE on ﬁnite projective Λ-modules.Using a version of Tannaka duality, one can then build a Hopf algebra in the Ind-category ofRep W IE (Λ) so that Sat IG (Λ) is given by its category of representations (internal in Rep W IE (Λ)). Forany ﬁnite set I , this is given by the tensor product of I copies of the corresponding Hopf algebrafor I = {∗} , which in turn is given by some aﬃne group scheme G (cid:86) over Λ with W E -action. Theorem

VI.0.2 (Theorem VI.11.1) . There is a canonical isomorphism G (cid:86) ∼ = (cid:98) G with the Lang-lands dual group, under which the action of W E on G (cid:86) agrees with the usual action of W E on (cid:98) G upto an explicit cyclotomic twist. If √ q ∈ Λ , the cyclotomic twist can be trivialized, and Sat IG (Λ) isnaturally equivalent to the category of ( (cid:98) G (cid:111) W E ) I -representations on ﬁnite projective Λ -modules. For the proof, one can restrict to Λ = Z /(cid:96) n Z ; passing to a limit over n , one can actuallybuild a group scheme over Z (cid:96) . Its generic ﬁbre is reductive, as the Satake category with Q (cid:96) -coeﬃcients is (geometrically) semisimple: For this, we use the degeneration to the Witt vectoraﬃne Grassmannian and the decomposition theorem for schemes. To identify the reductive group,we argue ﬁrst for tori, and then for rank 1 groups, where everything reduces to G = PGL which iseasy to analyze by using the minuscule Schubert cell. Here, the pinning includes a cyclotomic twistas of course the cohomology of the minuscule Schubert variety P of Gr PGL contains a cyclotomictwist. Afterwards, we apply hyperbolic localization in order to construct symmetric monoidalfunctors Sat G → Sat M for any Levi M of G , inducing dually maps M (cid:86) → G (cid:86) . This produces manyLevi subgroups of G (cid:86) Q (cid:96) from which it is easy to get the isomorphism with (cid:98) G Q (cid:96) , including a pinning.As these maps M (cid:86) → G (cid:86) are even deﬁned integrally, and (cid:98) G ( Z (cid:96) ) ⊂ (cid:98) G ( Q (cid:96) ) is a maximal compact opensubgroup by Bruhat–Tits theory, generated by the rank 1 Levi subgroups, one can then deducethat G (cid:86) ∼ = (cid:98) G integrally, again with an explicit (cyclotomic) pinning.We will also need the following addendum regarding a natural involution. Namely, the localHecke stack H ck IG has a natural involution sw given by reversing the roles of the two G -torsors; inthe presentation in terms of LG , this is induced by the inversion on LG . Then sw ∗ induces naturallyan involution of Sat IG (Λ), and thus involution can be upgraded to a symmetric monoidal functorcommuting with the ﬁbre functor, thus realizing a W E -equivariant automorphism of ˇ G ∼ = (cid:98) G . Proposition

VI.0.3 (Proposition VI.12.1) . The action of sw ∗ on Sat IG induces the automor-phism of (cid:98) G that is the Cartan involution of the split group (cid:98) G , conjugated by (cid:98) ρ ( − . VI.1. The Beilinson–Drinfeld Grassmannian

First, we deﬁne the base space of the Beilinson–Drinfeld Grassmannian for any d ≥ Definition

VI.1.1 . For any d ≥ , consider the small v-sheaves on Perf F q given by Div d Y = (Spd O E ) d / Σ d , Div dY = (Spd E ) d / Σ d , Div dX = Div d = (Spd E/ϕ Z ) d / Σ d , where Σ d is the symmetric group. Proposition

VI.1.2 . For any d ≥ , there is a functorial injection (i) from Div d Y ( S ) into the set of closed Cartier divisors on Y S , (ii) from Div dY ( S ) into the set of closed Cartier divisors on Y S , I.1. THE BEILINSON–DRINFELD GRASSMANNIAN 187 (iii) from

Div dX ( S ) into the set of closed Cartier divisors on X S .Moreover, in case (i) and (ii), if S = Spa( R, R + ) is aﬃnoid perfectoid, then for any closed Cartierdivisor D ⊂ Y S resp. D ⊂ Y S in the image of this embedding, the adic space D = Spa( Q, Q + ) isaﬃnoid. In case (iii), the same happens locally in the analytic topology on S . Proof.

We handle case (i) ﬁrst. Over (Spd O E ) d and for S = Spa( R, R + ) aﬃnoid, we get d untilts R (cid:93)i , i = 1 , . . . , d of R , and there are elements ξ i ∈ W O E ( R + ) generating the kernel of θ i : W O E ( R + ) → R (cid:93) + i . Each of the ξ i deﬁnes a closed Cartier divisor by Proposition II.1.4. Then ξ = (cid:81) i ξ i deﬁnes another closed Cartier divisor, given by Spa( A, A + ) for A = W O E ( R + )[ (cid:36) ] ] /ξ ,and A + the integral closure of W O E ( R + ) /ξ , where (cid:36) ∈ R is a pseudouniformizer.Now the ideal sheaf of this closed Cartier divisor is a line bundle, and by [ SW20 , Proposition19.5.3], line bundles on Y S satisfy v-descent. Thus, even if we are only given a map S → Div d Y =(Spd O E ) d / Σ d , we can still deﬁne a line bundle I ⊂ O Y S , and it still deﬁnes a closed Cartierdivisor as this can be checked v-locally. Also, V ( I ) ⊂ Y S is quasicompact over S , as this canagain be checked v-locally. This implies that it is contained in some aﬃnoid Y S, [0 ,n ] , and hence D = Spa( A, A + ) is aﬃnoid in general.The case (ii) now follows formally by passing to an open subset, and case (iii) by passing to thequotient under Frobenius. (cid:3) Remark

VI.1.3 . As in [

Far18a ] one checks that Div d Y ( S ) is the set of “relative Cartier divisors”of degree d , that is to say Cartier divisors that give degree d Cartier divisors when pulled back viaany geometric point Spa(

C, C + ) → S . The same holds for Div dY and Div dX .In the following we will consider a perfectoid space S equipped with a map f : S → Div d Y (resp. f : S → Div dY , resp. f : S → Div dX ). We denote by D S ⊂ Y S (resp. D S ⊂ Y S , D S ⊂ X S )the corresponding closed Cartier divisor. Let I S ⊂ O Y S (resp. I S ⊂ O Y S , I S ⊂ O X S ) be thecorresponding invertible ideal sheaf.Let us note the following descent result for vector bundles. Proposition

VI.1.4 . Sending S as above to the category of vector bundles on D S deﬁnes av-stack. Proof.

Any vector bundle on D S deﬁnes a v-sheaf on Perf S : This reduces formally to thestructure sheaf of D S , which then further reduces to the structure of O Y S (resp. O Y S , resp. O X S ).It remains to prove that v-descent of vector bundles is eﬀective. The case of X S reduces to Y S aslocally on S , the relevant D S is isomorphic; and clearly Y S reduces to Y S .Now assume ﬁrst that S = Spa( C, C + ) for some complete algebraically closed C . Then D S is given by a ﬁnite sum of degree 1 Cartier divisors on Y S , and one can reduce by induction tothe case of degree 1 Cartier divisors, where the result is [ SW20 , Lemma 17.1.8] applied to thecorresponding untilt of S .On the other hand, assume that T → S is an ´etale cover with a vector bundle E T on D T equipped with a descent datum to D S ; we want to prove descent to D S . By the argument of deJong–van der Put [ dJvdP96 , Proposition 3.2.2], cf. [ KL15 , Proposition 8.2.20], one can reduce to

88 VI. GEOMETRIC SATAKE the case that T → S is a ﬁnite ´etale cover. Then D T → D S is also ﬁnite ´etale (as Y T → Y S is),and the result follows from usual ﬁnite ´etale descent.Now let S be general and T → S a v-cover with a vector bundle E T on D T equipped with adescent datum to D S . For any geometric point Spa( C, C + ) → S , one can descent E T × S Spa(

C,C + ) toa vector bundle E Spa(

C,C + ) on D Spa(

C,C + ) . Now we follow the proof of [ SW20 , Lemma 17.1.8] tosee that one can in fact descend E T in an ´etale neighborhood of Spa( C, C + ), which is enough by theprevious paragraph. We can assume that S and T are aﬃnoid. As E Spa(

C,C + ) is necessarily free,also E T × S Spa(

C,C + ) is free, and by an ´etale localization we can assume that E T is free. Then thedescent datum is given by some matrix with coeﬃcients in O D T × ST . Moreover, by approximatingthe basis coming via pullback from E Spa(

C,C + ) , we can ensure that this matrix has coeﬃcients in1 + [ (cid:36) ] O + Y T × ST ( Y T × S T, [0 ,n ] ) /ξ for some n so that D S ⊂ Y S, [0 ,n ] ; here ξ is a generator of I S . Now oneuses that the v-cohomology group H v ( S, O ( Y [0 ,n ] ) + ) is almost zero, as follows from almost vanishingin the perfectoid case, and writing it as a direct summand of the positive structure sheaf of thebase change to O E [ π /p ∞ ] ∧ . Then the argument from [ SW20 , Lemma 17.1.8] applies, showing thatone can successively improve the basis to produce a basis invariant under the descent datum in thelimit. (cid:3)

Assuming that D S is aﬃnoid, as is the case locally on S , we let B +Div d Y ( S ) (resp . B +Div dY ( S ) , resp . B +Div dX ( S ))be (the global sections of) the completion of O Y S along I S (resp. of O Y S along I S , resp. of O X S along I S ), and B Div d ( − ) ( S ) = B +Div d ( − ) ( S )[ I S ] . This deﬁnes v-sheaves B +Div d ( − ) ⊂ B Div d ( − ) over Div d ( − ) in all three cases. In the case of d = 1, thoserings are the ones that are usually denoted B +dR , resp. B dR . Definition

VI.1.5 . Let Z be an aﬃne scheme over O E . The positive loop space L +Div d Y Z (resp. loop space L Div d Y Z ) of Z is the v-sheaf over Div d Y given by S (cid:55)→ L +Div d Y ( S ) = Z ( B +Div d Y ( S )) (resp . S (cid:55)→ L Div d Y ( S ) = Z ( B Div d Y ( S ))) . Similarly, if Z is an aﬃne scheme over E , one deﬁnes the positive loop space L +Div dY Z and L +Div dX Z (resp. loop space L Div dY Z and L Div dX Z ). We note that we use aﬃnity of Z to see that these are actually v-sheaves. Now we can deﬁnethe local Hecke stacks. Definition

VI.1.6 . Let G be a reductive group over O E (resp. over E , resp. over E ). Thelocal Hecke stack H ck G, Div d Y (resp. H ck G, Div dY , resp. H ck G, Div dX ) is the functor sending an aﬃnoidperfectoid S → Div d Y (resp. S → Div dY , resp. S → Div dX , assuming that D S is aﬃnoid) to thegroupoid of pairs of G -bundles E , E over B +Div d Y ( S ) (resp. over B +Div dY ( S ) , resp. over B +Div dX ( S ) )together with an isomorphism E ∼ = E over B Div d Y ( S ) (resp. over B Div dY ( S ) , resp. over B Div dX ( S ) ). I.1. THE BEILINSON–DRINFELD GRASSMANNIAN 189

Proposition

VI.1.7 . The local Hecke stack H ck G, Div d Y (resp. H ck G, Div dY , resp. H ck G, Div dX ) is asmall v-stack. There is a natural isomorphism of ´etale stacks over Div d Y (resp. over Div dY , resp. over Div dX ) H ck G, Div d Y ∼ = ( L +Div d Y G ) \ ( L Div d Y G ) / ( L +Div d Y G ) (resp. H ck G, Div dY ∼ = ( L +Div dY G ) \ ( L Div dY G ) / ( L +Div dY G ) , resp. H ck G, Div dX ∼ = ( L +Div dX G ) \ ( L Div dX G ) / ( L +Div dX G ) . ) Proof.

The category of vector bundles over B +Div d Y (resp. B +Div dY , B +Div dX ) satisﬁes v-descent:It is enough to check this modulo powers of the ideal I S , where the result is Proposition VI.1.4.By the Tannakian formalism, it follows that the category of G -bundles also satisﬁes v-descent, soone can descend E , E . The isomorphism between E and E over B Div d Y S (resp. over B Div dY ( S ),resp. over B Div dX ( S )) is then given by a section of an aﬃne scheme over the respective ring, whichagain satisﬁes v-descent. Smallness follows from the argument of Proposition III.1.3.Any G -bundle over B +Div d Y ( S ) is ´etale locally on S trivial. Indeed, if S is a geometric point then B +Div d Y ( S ) is a product of complete discrete valuation rings with algebraically closed residue ﬁeld,so that all G -torsors are trivial. In general, note that triviality of the G -torsor over B +Div d Y ( S ) isimplied by triviality modulo I S (as one can always lift sections over nilpotent thickenings). Thenthe result follows from [ GR03 , Proposition 5.4.21]. Trivializing E and E ´etale locally then directlyproduces the given presentations. (cid:3) Similarly, one can deﬁne the Beilinson–Drinfeld Grassmannians.

Definition

VI.1.8 . Let G be a reductive group over O E (resp. over E , resp. over E ). TheBeilinson–Drinfeld Grassmannian Gr G, Div d Y (resp. Gr G, Div dY , resp. Gr G, Div dX ) is the functor sendingan aﬃnoid perfectoid S → Div d Y (resp. S → Div dY , resp. S → Div dX , assuming again that D S isaﬃnoid) to the groupoid of G -bundles E over B +Div d Y ( S ) (resp. over B +Div dY ( S ) , resp. over B +Div dX ( S ) )together with a trivialization of E over B Div d Y ( S ) (resp. over B Div dY ( S ) , resp. over B Div dX ( S ) ). Proposition

VI.1.9 . The Beilinson–Drinfeld Grassmannian Gr G, Div d Y (resp. Gr G, Div dY , resp. Gr G, Div dX )is a small v-sheaf. There is a natural isomorphism of ´etale sheaves over Div d Y (resp. over Div dY ,resp. over Div dX ) Gr G, Div d Y ∼ = ( L Div d Y G ) / ( L +Div d Y G ) (resp. Gr G, Div dY ∼ = ( L Div dY G ) / ( L +Div dY G ) , resp. Gr G, Div dX ∼ = ( L Div dX G ) / ( L +Div dX G ) . ) Proof.

The proof is identical to the proof of Proposition VI.1.7. (cid:3)

90 VI. GEOMETRIC SATAKE

The positive loop group L +Div d Y G admits the natural ﬁltration by closed subgroups( L +Div d Y G ) ≥ m ⊂ L +Div d Y G deﬁned for all m ≥ G ( B +Div d Y ) → G ( B +Div Y / I mS );we refer to these as the principal congruence subgroups of L +Div Y G . Similar deﬁnitions of courseapply also over Div dY and Div dX . For d = 1, one can easily describe the graded pieces of thisﬁltration (and again the result also holds for Div Y and Div X ). Proposition

VI.1.10 . There are natural isomorphisms L +Div Y G/ ( L +Div Y G ) ≥ ∼ = G ♦ and ( L +Div Y G ) ≥ m / ( L +Div Y G ) ≥ m +1 ∼ = (Lie G ) ♦ { m } where { m } signiﬁes a “Breuil-Kisin twist” by I mS / I m +1 S . Proof.

The ﬁrst equality follows directly from the deﬁnitions, while the second comes fromthe exponential. (cid:3)

For general d , we still have the following result. Proposition

VI.1.11 . For any d and m ≥ , the quotient ( L +Div d Y G ) ≥ m / ( L +Div d Y G ) ≥ m +1 sends a perfectoid space S → Div d Y with corresponding Cartier divisor D S ⊂ Y S with ideal sheaf I S to (Lie G ⊗ O E I mS / I m +1 S )( S ) where I mS / I m +1 S is a line bundle on D S . This is representable in locally spatial diamonds, partiallyproper, and cohomologically smooth of (cid:96) -dimension equal to d times the dimension of G .Moreover, one can ﬁlter ( L +Div d Y G ) ≥ m / ( L +Div d Y G ) ≥ m +1 × Div d Y (Div Y ) d with subquotients given by twists of (Lie G ) ♦ × Div Y ,π i (Div Y ) d where π i : (Div Y ) d → Div Y is the projection to the i -th factor. Proof.

The description of the subquotient follows from the exponential sequence again. Tocheck that it is representable in locally spatial diamonds, partially proper, and cohomologicallysmooth, we can pull back to (Div Y ) d , and then these properties follow from the existence of thegiven ﬁltration. For this in turn, note that over (Div Y ) d , we have d ideal sheaves I , . . . , I d , andone can ﬁlter O D S by O D S / I , I / I I , . . . , I · · · I d − / I · · · I d , each of which is, after pullbackto an aﬃnoid perfectoid space S , isomorphic to O S (cid:93)i . (cid:3) I.1. THE BEILINSON–DRINFELD GRASSMANNIAN 191

One can also show that the ﬁrst quotient is cohomologically smooth, but this is slightly moresubtle.

Proposition

VI.1.12 . For any d , the quotient L +Div d Y G/ ( L +Div d Y G ) ≥ → Div d Y parametrizes over a perfectoid space S → Div d Y maps D S → G . For any quasiprojective smoothscheme Z over O E , the sheaf T Z → Div d Y taking a perfectoid S over Div d Y to maps D S → Z is representable in locally spatial diamonds, par-tially proper, and cohomologically smooth over Div d Y of (cid:96) -dimension equal to d times the dimensionof Z ; in particular, this applies to this quotient group. Proof.

The description of the quotient group is clear. To analyze T Z , we ﬁrst note that if Z is an aﬃne space, then the result holds true, as was proved in the previous proposition. In fact,after pullback via the quasi-pro-´etale surjective morphism (Div Y ) d → Div d Y , there is a sequence ofmorphisms T A n = W −→ W −→ · · · −→ W d +1 = (Div Y ) d where, for S aﬃnoid perfectoid with S → W i +1 giving rise to the untilts ( S (cid:93) , . . . , S (cid:93)d ) ∈ (Div Y ) d ( S ), W i × W i +1 S → S is locally on S isomorphic to A n, ♦ S (cid:93)i .If Z (cid:48) → Z is any separated ´etale map between schemes over O E , we claim that T Z (cid:48) → T Z isalso separated ´etale. For this, we analyze the pullback along any S → T Z given by some perfectoidspace S and a map D S → Z . Then D (cid:48) = D S × Z Z (cid:48) → D S is separated ´etale, and the ﬁbre product T Z (cid:48) × T Z S parametrizes S (cid:48) → S with a lift D S (cid:48) → D (cid:48) over D S . By Lemma VI.1.13 below, this isrepresentable by a perfectoid space separated ´etale over S . In case Z (cid:48) → Z is an open immersion,it follows that T Z (cid:48) → T Z is injective and ´etale, thus an open immersion.Now note that for any geometric point of T Z , the corresponding map D S → Z has ﬁniteimage, and is thus contained in some open aﬃne subscheme. It follows that T Z admits an opencover by T Z (cid:48) for aﬃne Z (cid:48) . If Z is aﬃne, then one sees directly that T Z is representable in locallyspatial diamonds and partially proper by taking a closed immersion into A n O E for some n . Forcohomological smoothness, we observe that we can in fact choose these aﬃnes so that they admit´etale maps to A d O E , as again we only need to arrange this at ﬁnitely many points at a time. Nowthe result follows from the discussion of A d O E and of separated ´etale maps. (cid:3) Lemma

VI.1.13 . Let S be a perfectoid space with a map S → Div d Y giving rise to the Cartierdivisor D S ⊂ Y S . Let D (cid:48) → D S be a separated ´etale map. Then there is a separated ´etale map S (cid:48) → S such that for T → S , maps T → S (cid:48) over S are equivalent to lifts D T → D (cid:48) over D S . Proof.

By descent of separated ´etale maps [

Sch17a , Proposition 9.7], we can assume that S is strictly totally disconnected. Exhausting D (cid:48) by a rising union of quasicompact subspaces, we canassume that D (cid:48) is quasicompact. In any geometric ﬁbre, D (cid:48) is then a disjoint union of open subsets(as any geometric ﬁbre is, up to nilpotents, a ﬁnite disjoint union of untilts Spa( C (cid:93)i , C (cid:93) + i )), and thisdescription spreads into a small neighborhood by [ Sch17a , Proposition 11.23, Lemma 15.6]. We

92 VI. GEOMETRIC SATAKE can thus reduce to the case that D (cid:48) → D S is an open immersion. Now the lemma follows from theobservation that the map | D S | → | S | is closed. (cid:3) VI.2. Schubert varieties

Now we recall the Schubert varieties. Assume in this section that G is a split reductive groupover O E (or over E , but in that case we can always choose a model over O E ). Fix a split torusand Borel T ⊂ B ⊂ G . Note that we can always pass to the situation of split G by making a ﬁnite´etale extension of O E resp. E ; this way, the results of this section are useful in the general case.Similarly, the cases of X S and Y S reduce easily to the case of Y S , so we only do the latter caseexplicitly here.Assume ﬁrst that d = 1. In that case, for every geometric point S = Spa( C, C + ) → Div Y =Spd O E given by an untilt S (cid:93) = Spa( C (cid:93) , C (cid:93) + ) of S , one has B +Div Y = B +dR ( C (cid:93) ) and B Div Y = B dR ( C (cid:93) )for the usual deﬁnition of B +dR and B dR (relative to O E ). Recall that B +dR ( C (cid:93) ) is a complete discretevaluation ring with residue ﬁeld C (cid:93) , fraction ﬁeld B dR ( C (cid:93) ), and uniformizer ξ . It follows that bythe Cartan decomposition G ( B dR ( C (cid:93) )) = (cid:71) µ ∈ X ∗ ( T ) + G ( B +dR ( C (cid:93) )) µ ( ξ ) G ( B +dR ( C (cid:93) )) , so as a set H ck G, Div Y ( S ) / ∼ = = X ∗ ( T ) + , the dominant cocharacters of T . Recall that on X ∗ ( T ) + , we have the dominance order, where µ ≥ µ (cid:48) if µ − µ (cid:48) is a sum of positive coroots with Z ≥ -coeﬃcients. Remark

VI.2.1 . Since we work over Y and do not restrict ourselves to Y , we include the caseof the Cartier divisor π = 0. For this divisor, C (cid:93) = C and B +dR ( C (cid:93) ) = W O E ( C ). Definition

VI.2.2 . For any µ ∈ X ∗ ( T ) + , let H ck G, Div Y , ≤ µ ⊂ H ck G, Div Y be the subfunctor of all those maps S → H ck G, Div Y such that at all geometric points S (cid:48) = Spa( C, C + ) → S , the corresponding S (cid:48) -valued point is given by some µ (cid:48) ∈ X ∗ ( T ) + with µ (cid:48) ≤ µ . Moreover, Gr G, Div Y , ≤ µ ⊂ Gr G, Div Y is the preimage of H ck G, Div Y , ≤ µ ⊂ H ck G, Div Y . Recall the following result.

Proposition

VI.2.3 ([

SW20 , Proposition 20.3.6]) . The inclusion H ck G, Div Y , ≤ µ ⊂ H ck G, Div Y is a closed subfunctor and H ck G, Div Y = lim −→ µ H ck G, Div Y , ≤ µ ; I.2. SCHUBERT VARIETIES 193 thus, similar properties hold for Gr G, Div Y . Here, the index category is the partially ordered set of µ ’s under the dominance order, which is a disjoint union (over π ( G ) ) of ﬁltered partially orderedsets.The map Gr G, Div Y , ≤ µ → Div Y is proper and representable in spatial diamonds. Proof.

It is enough to prove the assertions over Gr G, Div Y as this is a v-cover of H ck G, Div Y .Then [ SW20 , Proposition 20.3.6] gives the results, except for the assertion thatGr G, Div Y = lim −→ µ Gr G, Div Y , ≤ µ . For this, note that the map from right to left is clearly an injection. For surjectivity, note thatfor any quasicompact S with a map S → Gr G, Div Y , only ﬁnitely many strata can be met, as themeromorphic isomorphism of G -bundles necessarily has bounded poles. This, coupled with thefact Gr G, Div Y → Div Y is separated while Gr G, Div Y , ≤ µ → Div Y is proper, implies that the map (cid:70) µ Gr G, Div Y , ≤ µ → Gr G, Div Y is a v-cover, whence we get the desired surjectivity. (cid:3) In particular, H ck G, Div Y ,µ = H ck G, Div Y , ≤ µ \ (cid:91) µ (cid:48) <µ H ck G, Div Y , ≤ µ (cid:48) ⊂ H ck G, Div Y , ≤ µ is an open subfunctor, and similarly its preimage Gr G, Div Y ,µ ⊂ Gr G, Div Y , ≤ µ . By the Cartan decom-position, the space H ck G, Div Y ,µ has only one point in every geometric ﬁbre over Div Y . This pointcan in fact be deﬁned as a global section[ µ ] : Div Y → Gr G, Div Y ,µ given by µ ( ξ ) ∈ ( L Div d Y G )( S ) whenever ξ is a local generator of I S ; up to the action of L +Div d Y G ,this is independent of the choice of ξ . Proposition

VI.2.4 . The map [ µ ] : Div Y → H ck G, Div Y ,µ given by µ is a v-cover. This gives an isomorphism H ck G, Div Y ,µ ∼ = [Div Y / ( L +Div Y G ) µ ] where ( L +Div Y G ) µ ⊂ L +Div Y G is the closed subgroup stabilizing [ µ ] ∈ Gr G, Div Y ( S ) . Recalling theprincipal congruence subgroups ( L +Div Y G ) ≥ m ⊂ L +Div Y G, we let ( L +Div Y G ) ≥ mµ = ( L +Div Y G ) µ ∩ ( L +Div Y G ) ≥ m ⊂ ( L +Div Y G ) µ . Then ( L +Div Y G ) µ / ( L +Div Y G ) ≥ µ ∼ = ( P − µ ) ♦ ⊂ L +Div Y G/ ( L +Div Y G ) ≥ ∼ = G ♦

94 VI. GEOMETRIC SATAKE and ( L +Div Y G ) ≥ mµ / ( L +Div Y G ) ≥ m +1 µ ∼ = (Lie G ) ♦ µ ≤ m { m } ⊂ ( L +Div Y G ) ≥ m / ( L +Div Y G ) ≥ m +1 ∼ = (Lie G ) ♦ { m } , where P − µ ⊂ G is the parabolic with Lie algebra (Lie G ) µ ≤ , and (Lie G ) µ ≤ m ⊂ Lie G is the subspaceon which µ acts via weights ≤ m via the adjoint action.In particular, Gr G, Div Y ,µ ∼ = L +Div Y G/ ( L +Div Y G ) µ is cohomologically smooth of (cid:96) -dimension (cid:104) ρ, µ (cid:105) over Div Y . Proof.

We ﬁrst handle the case G = GL n with its standard upper-triangular Borel and di-agonal torus. In that case, µ is given by some sequence k ≥ . . . ≥ k n of integers, and Gr G, Div Y ,µ parametrizes B +Div Y -lattices Ξ ⊂ B n Div Y that are of relative position µ at all points. Let S = Spa( R, R + ) be an aﬃnoid perfectoid spacewith a map S → Div Y = Spd O E given by an untilt S (cid:93) = Spa( R (cid:93) , R (cid:93) + ) over O E of S . By the proofof [ SW20 , Proposition 19.4.2], the R (cid:93) -modulesFil i Ξ ( R (cid:93) ) n = ( ξ i Ξ ∩ B +dR ( R (cid:93) ) n ) / ( ξ i Ξ ∩ ξB +dR ( R (cid:93) ) n )are ﬁnite projective of rank equal to the number of occurrences of − i among k , . . . , k n . Localizing,we may assume that they are ﬁnite free. We may then pick a basis e , . . . , e n of ( R (cid:93) ) n so thatany Fil i Ξ ( R (cid:93) ) n is freely generated by a subset e , . . . , e n i of e , . . . , e n . Lifting e n i − +1 , . . . , e n i toelements of f n i − +1 , . . . , f n i ∈ ξ i Ξ ∩ B +dR ( R (cid:93) ) n , and setting g n i − +1 = ξ − i f n i − +1 , . . . , g n i = ξ − i f n i , orequivalently g j = ξ k j f j for j = 1 , . . . , n , one sees that f , . . . , f n form a B +dR ( R (cid:93) )-basis of B +dR ( R (cid:93) ) n ,and g , . . . , g n will form a B +dR ( R (cid:93) )-basis of Ξ. Thus, changing basis to the f i ’s, one has moved Ξto the lattice ξ k B +dR ( R (cid:93) ) ⊕ . . . ⊕ ξ k n B +dR ( R (cid:93) ) . This is the lattice corresponding to [ µ ] ∈ Gr GL n , Div Y , showing that the mapDiv Y → H ck GL n , Div Y ,µ is indeed surjective.Moreover, the stabilizer ( L +Div Y GL n ) µ of ξ k B +dR ( R (cid:93) ) ⊕ . . . ⊕ ξ k n B +dR ( R (cid:93) ) in L +Div Y GL n is theset of all matrices A = ( A ij ) ∈ GL n ( B +Div Y ) such that for i < j , A ij ∈ ξ k i − k j B +Div Y . This easilyimplies the description of ( L +Div Y GL n ) µ / ( L +Div Y GL n ) ≥ µ ∼ = ( P − µ ) ♦ ⊂ GL ♦ n and ( L +Div Y GL n ) ≥ mµ / ( L +Div Y GL n ) ≥ m +1 µ ∼ = (Lie GL n ) ♦ µ ≤ m { m } ⊂ (Lie GL n ) ♦ { m } . The description also implies that ( L +Div Y GL n ) µ contains L +Div Y P − µ and ( L +Div Y U a ) ≥ µ ( a ) for any pos-itive root a . I.2. SCHUBERT VARIETIES 195

In general, picking a closed immersion of G into GL n (compatible with the torus and the Borel),one sees that ( L +Div Y G ) µ / ( L +Div Y G ) ≥ µ ⊂ ( P − µ ) ♦ ⊂ G ♦ and ( L +Div Y G ) ≥ mµ / ( L +Div Y G ) ≥ m +1 µ ⊂ (Lie G ) ♦ µ ≤ m { m } ⊂ (Lie G ) ♦ { m } as these subquotients embed into the similar subquotient for GL n . Moreover, one sees that( L +Div Y G ) µ contains L +Div Y P − µ and ( L +Div Y U a ) ≥ µ ( a ) for any positive root a . These imply that the twodisplayed inclusions are actually equalities.A consequence of these considerations is that the map L +Div Y G/ ( L +Div Y G ) µ → L +Div Y GL n / ( L +Div Y GL n ) µ is a closed immersion (as this happens on all subquotients for the principal congruence ﬁltra-tion). The target is isomorphic to Gr GL n , Div Y ,µ , which contains Gr G, Div Y ,µ as a closed subspace (by[ SW20 , Proposition 20.3.7]). We see that we get an inclusion L +Div Y G/ ( L +Div Y G ) µ (cid:44) → Gr G, Div Y ,µ ⊂ Gr GL n , Div Y ,µ of closed subspaces, with the same geometric points: This implies that it is an isomorphism (e.g.,as the map is then necessarily a closed immersion, thus qcqs, so one can apply [ Sch17a , Lemma12.5]). From here, all statements follow. (cid:3)

Remark

VI.2.5 . The mapGr G, Div Y ,µ = L +Div Y G/ ( L +Div Y G ) µ → L +Div Y G/ ( L +Div Y G ) ≥ µ ∼ = ( G/P − µ ) ♦ is the Bia(cid:32)lynicki-Birula map, see [ CS17 ].Passing to general d , we ﬁrst note that any geometric ﬁbre of H ck G, Div d Y → Div d Y is isomorphic to a product of geometric ﬁbres of H ck G, Div Y → Div Y . More precisely, if f :Spa( C, C + ) → Div d Y is a geometric point, it is given by an unordered tuple Spa( C (cid:93)i , C (cid:93) + i ), i ∈ I with | I | = d , of untilts over O E . Some of these may be equal, so one can partition I into sets I , . . . , I r of equal untilts. Then we really have r untilts, given by maps f , . . . , f r : Spa( C, C + ) → Div Y , andone has an isomorphism H ck G, Div d Y × Div d Y ,f Spa(

C, C + ) ∼ = r (cid:89) i =1 H ck G, Div Y × Div Y ,f i Spa(

C, C + ) ,

96 VI. GEOMETRIC SATAKE and similarly L +Div d Y G × Div d Y ,f Spa(

C, C + ) ∼ = r (cid:89) i =1 L +Div Y G × Div Y ,f i Spa(

C, C + ) ,L Div d Y G × Div d Y ,f Spa(

C, C + ) ∼ = r (cid:89) i =1 L Div Y G × Div Y ,f i Spa(

C, C + ) , Gr G, Div d Y × Div d Y ,f Spa(

C, C + ) ∼ = r (cid:89) i =1 Gr G, Div Y × Div Y ,f i Spa(

C, C + ) . Indeed, it suﬃces to prove this on the level of the positive loop and loop group, where in turn itfollows from a similar decomposition of B +Div d Y after pullback, which is clear.In particular, we can deﬁne the following version of Schubert varieties. Definition

VI.2.6 . For any unordered collection µ • = ( µ j ) j ∈ J of elements µ j ∈ X ∗ ( T ) + with | J | = d , let H ck G, Div d Y , ≤ µ • ⊂ H ck G, Div d Y be the subfunctor of all those S → H ck G, Div d Y such that at all geometric points Spa(

C, C + ) → S ,then equipped with an (unordered) tuple of d untilts Spa( C (cid:93)i , C (cid:93) + i ) , i ∈ I with | I | = d , there is somebijection between ψ : I ∼ = J such that the relative position of E and E at Spa( C (cid:93)i , C (cid:93) + i ) is boundedby (cid:88) j ∈ J,C (cid:93)ψ ( j ) ∼ = C (cid:93)i µ j . Let Gr G, Div d Y , ≤ µ • ⊂ Gr G, Div d Y be the preimage of H ck G, Div d Y , ≤ µ • ⊂ H ck G, Div d Y . Proposition

VI.2.7 . The inclusion H ck G, Div d Y , ≤ µ • ⊂ H ck G, Div d Y is a closed subfunctor. The map Gr G, Div d Y , ≤ µ • → Div d Y is proper, representable in spatial diamonds,and of ﬁnite dim . trg . Proof.

This can be checked after pullback to (Div Y ) d . Then it follows from [ SW20 , Propo-sition 20.5.4]. (cid:3)

Moreover, we have the following result. Here, we let( L +Div d Y G )

Proposition

VI.2.8 . For any µ • = ( µ j ) j ∈ J as above, the action of L +Div d Y G on Gr G, Div d Y , ≤ µ • factors over ( L +Div d Y G )

We need to see that the action of ( L +Div d Y G ) ≥ m is trivial. As everything is separated,this can be checked on geometric points, where one reduces to d = 1 by a decomposition intoproducts. Then it follows from Proposition VI.2.4. (cid:3) VI.3. Semi-inﬁnite orbits

For this section, we continue to assume that G is split, and again we only spell out the case ofDiv d Y ; analogous results hold for Div dY and Div dX , and follow easily from the case presented.Previously, we stratiﬁed the aﬃne Grassmanian using the Cartan decomposition, the stratabeing aﬃne Schubert cells. We now use the Iwasawa decomposition to obtain another stratiﬁcationby semi-inﬁnite orbits.Fix a cocharacter λ : G m → T ⊂ G , inducing a Levi M λ with Lie algebra (Lie G ) λ =0 , a parabolic P λ = P + λ with Lie algebra (Lie G ) λ ≥ and its unipotent radical U λ with Lie algebra (Lie G ) λ> . Weget an action of the v-sheaf G m (taking an aﬃnoid perfectoid space S = Spa( R, R + ) of characteristic p to R × ) on Gr G, Div d Y via the composition of the Teichm¨uller map[ · ] : G m → L +Div d Y G m , the map L +Div d Y λ : L +Div d Y G m → L +Div d Y G and the action of L +Div d Y G on Gr G, Div d Y . We wish to apply Braden’s theorem in this setup. For thispurpose, we need to verify Hypothesis IV.6.1. To construct the required stratiﬁcation, we use theaﬃne Grassmannian Gr P λ , Div d Y associated to the parabolic P λ . Note that this admits a mapGr P λ , Div d Y → Gr M λ , Div d Y → Gr M λ , Div d Y where M λ is the Levi quotient of P λ and M λ is the maximal torus quotient of M λ (the cocenter).Then Gr M λ , Div d Y admits a surjection from a disjoint union of copies of (Div Y ) d parametrized by X ∗ ( M λ ) d . While there are many identiﬁcations between these copies, the sum µ := (cid:80) di =1 µ i ∈ X ∗ ( M λ ) deﬁnes a well-deﬁned locally constant function(VI.3.1) Gr M λ , Div d Y → X ∗ ( M λ ) . More precisely, for Spa(

C, C + ) → Gr M λ , Div d Y a geometric point, let C (cid:93) , . . . , C (cid:93)r be the correspondingdistinct untilts with 1 ≤ r ≤ d . ThenGr M λ , Div d Y × Div d Y Spa(

C, C + ) ∼ = r (cid:89) i =1 Gr M λ , Div Y × Div Y Spa(

C, C + )

98 VI. GEOMETRIC SATAKE where the morphism Spa(

C, C + ) → Div Y is given by C (cid:93)i on the i -th component of the product.This is identiﬁed with r (cid:89) i =1 X ∗ ( M λ ) × Spa(

C, C + )and the weighted sum morphism X ∗ ( M λ ) r → X ∗ ( M λ ) (weighing each term with the multiplicityof C (cid:93)i as an untilt of C in the morphism Spa( C, C + ) → Div d Y ) deﬁnes thus a functionGr M λ , Div d Y × Div d Y Spa(

C, C + ) −→ X ∗ ( M λ ) . This deﬁnes the locally constant function of (VI.3.1).For ν ∈ X ∗ ( M λ ) let Gr νP λ , Div d Y ⊂ Gr P λ , Div d Y be the corresponding open and closed subset obtained as the preimage. Proposition

VI.3.1 . The map Gr P λ , Div d Y = (cid:71) ν Gr νP λ , Div d Y → Gr G, Div d Y is bijective on geometric points, and it is a locally closed immersion on each Gr νP λ , Div d Y . The union (cid:83) ν (cid:48) ≤ ν Gr ν (cid:48) P λ , Div d Y has closed image in Gr G, Div d Y . The action of G m via L + λ on Gr P λ , Div d Y extends toan action of the monoid A , and the G m -ﬁxed points agree with Gr M λ , Div d Y . Applying this proposition also in the case of the inverse G m -action, and pulling back to arelative Schubert variety, veriﬁes Hypothesis IV.6.1 in this situation. Proof.

The action of G m on P λ via conjugation extends to an action of the monoid A .Applying loop spaces to this observation and the observation that the map L Div d Y P λ → Gr P λ , Div d Y isequivariant for the action of L +Div d Y G m on the source via conjugation and on the target via the givenaction (as we quotient by the right action of L +Div d Y P λ ) gives the action of the monoid L +Div d Y A ,and thus of A via restricting to Teichm¨uller elements. As everything is separated, this also showsthat G m -ﬁxed points necessarily lie in the image of L Div d Y M λ , thus the G m -ﬁxed points agree withGr M λ , Div d Y .Bijectivity of the map Gr P λ , Div d Y → Gr G, Div d Y on geometric points follows from the Iwahori decomposition. It remains to prove that the map isa locally closed immersion on each Gr νP λ , Div d Y , and the union over ν (cid:48) ≤ ν is closed. Picking a closedembedding into GL n , this reduces to the case G = GL n , and by writing any standard parabolicas an intersection of maximal parabolics, we can assume that P λ ⊂ GL n is a maximal parabolic.Passing to a higher exterior power, we can even assume that P λ ⊂ GL n is the mirabolic, ﬁxinga one-dimensional quotient of the standard representation. In that case, Gr GL n , Div d Y parametrizesﬁnite projective B +Div d Y -modules M with an identiﬁcation M ⊗ B +Div d Y B Div d Y ∼ = B n Div d Y , and Gr P λ , Div d Y I.3. SEMI-INFINITE ORBITS 199 parametrizes such M for which the image L ⊂ B Div d Y of M in the quotient B n Div d Y → B Div d Y :( x , . . . , x n ) (cid:55)→ x n is a line bundle over B +Div d Y . It also suﬃces to prove the result after pullbackalong (Div Y ) d → Div d Y . Now the result follows from the next lemma. (cid:3) In the following, the “relative position” of a B + -lattice L ⊂ B to the standard lattice B + ⊂ B refers to the image under the map Gr G m , Div d Y → X ∗ ( G m ) = Z deﬁned above. Lemma

VI.3.2 . Let S = Spa( R, R + ) be an aﬃnoid perfectoid space over F q with untilts S (cid:93)i =Spa( R (cid:93)i , R (cid:93) + i ) over O E for i = 1 , . . . , n . Let ξ i ∈ W O E ( R + ) generate the kernel of θ i : W O E ( R + ) → R (cid:93) + i and let ξ = ξ · · · ξ n . Let B + be the ξ -adic completion of W O E ( R + )[ (cid:36) ] ] where (cid:36) ∈ R is apseudouniformizer, and let B = B + [ ξ ] . Finally, let L ⊂ B be a ﬁnitely generated B + -module that is open and bounded, i.e. there is some integer N such that ξ N B + ⊂ L ⊂ ξ − N B + ⊂ B .For any m ∈ Z , let S m ⊂ S be the subset of those points at which the relative position of L to B + ⊂ B is given by m . Then (cid:83) m (cid:48) ≥ m S m (cid:48) is closed, and if S m = S then the B + -module L is a linebundle. Proof.

We can assume that L ⊂ B + via multiplying by a power of ξ . Let s ∈ S be anypoint, corresponding to a map Spa( K ( s ) , K ( s ) + ) → S m . Let B + s be the version of B + constructedfrom ( K ( s ) , K ( s ) + ). Then B + s is a ﬁnite product of discrete valuation rings, and the image L s of L ⊗ B + B + s in B + s is necessarily free of rank 1. Then s ∈ S m if and only if the length of B + s /L s as B + s -module is given by m . Localizing on S if necessary, we can ﬁnd an element l ∈ L ⊂ B whose image in L s is a generator. In a neighborhood of s , the element l generates a submodule L (cid:48) = B + · l ⊂ L whose relative position to B + is bounded above by m at all points by the nextlemma, and then the relative position of L ⊂ B is also bounded above by m at all points as L (cid:48) ⊂ L .This gives the desired semicontinuity of the stratiﬁcation (noting that as L is open and bounded,only ﬁnitely many values of m can appear). If S m = S , then the containment L (cid:48) ⊂ L has to be anequality, and hence L = L (cid:48) is generated by l , so L is a line bundle. (cid:3) Lemma

VI.3.3 . In the situation of the previous lemma, let f ∈ B + be any element, and considerthe map | S | → Z ≥ ∪ {∞} sending any point s of S to the length of B + s /f as B + s -module. This map is semicontinuous in thesense that for any m ≥ , the locus where it is ≤ m is open. Proof.

For any i = 1 , . . . , n , one can look at the closed subspace S i ⊂ S where the image of f in R (cid:93)i vanishes. On the open complement of all S i , the function is identically 0. By induction, wecan thus pass to a closed subspace S i ⊂ S , where we can consider the function f i = fξ i ; the lengthfunction for f is then the length function for f i plus one. This gives the result. (cid:3)

00 VI. GEOMETRIC SATAKE

Example

VI.3.4 . Suppose λ ∈ X ∗ ( T ) is regular dominant. Then P λ = B . We then obtain thestratiﬁcation by semi-inﬁnite orbits S ν = Gr νB, Div d Y for ν ∈ X ∗ ( T ). One has S ν (cid:44) → Gr G, Div Y , a locally closed immersion, andGr G, Div Y = (cid:91) ν ∈ X ∗ ( T ) Gr νB, Div d Y (disjoint union) at the level of points.We can now apply Theorem IV.6.5. Here, we also use the opposite parabolic P − λ ⊂ G . If S → Div d Y is a small v-stack, we denoteGr G,S/

Div d Y = Gr G, Div d Y × Div d Y S and similarly H ck G,S/

Div d Y = H ck G, Div d Y × Div d Y S. For any A ∈ D ´et (Gr G,S/

Div d Y , Λ), we call A bounded if it arises via pushforward from some ﬁniteunion Gr G,S/

Div d Y , ≤ µ • ⊂ Gr G,S/

Div d Y . We let D ´et (Gr G,S/

Div d Y , Λ) bd ⊂ D ´et (Gr G,S/

Div d Y , Λ)be the corresponding full subcategory.

Corollary

VI.3.5 . Let S → Div d Y be any small v-stack. Consider the diagram Gr P λ , Div d Y q + (cid:120) (cid:120) p + (cid:38) (cid:38) Gr G, Div d Y Gr M λ , Div d Y Gr P − λ , Div d Y q − (cid:102) (cid:102) p − (cid:56) (cid:56) and denote by q + S etc. the base change along S → Div d Y . Consider the full subcategory D ´et (Gr G,S/

Div d Y , Λ) G m - mon , bd ⊂ D ´et (Gr G,S/

Div d Y , Λ) bd of all A ∈ D ´et (Gr G,S/

Div d Y , Λ) that are bounded and G m -monodromic in the sense of Deﬁnition IV.6.11.On D ´et (Gr G,S/

Div d Y , Λ) G m - mon , bd , the natural map R ( p − S ) ∗ R ( q − S ) ! → R ( p + S ) ! ( q + S ) ∗ is an equivalence, inducing a “constant term” functor CT P λ : D ´et (Gr G,S/

Div d Y , Λ) G m - mon , bd → D ´et (Gr M λ ,S/ Div d Y , Λ) bd . This functor commutes with any base change in S and preserves the condition of being universallylocally acyclic over S (which is well-deﬁned for bounded A ). I.3. SEMI-INFINITE ORBITS 201

Proof.

This follows from Proposition VI.3.1 and Theorem IV.6.5, Proposition IV.6.12 andProposition IV.6.14. (cid:3)

Example

VI.3.6 (Follow-up to Example VI.3.4) . In the context of Example VI.3.4, suppose d = 1. Then, Gr T,S/

Div Y = X ∗ ( T ) × S , and the corresponding semi-inﬁnite orbits are denoted by S ν ⊂ Gr G, Div Y for ν ∈ X ∗ ( T ). Thus,CT B ( A ) = (cid:77) ν ∈ X ∗ ( T ) R ( p ν ) ! ( A | S ν )with p ν : S ν × Div Y S → S ⊂ Gr T,S/

Div Y the embedding indexed by ν .As a ﬁnal topic here, let us analyze more closely the semi-inﬁnite orbits in the special ﬁbre,i.e. for the Witt vector aﬃne Grassmannian Gr Witt G (so that (Gr Witt G ) ♦ ∼ = Gr G, Spd F q / Div Y ), which isan increasing union of perfections of projective varieties over F q by [ BS17 ], cf. also [

Zhu17 ]. Forany λ ∈ X ∗ ( T ) as above, we have the semi-inﬁnite orbit S λ = LU · [ λ ] ⊂ Gr Witt G . Proposition

VI.3.7 . For any µ ∈ X ∗ ( T ) + , the intersection S λ ∩ Gr Witt G, ≤ µ is representable by anaﬃne scheme. Proof.

Picking a closed immersion

G (cid:44) → GL n , one can reduce to G = GL n . In that case, thereis an ample line bundle L on Gr Witt G constructed in [ BS17 ]. We ﬁrst claim that the pullback of L toGr Witt B is trivial. Indeed, recall that if Spec R → Gr Witt G corresponds to a lattice Ξ ⊂ W O E ( R )[ π ] n ,then L is given by det( π − m W O E ( R ) / Ξ) for any large enough m , using the determinantdet : Perf( W O E ( R ) on R ) → Pic( R ) , which is multiplicative in exact triangles. On Gr Witt B , one has a universal ﬁltration of Ξ com-patible with the standard ﬁltration on the standard lattice, which induces a similar ﬁltration onΞ /π m W O E ( R ), where all the graded quotients are locally constant (and constant on S λ ). Thismeans that the line bundle is naturally trivialized over each connected component S λ of Gr Witt B .We claim that this section over S λ extends uniquely to a section over the closed subset (cid:83) λ (cid:48) ≤ λ S λ (cid:48) that vanishes over the complement of S λ , showing that the intersection of S λ with each Gr Witt G, ≤ µ must be aﬃne. To see this, by the v-descent results of [ BS17 ], it suﬃces to check that for anyrank 1 valuation ring V with a map Spec V → Gr Witt G whose generic point Spec K maps into S λ ,the section of L over Spec K extends to Spec V and is nonzero in the special ﬁbre precisely whenall of Spec V maps into S λ . Now the ﬁltration0 = Ξ K, ⊂ Ξ K, ⊂ . . . ⊂ Ξ K,n = Ξ K with Ξ K,i = Ξ K ∩ W O E ( K )[ π ] i ⊂ W O E ( K )[ π ] i has the property that Ξ K,i / Ξ K,i − = π λ i W O E ( K )for the cocharacter λ = ( λ , . . . , λ n ). Moreover, the ﬁltration by the Ξ K,i extends integrally to theﬁltration 0 = Ξ ⊂ Ξ ⊂ . . . ⊂ Ξ n = Ξ

02 VI. GEOMETRIC SATAKE with Ξ i = Ξ ∩ W O E ( V )[ π ] i ⊂ W O E ( V )[ π ] i , which is still a ﬁltration by ﬁnite projective W O E ( V )-modules by [ SW20 , Lemma 14.2.3]. Theinjection of Ξ i / Ξ i − into W O E ( V )[ π ] (projecting to the i -th coordinate) has image contained in W O E ( V )[ π ] ∩ π λ i W O E ( K ) = π λ i W O E ( V ) , so we get natural injections Ξ i / Ξ i − (cid:44) → π λ i W O E ( V ), that are isomorphisms after inverting π or [ a ]for a pseudouniformizer a ∈ V . Now the relevant line bundle can be written as the tensor productof the line bundles given by the determinants of the complexes π λ i W O E ( V ) / (Ξ i / Ξ i − ) ∈ Perf( W O E ( R ) on R ) . These line bundles are indeed naturally trivial over K as the perfect complex is acyclic there. Nowthis complex is concentrated in degree 0, and is torsion, so admits a ﬁltration by complexes of theform V /a ∼ = [ aV (cid:44) → V ] for pseudouniformizers a ∈ V . The associated line bundle on V is then givenby the alternating tensor product V ⊗ V ( aV ) − = a − V , and the natural section by 1 ∈ a − V . Wesee that the section is indeed integral, and that it is nonzero in the special ﬁbre if and only if allthe above complexes are acyclic, equivalently if Ξ i / Ξ i − → π λ i W O E ( V ) is an isomorphism. Butthis is precisely the condition that all of Spec V maps into S λ . (cid:3) The union (cid:83) λ, (cid:104) ρ,λ (cid:105)≤ d S λ ⊂ Gr Witt G is closed, thus so is (cid:91) λ, (cid:104) ρ,λ (cid:105)≤ d S λ ∩ Gr Witt G, ≤ µ ⊂ Gr Witt G, ≤ µ . For d = (cid:104) ρ, µ (cid:105) , this is all of Gr Witt G, ≤ µ , while for d = −(cid:104) ρ, µ (cid:105) it contains only a point, correspondingto [ λ ] for λ the antidominant representative of the Weyl orbit of µ . Also, only d of the same parityas (cid:104) ρ, µ (cid:105) are relevant. By Proposition VI.3.7, the successive complements (cid:91) λ, (cid:104) ρ,λ (cid:105)≤ d S λ ∩ Gr Witt G, ≤ µ \ (cid:91) λ, (cid:104) ρ,λ (cid:105)≤ d − S λ ∩ Gr Witt G, ≤ µ = (cid:71) nu, (cid:104) ρ,λ (cid:105) = d S λ ∩ Gr Witt G, ≤ µ are aﬃne. This means that at each step, the dimension can drop by at most 1. However, in (cid:104) ρ, µ (cid:105) steps, it drops by (cid:104) ρ, µ (cid:105) . We get the following corollary on Mirkovi´c–Vilonen cycles, cf. [ MV07 ,Theorem 3.2], and [

GHKR10 ], [

Zhu17 , Corollary 2.8] for a diﬀerent proof based on point counting,the classical Satake isomorphism, and the Kato-Lusztig formula [

Kat82 ], [

Lus83 ]. Corollary

VI.3.8 . The scheme S λ ∩ Gr Witt G, ≤ µ is equidimensional of dimension (cid:104) ρ, µ + λ (cid:105) . VI.4. Equivariant sheaves

Now we go back to the setting of general reductive groups G over O E (resp. over E if we workover Div dY or Div dX ). As usual, let Λ be some coeﬃcient ring killed by some integer n prime to p .We want to study D ´et ( − , Λ) for the local Hecke stack H ck G, Div d Y = L +Div d Y G \ Gr G, Div d Y I.4. EQUIVARIANT SHEAVES 203 or its versions for Div dY and Div dX . Neither this nor its bounded versions H ck G, Div d Y , ≤ µ • (say, when G is split) is an Artin stack as L +Div d Y G is not ﬁnite-dimensional. However, Proposition VI.2.8 showsthat on the bounded version, the action factors over a ﬁnite-dimensional quotient.First, we observe that on the level of D ´et ( − , Λ), one can then forget about the rest of the action.

Proposition

VI.4.1 . Let H be a group small v-sheaf over a small v-sheaf S that admits aﬁltration H ≥ m ⊂ H by closed subgroups such that, v-locally on S , for each m ≥ each quotient H ≥ m /H ≥ m +1 admits a further ﬁnite ﬁltration with graded pieces given by ( A S (cid:93) ) ♦ for some untilt S (cid:93) of S (that may vary). Let X be some small v-sheaf over S with an action of H that factors over H . Then the pullback functor D ´et ( H

Both stacks live over the classifying stack [ H

Proposition

VI.4.2 . Assume that B ⊂ G is a Borel. Let S → Div d Y be any small v-sheaf. Let A ∈ D ´et ( H ck G,S/

Div d Y , Λ) with support quasicompact over S . Assume that the hyperbolic localization CT B ( A ) = 0 of the pullback of A to Gr G,S/

Div d Y vanishes. Then A = 0 .The similar assertion holds with Div dY and Div dX in place of Div d Y .

04 VI. GEOMETRIC SATAKE

Proof.

Note that the formation of CT B commutes with any base change in S , by Corol-lary VI.3.5. We can thus assume that S = Spa( C, C + ) is strictly local. Up to replacing d by asmaller integer, removing double points, we can assume that the map S → Div d Y is given by d distinct untilts S (cid:93)i over O E , i = 1 , . . . , d . Let E (cid:48) | E be an extension splitting G , assumed unramiﬁedin our situation where we work over Y . We can then lift all S (cid:93)i to O E (cid:48) , and thereby reduce to thecase of split G . The corresponding geometric ﬁbre H ck G,S/

Div d Y has a stratiﬁcation enumerated by µ , . . . , µ d ∈ X ∗ ( T ) + , with strata[ S/ ( d (cid:89) i =1 ( L +Div Y G ) µ i × Div Y S )] . If A is nonzero, we can ﬁnd a maximal such stratum on which A is nonzero. Now we applyCorollary VI.3.5, see Example VI.3.6. One has an isomorphism S × X ∗ ( T ) d ∼ = Gr T,S/

Div d Y . Over the copy of S enumerated by the antidominant representatives of (the Weyl group orbitsof) µ , . . . , µ n the functor CT B is the pullback of A to a section of the stratum corresponding to µ , . . . , µ n ∈ X ∗ ( T ) + (which, as we recall, correspond to a maximal stratum where A is nonzero).This shows that the restriction of A to a section over this maximal stratum is zero. This gives thedesired contradiction, so A = 0. (cid:3) VI.5. Aﬃne ﬂag variety

At a few isolated spots, it will be useful to use the aﬃne ﬂag variety, the main point being thatthe Schubert varieties in the aﬃne ﬂag variety admit explicit resolutions of singularities, given byDemazure resolutions (also known as Bott–Samelson resolutions). It will be enough to appeal tothese in the setting of a split reductive group G , with a reductive model over O E and Borel B ⊂ G deﬁned over O E , for d = 1, and for a small v-stack S → Div Y factoring over Spd O C where C = (cid:98) E ,so we restrict attention to this setting.Consider the base change G A of G to A = W O E ( O C (cid:91) ). We have Fontaine’s map θ : A → O C ,and we can deﬁne an “Iwahori” group scheme I → G A , ﬂat over A , whose points in a ker θ -torsionfree A -algebra R are given those elements g ∈ G ( R ) such that θ ( g ) ∈ G ( R ⊗ A O C ) lies in B ( R ⊗ A O C ). Similarly, for any parabolic P ⊂ G containing B , we get a “parahoric” group scheme P → G A , ﬂat over A , whose points in a ker θ -torsionfree A -algebra R are those g ∈ G ( R ) suchthat θ ( g ) ∈ P ( R ⊗ A O C ). In particular, this applies to the parabolics P i corresponding to thesimple reﬂections s i ; let P i be the corresponding parahorics. Still more generally, for any aﬃnesimple reﬂection s i , one can deﬁne a parabolic P i → G ﬂat over A , and such that I → G factorsover P i . (The construction of these parahoric group schemes over A can be reduced to the caseof W O E ( k )[[ u ]] via a faithfully ﬂat embedding W O E ( k )[[ u ]] (cid:44) → A along which everything arises viabase change, and then one can appeal to the work of Bruhat–Tits [ BT84 , Section 3.9.4].)

I.5. AFFINE FLAG VARIETY 205

Definition

VI.5.1 . In the situation above, including a small v-stack S over Spd O C , mappingto Div Y , let F (cid:96) G,S → S be the ´etale quotient LG/L + I , where L + I ( R, R + ) = I ( B +dR ( R (cid:93) )) . Note here that as S lives over Spd O C , any Spa( R, R + ) over S comes with an untilt R (cid:93) over O C , in which case B +dR ( R (cid:93) ) is an A = W O E ( O C )-algebra, so that I ( B +dR ( R (cid:93) )) is well-deﬁned. Proposition

VI.5.2 . There is a natural projection map F (cid:96) G,S → Gr G,S/

Div Y that is a torsor under ( G/B ) ♦ . In particular, it is proper, representable in spatial diamonds, andcohomologically smooth. Proof.

This follows from the identiﬁcation L + G/L + I ∼ = (

G/B ) ♦ , which follows from thedeﬁnition, and the similar properties of ( G/B ) ♦ → Spd O E . (cid:3) We analyze the stratiﬁcation of F (cid:96) G,S into L + I -orbits. Let N ( T ) ⊂ T be the normalizer of T ,and (cid:102) W = N ( T )( B dR ( C (cid:48) )) /T ( B +dR ( C (cid:48) ))be the aﬃne Weyl group, for any complete algebraically closed ﬁeld C (cid:48) over O E with a map C (cid:91) → C (cid:48) (cid:91) ; this is naturally independent of the choice of C (cid:48) . As T ( B dR ( C (cid:48) )) /T ( B +dR ( C (cid:48) )) ∼ = X ∗ ( T ),there is a short exact sequence 0 → X ∗ ( T ) → (cid:102) W → W → , where W is the usual Weyl group of G . Proposition

VI.5.3 . The decomposition of F (cid:96) G,S ( C (cid:48) ) into L + I ( C (cid:48) ) -orbits is given by F (cid:96) G,S ( C (cid:48) ) = (cid:71) w ∈ (cid:102) W L + I ( C ) · w. Proof. If C (cid:48) lives over E , we can choose an isomorphism B dR ( C (cid:48) ) ∼ = C (cid:48) (( ξ )) and the resultfollows from the classical result. If C (cid:48) lives over the residue ﬁeld F q of E , this reduces to theassertion for the Witt vector aﬃne ﬂag variety, for which we refer to [ Zhu17 ]. (cid:3) Recall that (cid:102) W acts on X ∗ ( T ). Fixing the alcove a corresponding to the Iwahori group I , onegets a set of aﬃne simple reﬂections s i as the reﬂections along the faces of the alcove; these generatea normal subgroup W aﬀ ⊂ (cid:102) W . Letting Ω ⊂ (cid:102) W denote the stabilizer of the alcove, there is a splitshort exact sequence 1 → W aﬀ → (cid:102) W → Ω → . One gets the Bruhat order on (cid:102) W : If w i = w i, ω i ∈ (cid:102) W = W aﬀ (cid:111) Ω for i = 1 , w ≤ w if ω = ω and in one (hence every) presentation of w as a product of aﬃne simplereﬂections, w is obtained by removing some factors.

06 VI. GEOMETRIC SATAKE

Definition

VI.5.4 . For w ∈ (cid:102) W , the aﬃne Schubert cell is the subfunctor F (cid:96) G,w,S ⊂ F (cid:96)

G,S ofall maps

Spa(

R, R + ) → F (cid:96) G,S that on all geometric points lie in the L + I -orbit of w . The aﬃneSchubert variety is the subfunctor F (cid:96) G, ≤ w,S of all maps Spa(

R, R + ) → F (cid:96) G,S that on all geometricpoints lie in the L + I -orbit of w (cid:48) for some w (cid:48) ≤ w . Theorem

VI.5.5 . For each w ∈ (cid:102) W , the subfunctor F (cid:96) G, ≤ w,S ⊂ F (cid:96) G,S is closed, and F (cid:96) G, ≤ w,S → Spd O C is proper and representable in spatial diamonds, of ﬁnite dim . trg . The subfunctor F (cid:96) G,w,S ⊂F (cid:96) G, ≤ w,S is open and dense. Proof.

We will prove the theorem by constructing the Demazure resolution of F (cid:96) G, ≤ w,S . Write w = w ω ∈ (cid:102) W = W aﬀ (cid:111) Ω, and ﬁx a decomposition w = (cid:81) lj =1 s i j as a product of aﬃne simplereﬂections of minimal length, so l ( w ) = l ( w ) = l . We write ˙ w for the element w with such a choiceof decomposition.For each aﬃne simple reﬂection s i , we have a corresponding parahoric group P i → G A corre-sponding to the face of a ; one has L + P i /L + I ∼ = ( P ) ♦ . Definition

VI.5.6 . The Demazure variety corresponding to ˙ w is the ´etale sheaf Dem ˙ w,S = L + P s i × L + I L + P s i × L + I . . . × L + I L + P s il /L + I → S, equipped with the left L + I -action and the L + I -equivariant map Dem ˙ w,S → F (cid:96) G,S given by ( p , . . . , p l ) (cid:55)→ p · · · p l · ω . It is clear from the deﬁnition that Dem ˙ w → S is a successive ( P ) ♦ -ﬁbration over S , and inparticular is a spatial diamond, proper over S of ﬁnite dim . trg. As F (cid:96) G,S → S is partially proper,it follows that the image of Dem ˙ w,S → F (cid:96) G,S is proper. Moreover, the image can be identiﬁed ongeometric points, and we see that Dem ˙ w,S → F (cid:96) G, ≤ w,S is surjective, F (cid:96) G, ≤ w,S ⊂ F (cid:96) G,S is closed,and F (cid:96) G, ≤ w,S is proper over S . In particular, F (cid:96) G,w,S ⊂ F (cid:96) G, ≤ w,S is open, as the complement is aﬁnite union of closed subfunctors. As F (cid:96) G,S → Gr G,S is locally a product with (

G/B ) ♦ , it followsfrom [ SW20 , Theorem 19.2.4] that F (cid:96) G,S × Gr G,S Gr G, ≤ µ,S is a spatial diamond, and thus so is F (cid:96) G, ≤ w,S , as it is a closed subspace for µ large enough.Also, by checking on geometric points and reducing to the classical case, the map Dem ˙ w,S →F (cid:96) G, ≤ w,S is an isomorphism over F (cid:96) G,w,S whose preimage is given by( L + P s i \ L + I ) × L + I ( L + P s i \ L + I ) × L + I . . . × L + I ( L + P s il \ L + I ) /L + I . This implies that F (cid:96) G,w,S ⊂ F (cid:96) G, ≤ w,S is dense, as desired. As usual, a consequence of this discussionis that the Bruhat order is independent of the choice of ˙ w . (cid:3) Using Demazure resolutions, one can prove that the standard sheaves on the aﬃne ﬂag varietyare universally locally acyclic.

Proposition

VI.5.7 . For any w ∈ (cid:102) W , let j w : F (cid:96) G,w,S (cid:44) → F (cid:96) G, ≤ w,S be the open embedding.Then j w ! Λ ∈ D ´et ( F (cid:96) G, ≤ w,S , Λ) is universally locally acyclic over S . I.6. ULA SHEAVES 207

Proof.

Using Proposition IV.2.11, it suﬃces to prove the same for (cid:101) j w : F (cid:96) G,w,S (cid:44) → Dem ˙ w,S and (cid:101) j w ! Λ. Then (cid:101) j w ! Λ can be resolved in terms of Λ and all i ˙ w (cid:48) , ˙ w, ∗ Λ for i ˙ w (cid:48) , ˙ w : Dem ˙ w (cid:48) ,S → Dem ˙ w,S the closed immersion from another Demazure variety, corresponding to a subword of ˙ w (cid:48) of ˙ w ; notethat combinatorially, we are dealing with the situation of a normal crossing divisor at the boundary.By cohomological smoothness of all Dem ˙ w (cid:48) ,S → S and Proposition IV.2.11, the result follows. (cid:3) VI.6. ULA sheaves

We will be interested in universally locally acyclic sheaves on the local Hecke stack.

Definition

VI.6.1 . Let S → Div d Y be any small v-stack. An object A ∈ D ´et ( H ck G,S/

Div d Y , Λ) is universally locally acyclic over S if it is bounded, and its pullback to Gr G,S/

Div d Y is universally locally acyclic over S .Let D ULA´et ( H ck G,S/

Div d Y , Λ) ⊂ D ´et ( H ck G,S/

Div d Y , Λ) be the corresponding full subcategory. This deﬁnition is a priori not symmetric in the two bundles E , E parametrized by the localHecke stack. However, we can check that it actually is. Proposition

VI.6.2 . Consider the automorphism sw : H ck G,S/

Div d Y ∼ = H ck G,S/

Div d Y switching E and E . Then A ∈ D ´et ( H ck G,S/

Div d Y , Λ) is universally locally acyclic over S if and onlyif sw ∗ A is universally locally acyclic over S . Proof.

Fix any large enough substack U ⊂ H ck G,S/

Div d Y quasicompact over S containing thesupport of A . Let ( L Div d Y G ) U ⊂ L Div d Y G be the preimage of U . Universal local acyclicity afterpullback to Gr G, Div d Y is equivalent to universal local acyclicity after pullback to( L Div d Y G ) U / ( L +Div d Y G ) ≥ m for any m >

0, by Proposition VI.1.11 and Proposition VI.1.12. We need to see that this isequivalent to universal local acyclicity after pullback to( L +Div d Y G ) ≥ m \ ( L Div d Y G ) U for any m >

0. For this, we note that these two pro-systems in m are pro-isomorphic. By thenext lemma, the transition maps back and forth are also cohomologically smooth, which impliesthe desired equivalence. (cid:3)

08 VI. GEOMETRIC SATAKE

In the following lemma, we call a map f universally locally acyclic if Λ is f -universally locallyacyclic. Lemma

VI.6.3 . Let X f −→ X f −→ X f −→ X f −→ X be surjective maps of locally spatial diamonds that are compactiﬁable and of locally ﬁnite dim . trg .Assume that f ◦ f and f ◦ f are cohomologically smooth. Then f is universally locally acyclic.If f is universally locally acyclic and f ◦ f is cohomologically smooth, then f is cohomologicallysmooth. Thus, if f ◦ f , f ◦ f and f ◦ f are cohomologically smooth, then f and f arecohomologically smooth. We would expect that f : X → X should be unnecessary in order for f to be cohomologicallysmooth. Proof.

We claim that for any map g : Y → X , with pullbacks g i : Y i → X i and (cid:101) f i : Y i +1 → Y i , the natural transformation f ∗ Rg !0 → Rg !1 (cid:101) f ∗ is an isomorphism. Indeed, we have natural maps f ∗ f ∗ f ∗ Rg !0 → f ∗ f ∗ Rg !1 (cid:101) f ∗ → f ∗ Rg !2 (cid:101) f ∗ (cid:101) f ∗ → Rg !3 (cid:101) f ∗ (cid:101) f ∗ (cid:101) f ∗ and the composite of any two maps is an isomorphism. By the two-out-of-six-lemma, this impliesthat all maps are isomorphisms. By surjectivity of f and f , this implies that f ∗ Rg !0 → Rg !1 (cid:101) f ∗ is an isomorphism. Applying this with Y = X and to the constant sheaf Λ then shows, by thecriterion of Theorem IV.2.23, that Λ is f -universally locally acyclic.Now assume that f is universally locally acyclic and f ◦ f is cohomologically smooth, then R ( f ◦ f ) ! Λ ∼ = f ∗ Rf !0 Λ ⊗ L Λ Rf !1 Λis invertible. This implies that both tensor factors are invertible, and in particular Rf !1 Λ is invertible,so f is cohomologically smooth. For the ﬁnal statement, we now know that the hypotheses implythat f and f are universally locally acyclic, so the displayed equation implies that f and f arecohomologically smooth. (cid:3) Using the conversativity result Proposition VI.4.2, we can characterize universally locally acyclicsheaves in terms of their hyperbolic localization. Note that we can always reduce to the case ofquasisplit G by ´etale localization on S . Proposition

VI.6.4 . Let B ⊂ G be a Borel with torus quotient T . Let S be a small v-stackwith a map S → Div d Y , and let A ∈ D ´et ( H ck G,S/

Div d Y , Λ) bd . Then A is universally locally acyclic over S if and only if the hyperbolic localization CT B ( A ) ∈ D ´et (Gr T,S/

Div d Y , Λ) bd is universally locally acyclic over S . This, in turn, is equivalent to the property that Rπ T,S, ∗ CT B ( A ) ∈ D ´et ( S, Λ) is locally constant with perfect ﬁbres. I.6. ULA SHEAVES 209

Here π T,S : Gr

T,S/

Div d Y → S is the projection. Proof.

The forward direction follows from Corollary VI.3.5 and the ind-properness of π T,S andCorollary IV.2.12. For the converse direction, we may assume that S is strictly totally disconnectedand G is split. Note that to prove universal local acyclicity of A , it is enough to prove that themap p ∗ R H om( A, Rπ ! G,S Λ) ⊗ L Λ p ∗ A → R H om( p ∗ A, Rp !2 A )is an isomorphism (by Theorem IV.2.23). (Implicitly, we pass here to a bounded part of H ck G, Div d Y and replace the quotient by L +Div d Y G by a ﬁnite-dimensional quotient in order to be in the settingof Artin stacks.) By Proposition VI.4.2 applied to G × G , it is enough to prove this after ap-plying CT B − × B , where B − is the opposite Borel. Using that hyperbolic localization commuteswith exterior tensor products, and Proposition IV.6.13, this translates exactly into the similar iso-morphism characterizing universal local acyclicity of CT B ( A ). The ﬁnal statement follows fromProposition IV.2.28. (cid:3) In the case of one leg, one can completely characterize universally locally acyclic sheaves.

Proposition

VI.6.5 . Assume that G is split. Let S → Div Y be any small v-stack. Consider A ∈ D ´et ( H ck G,S/

Div Y , Λ) bd . Then A is universally locally acyclic over S if and only if for all µ ∈ X ∗ ( T ) + , the restriction of A to the section [ µ ] : S → H ck G,S/

Div Y is locally constant with perfect ﬁbres in D ´et ( S, Λ) . If G is not split, a similar characterization holds, by applying the result ´etale locally to reduceto the case of split G . Again, there is also the obvious version for Div Y and Div X . Proof.

First, we prove that if all ﬁbres are locally constant with perfect ﬁbres, then A isuniversally locally acyclic. This easily reduces to the case of j µ ! Λ where j µ : H ck G, Div Y ,µ (cid:44) → H ck G, Div Y is the inclusion of an open Schubert cell, and S = Div Y . We can also argue v-locally on Div Y andso base change to the case S = Spd O C . In that case, Proposition VI.5.2 and Proposition IV.2.13show that it suﬃces to prove the similar assertion for the aﬃne ﬂag variety, where it follows fromProposition VI.5.7.Now for the converse, we argue by induction on the support of A . On a maximal Schubert cellGr G,S/

Div Y ,µ where A is nonzero, its restriction is universally locally acyclic, and as on the Heckestack this stratum is the classifying space of a (pro-)cohomologically smooth group, it follows thatthe restriction of A along the section [ µ ] : S → H ck G,S/

Div Y is locally constant with perfect ﬁbres.Replacing A by the cone of j µ ! A | H ck G,S/

Div1 Y ,µ → A, the claim follows. (cid:3)

10 VI. GEOMETRIC SATAKE

In the following corollaries, we no longer assume that G is split. Corollary

VI.6.6 . Let S → Div Y be any small v-stack. Then D ULA´et ( H ck G,S/

Div Y , Λ) is stable under Verdier duality and − ⊗ L Λ − , R H om Λ ( − , − ) as well as j ! j ∗ , Rj ∗ j ∗ , j ! Rj ! , Rj ∗ Rj ! where j is the locally closed immersion of a Schubert cell. Moreover, all of these operations commutewith all pullbacks in S . Proof.

Stability under Verdier duality and compatibility with base change in S follow fromCorollary IV.2.25. For the other assertions, one can reduce to the case that G is split by workinglocally on S , where it follows from the previous proposition, and juggling with the six functors, andthe stalkwise characterization of the previous proposition. (cid:3) Corollary

VI.6.7 . For a complete algebraically closed extension C of E with residue ﬁeld k ,taking S = Spd O C , S = Spd C and S = Spd k , the functors D ULA´et ( H ck G, Spd C/ Div Y , Λ) ← D ULA´et ( H ck G, Spd O C / Div Y , Λ) → D ULA´et ( H ck G, Spd k/ Div Y , Λ) are equivalences. Proof.

Use that the formation of R H om commutes with any base change in S , and that thecategory of locally constant sheaves with perfect ﬁbres on any such S is equivalent to the categoryof perfect Λ-modules. (cid:3) In fact, the previous results extend to the case of general d as long as S → Div d Y has image inthe open subset (Div d Y ) (cid:54) = ⊂ Div d Y where all untilts are distinct. After passing to a ﬁnite ´etale coverof S , we can then in fact assume that S maps to (Div Y ) d (cid:54) = . Proposition

VI.6.8 . Assume that G is split. Let S → (Div Y ) d (cid:54) = → Div d Y be a small v-stack.Consider A ∈ D ´et ( H ck G,S/

Div d Y , Λ) bd . Then A is universally locally acyclic over S if and only if for all µ , . . . , µ d ∈ X ∗ ( T ) + , the restrictionof A to the section [ µ • ] : S → H ck G,S/

Div d Y is locally constant with perfect ﬁbres in D ´et ( S, Λ) .The category D ULA´et ( H ck G,S/

Div d Y , Λ) is stable under Verdier duality and − ⊗ L Λ − , R H om Λ ( − , − ) as well as j ! j ∗ , Rj ∗ j ∗ , j ! Rj ! , Rj ∗ Rj ! where j is the locally closed immersion of a Schubert cell. Moreover, all of these operations commutewith all pullbacks in S . Proof.

We have the decomposition H ck G,S/

Div d Y ∼ = d (cid:89) i =1 H ck G,S/ πi Div Y where π , . . . , π d : S → Div Y are the d projections, and the product on the right is taken over S . Onecan then stratify according to Schubert cells parametrized by tuples µ • = ( µ , . . . , µ d ) and the above I.7. PERVERSE SHEAVES 211 arguments imply the result. Here, in the beginning, to see that j µ • ! Λ is universally locally acyclic,one uses that exterior tensor products preserve universal local acyclicity, see Corollary IV.2.25, toreduce to the case of one leg. (cid:3)

VI.7. Perverse Sheaves

For any small v-stack S → Div d Y , we deﬁne a (relative) perverse t -structure on D ´et ( H ck G,S/

Div d Y , Λ) bd . Definition/Proposition

VI.7.1 . Let S → Div d Y be a small v-stack. There is a unique t -structure ( p D ≤ , p D ≥ ) on D ´et ( H ck G,S/

Div d Y , Λ) bd such that A ∈ p D ≤ ( H ck G,S/

Div d Y , Λ) bd if and only if for all geometric points Spa(

C, C + ) → S and open Schubert cells of H ck G, Spa(

C,C + ) / Div d Y , parametrized by some µ , . . . , µ r ∈ X ∗ ( T ) + (where r is the number of distinct untilts at Spa(

C, C + ) → Div d Y ), the pullback of A to this open Schubert cell sits in cohomological degrees ≤ − (cid:80) ri =1 (cid:104) ρ, µ i (cid:105) . Proof.

We note that on any bounded closed subset of Z ⊂ H ck G, Div d Y there is a presentablestable ∞ -category D ´et ( Z × Div d Y S, Λ) reﬁning the derived category, and the given class of objects isstable under all colimits and extensions (and is generated by a set of objects). Thus, the existenceand uniqueness of the t -structure follow from [ Lur16 , Proposition 1.4.4.11]. Moreover, one easilychecks that when enlarging Z , the inclusion functors are t -exact, so these glue to a t -structure inthe direct limit. (cid:3) Let Perv( H ck G,S/

Div d Y , Λ) ⊂ D ´et ( H ck G,S/

Div d Y , Λ) bd be the heart of the perverse t -structure. On it, pullback to the aﬃne Grassmannian is fully faithful. Proposition

VI.7.2 . The pullback functor

Perv( H ck G,S/

Div d Y , Λ) → D ´et (Gr G,S/

Div d Y , Λ) bd is fully faithful.Moreover, if A ∈ p D ≤ ( H ck G,S/

Div d Y , Λ) bd and B ∈ p D ≥ ( H ck G,S/

Div d Y , Λ) bd , then R H om Λ ( A, B ) ∈ D ≥ ( H ck G,S/

Div d Y , Λ) bd . Proof.

For the ﬁnal statement, we need to see that if C ∈ D ≤− ( H ck G,S/

Div d Y , Λ) bd , then thereare no nonzero maps C → R H om Λ ( A, B ); equivalently, there are no nonzero maps C ⊗ L Λ A → B .But this follows from the simple observation that C ⊗ L Λ A lies in p D ≤− .

12 VI. GEOMETRIC SATAKE

Now using this property of R H om Λ ( A, B ), descent implies that it is enough to see that if

A, B ∈ Perv( H ck G,S/

Div d Y , Λ), then any map between their pullbacks to Gr

G,S/

Div d Y is automati-cally invariant under the action of L +Div d Y G . This follows from Lemma VI.7.3 applied to a ﬁnite-dimensional approximation ofGr G,S/

Div d Y × Div d Y L +Div d Y G → Gr G,S/

Div d Y . (cid:3) We used the following lemma on actions of connected groups on ´etale sheaves.

Lemma

VI.7.3 . Let f : Y → X be a compactiﬁable cohomologically smooth map of locally spatialdiamonds with a section s : X → Y . Assume that all geometric ﬁbres of f are connected. Then forall A ∈ D ≥ ( X, Λ) , the map H ( Rf ∗ f ∗ A ) → H ( A ) given by evaluation at the section s is an isomorphism. Proof.

Note that Rf ∗ f ∗ A ∼ = R H om( Rf ! Rf ! Λ , A )where Rf ! Rf ! Λ sits in cohomological degrees ≤ H ∼ = Λ. Indeed, this reduces easily tothe case of discrete Λ, and then to Λ = F (cid:96) , and can be checked on geometric stalks. But if S = Spa( C, C + ) and i : { s } (cid:44) → S is the closed point, then i ∗ Rf ! Rf ! F (cid:96) ∈ D ( F (cid:96) ) with dual R Hom( i ∗ Rf ! Rf ! F (cid:96) , F (cid:96) ) ∼ = R Γ( Y, f ∗ i ∗ F (cid:96) )which sits in degrees ≥ F (cid:96) in degree 0, as the geometric ﬁbres are connected.Using the section, we get Rf ! Rf ! Λ ∼ = Λ ⊕ B for some B that sits in cohomological degrees ≤ − (cid:3) Unfortunately, it is a priori not easy to describe the category p D ≥ . It is however possible todescribe it via hyperbolic localization. This also implies that pullbacks in S are t -exact. Proposition

VI.7.4 . For any S (cid:48) → S → Div d Y , pullback along H ck G,S (cid:48) / Div d Y → H ck G,S/

Div d Y is t -exact for the perverse t -structure. Moreover, if G is split, then CT B : D ´et ( H ck G,S/

Div d Y , Λ) bd → D ´et (Gr T,S/

Div d Y , Λ) satisﬁes the following exactness property. There is the natural locally constant map Gr T, Div d Y → X ∗ ( T ) measuring the sum of relative positions, and by pairing with ρ , we get a locally constantmap deg : Gr T, Div d Y → Z . Then CT B [deg] is t -exact for the perverse t -structure on the source, andthe standard t -structure on the right. As CT B [deg] is conservative, this implies in particular that A ∈ p D ≤ ( H ck G,S/

Div d Y , Λ) bd (resp . A ∈ p D ≥ ( H ck G,S/

Div d Y , Λ) bd ) if and only if CT B ( A )[deg] ∈ D ≤ (Gr T,S/

Div d Y , Λ) (resp . CT B ( A )[deg] ∈ D ≥ (Gr T,S/

Div d Y , Λ)) . I.7. PERVERSE SHEAVES 213

Proof.

To prove t -exactness of pullbacks, we need to see that pullback commutes with t -truncations. By descent, it is enough to check that this holds v-locally on S ; this allows us toreduce to the case that G is split. It is then enough to prove t -exactness of CT B [deg], as byconservativity of CT B [deg] (see Proposition VI.4.2) this gives the characterization in terms of the t -structure in terms of CT B [deg], and the latter characterization is clearly preserved under pullback(as hyperbolic localization commutes with pullback, see Corollary VI.3.5).We can assume that S → Div d Y lifts to S → (Div Y ) d . We have a stratiﬁcation of S into ﬁnitelymany strata i a : S a (cid:44) → S , obtained by pulling back the partial diagonals of (Div Y ) d . Accordingly,we get triangles expressing A as a successive extension of i a ! i ∗ a A resp. Ri a ∗ Ri ! a A ; if A ∈ p D ≤ (resp. A ∈ p D ≥ ), then also all i a ! i ∗ a A ∈ p D ≤ (resp. Ri a ∗ Ri ! a A ∈ p D ≥ ). This allows us to reduceto the cases of i a ! i ∗ a A and Ri a ∗ Ri ! a A . As hyperbolic localization commutes with all functors byProposition IV.6.12, we can then reduce to the case that S = S a for some a . Reducing d ifnecessary, we can then assume that S maps into the locus of distinct untilts (Div Y ) d (cid:54) = ⊂ (Div Y ) d .There is then a stratiﬁcation in terms of open Schubert cells j µ • : H ck G,S/

Div d Y ,µ • (cid:44) → H ck G,S/

Div d Y parametrized by µ • = ( µ , . . . , µ d ), µ i ∈ X ∗ ( T ) + . Now A ∈ p D ≤ if and only if all j ∗ µ • A ∈ D ≤− d µ • for d µ • = (cid:80) di =1 (cid:104) ρ, µ i (cid:105) , and dually A ∈ p D ≥ if and only if all Rj ! µ • A ∈ D ≥− d µ • . Using excision triangles, we can then assume that A = j µ • ! A µ • , A µ • ∈ D ≤− d µ • ( H ck G,S/

Div d Y ,µ • , Λ)resp. A = Rj µ • ∗ A µ • , A µ • ∈ D ≥− d µ • ( H ck G,S/

Div d Y ,µ • , Λ) . Moreover, ﬁltering by cohomology sheaves, we can actually assume that A µ • is concentrated indegree − d µ • . Recall that [ µ • ] : S → H ck G,S/

Div d Y ,µ • is a v-cover, and the automorphism group of the stratum is an inverse limit of smooth and connectedgroups (as follows from Proposition VI.2.4 and the K¨unneth formula); this implies that for com-plexes concentrated in one degree, pullback under [ µ • ] ∗ is fully faithful, cf. Lemma VI.7.3. We canthus assume that A µ • comes via pullback from some B ∈ D ( S, Λ) concentrated in cohomologicaldegree − d µ • . Note that at this point, the desired statement (that CT B ( A )[deg] sits in the correctdegrees) can be checked after pullback along Spa( C, C + ) → S , so we can assume S = Spa( C, C + )is strictly local, and it is enough to check that CT B ( A )[deg] sits in the correct degrees in the ﬁbreover the closed point of S . This ﬁbre in turn depends only on the restriction of A to the ﬁbre overthe closed point of S , by Proposition IV.6.12. We can thus assume that B is in fact constant. Wecan assume Λ = Z /n Z for some n prime to p , and then by d´evissage that Λ = F (cid:96) for some (cid:96) (cid:54) = p .One can then further reduce to the case B = F (cid:96) [ d µ • ]. Also, by the K¨unneth formula, we can thenreduce to the case d = 1. Thus, ﬁnally A = j µ ! F (cid:96) [ (cid:104) ρ, µ (cid:105) ]

14 VI. GEOMETRIC SATAKE resp. A = Rj µ ∗ F (cid:96) [ (cid:104) ρ, µ (cid:105) ] , and we want to see that CT B ( A )[deg] ∈ D ≤ (resp. CT B ( A )[deg] ∈ D ≥ ). By Proposition IV.6.13,it suﬃces to handle the ﬁrst case. Note that A is now universally locally acyclic, and the claimcan be checked in the universal case S = Div Y . As Gr T, Div Y → Div Y is a disjoint union of X ∗ ( T )many copies of Div Y , and the image is universally locally acyclic, thus locally constant, it is in factenough to check the result after pullback to the special ﬁbre S = Spd F q → Div Y , whereGr G, Spd F q / Div Y ∼ = (Gr Witt G ) ♦ . Using [

Sch17a , Section 27], we can now translate all computations to the setting of schemes.Let λ ∈ X ∗ ( T ) be any element, giving rise to the semi-inﬁnite orbit S λ ⊂ Gr Witt G , i.e. S λ = Gr Witt B × Gr Witt T [ λ ] . By Corollary VI.3.8, the dimension of S λ ∩ Gr Witt

G,µ is bounded by (cid:104) ρ, λ + µ (cid:105) . The restriction ofCT B ( j µ ! F (cid:96) [ (cid:104) ρ, µ (cid:105) ])to [ λ ] ∈ Gr Witt T is given by R Γ c (( S λ ∩ Gr Witt

G,µ ) F q , F (cid:96) )[ (cid:104) ρ, µ (cid:105) ]and thus sits in degrees ≤ (cid:104) ρ, λ + µ (cid:105) − (cid:104) ρ, µ (cid:105) = (cid:104) ρ, λ (cid:105) , giving the desired bound. (cid:3) We note that if d = 1, G is split, and S = Spd k → Div Y for k = F q , then under the fullinclusion Perv( H ck G, Spd k/ Div Y , Λ) ⊂ D ´et (Gr G, Spd k/ Div Y , Λ) bd , the identiﬁcation Gr G, Spd k/ Div Y ∼ = (Gr Witt

G,k ) ♦ and the full embedding D ´et (Gr Witt

G,k , Λ) bd (cid:44) → D ´et (Gr G, Spd k/ Div Y , Λ) bd from [ Sch17a , Proposition 27.2], the categoryPerv( H ck G, Spd k/ Div Y , Λ)identiﬁes with the full subcategoryPerv L + G (Gr Witt

G,k , Λ) ⊂ D ´et (Gr Witt

G,k , Λ) bd of L + G -equivariant perverse sheaves on Gr Witt

G,k ; this was considered by Zhu [

Zhu17 ] and Yu [

Yu19 ].In particular, this discussion implies the following result that we will need later.

Proposition

VI.7.5 . Assume that G is split, so that for any µ ∈ X ∗ ( T ) + we have the inclusion j µ : H ck G, Div Y ,µ (cid:44) → H ck G, Div Y of the open Schubert cell, of dimension d µ = (cid:104) ρ, µ (cid:105) . Then p j µ ! Λ[ d µ ] = p H ( j µ ! Λ[ d µ ]) , p Rj µ ∗ Λ[ d µ ] = p H ( Rj µ ∗ Λ[ d µ ]) are universally locally acyclic, and their image under CT B [deg] is locally ﬁnite free over Λ . Theirformation commutes with any base change in Λ . The natural map p Rj µ ∗ Λ[ d µ ]( d µ ) → D ( p j µ ! Λ[ d µ ]) I.7. PERVERSE SHEAVES 215 is an isomorphism.Moreover, if Λ is a Z (cid:96) -algebra, then there is some integer a = a ( µ ) (independent of Λ ) suchthat the kernel and cokernel of the map p H ( j µ ! Λ[ d µ ]) → p H ( Rj µ ∗ Λ[ d µ ]) are killed by (cid:96) a . We remark that the ﬁnal statement ultimately makes use of the decomposition theorem (andthus requires the degeneration to the Witt vector aﬃne Grassmannian).

Proof.

Consider A = j µ ! Λ[ d µ ] ∈ p D ≤ , which is universally locally acyclic. Then CT B ( A )[deg]sits in degrees ≤

0, and is universally locally acyclic. Moreover, its degree 0 part is locally ﬁnite freeover Λ. Indeed, this can be computed in terms of the top compactly supported cohomology groupof the Mirkovi´c–Vilonen cycles S λ ∩ Gr G,µ , which (as for any separated variety) is ﬁnite free overΛ. As CT B ( A )[deg] is t -exact, this implies that A (cid:48) = p H ( A ) has the property that CT B ( A (cid:48) )[deg]is locally ﬁnite free over Λ. Applying Verdier duality and using Proposition IV.6.13, we see thatCT B ( D ( A ))[deg] ∼ = D ( w ∗ CT B ( A )[deg]) (where w is the longest Weyl group element), which thensits in cohomological degrees ≥

0, and is ﬁnite free in degree 0, with degree 0 parts also underVerdier duality. This shows that the natural map p Rj µ ∗ Λ[ d µ ]( d µ ) → D ( p j µ ! Λ[ d µ ])is an isomorphism. The proof also shows that the formation commutes with any base change in Λ.For the ﬁnal statement, we can ﬁrst of all reduce by universal local acyclicity and Corol-lary VI.6.7 to the same statement on Gr Witt

G,k . By base change, we can assume that Λ = Z /(cid:96) N Z ,and we can even formally pass to the inverse limit over N , and then invert (cid:96) ; it is thus enough toshow that on the perfectly projective scheme Gr Witt

G,k, ≤ µ , the map p j µ ! Q (cid:96) [ d µ ] → p Rj µ ∗ Q (cid:96) [ d µ ]is an isomorphism. This follows from [ Zhu17 , Lemma 2.1], cf. also [

Gai01 , Proposition 1], [

Lus83 ].Let us recall the argument. It is enough to prove injectivity, as then surjectivity follows by Poincar´eduality (as the two sheaves are Verdier dual, as we have already proved), using that we are workingwith ﬁeld coeﬃcients now. Let j µ ! ∗ Q (cid:96) [ d µ ] be the image of the displayed map (i.e., the intersectioncomplex of Gr Witt

G,k, ≤ µ ). It is enough to see that for i : Gr Witt

G,k,<µ (cid:44) → Gr Witt

G,k, ≤ µ the complementaryclosed, that i ∗ j µ ! ∗ Q (cid:96) [ d µ ] lies in p D ≤− . Indeed, we have a short exact sequence0 → i ∗ K → p j µ ! Q (cid:96) [ d µ ] → j µ ! ∗ Q (cid:96) [ d µ ] → K on Gr Witt

G,k,<µ ; but this gives a map i ∗ j µ ! ∗ Q (cid:96) [ d µ ] → K [1], so if i ∗ j µ ! ∗ Q (cid:96) [ d µ ] ∈ p D ≤− , then necessarily K = 0.To prove that i ∗ j µ ! ∗ Q (cid:96) [ d µ ] ∈ p D ≤− , it suﬃces to prove that all geometric ﬁbres of j µ ! ∗ Q (cid:96) [ d µ ]are concentrated in degrees of the same parity as d µ ; indeed, any other stratum in Gr G,k, ≤ µ hasdimension of the same parity as d µ , so the trivial bound i ∗ j µ ! ∗ Q (cid:96) [ d µ ] ∈ p D ≤− gets ampliﬁed byone on each stratum for parity reasons. This parity claim about the intersection complex can bechecked smooth locally. We have the smooth map F (cid:96) Witt

G,k → Gr Witt

G,k from the Witt vector aﬃneﬂag variety, and choosing the element w in the Iwahori-Weyl group corresponding to the generic

16 VI. GEOMETRIC SATAKE stratum on the preimage of the Schubert cell, we get the smooth map F (cid:96) Witt

G,k, ≤ w → Gr Witt

G,k, ≤ µ . Itis thus enough to prove the similar claim about the intersection complex of F (cid:96) Witt

G,k, ≤ w . Choosing areduced expression ˙ w = s · · · s r · ω as above, we get the Demazure-Bott-Samuelson resolution π ˙ w : Dem Witt˙ w → F (cid:96) Witt

G,k, ≤ w . This has the property that all geometric ﬁbres admit stratiﬁcations into aﬃne spaces, cf. [

Zhu17 ,Section 1.4.2]. In particular, all geometric ﬁbres of Rπ ˙ w ∗ Q (cid:96) sit only in even degrees. On the otherhand, by the decomposition theorem, the intersection complex is a direct summand of Rπ ˙ w ∗ Q (cid:96) [ d µ ],giving the claim. (cid:3) As a consequence of Proposition VI.7.4, we see, perhaps surprisingly, that containment in p D ≥ can be checked in geometric ﬁbres over S . (Note however that we are using a relative perverse t -structure.) This gives a complete justiﬁcation for calling it a relative perverse t -structure. Corollary

VI.7.6 . Let S → Div d Y be any small v-stack and let A ∈ D ´et ( H ck G,S/

Div d Y , Λ) bd . Then A ∈ p D ≥ if and only if this holds true after pullback to all strictly local Spa(

C, C + ) → S . Inparticular, A ∈ Perv( H ck G,S/

Div d Y , Λ) if and only if for all strictly local Spa(

C, C + ) → S , the pullback of A to H ck G, Spa(

C,C + ) / Div d Y isperverse. Also note that over geometric points, we are simply considering the usual perverse t -structurecorresponding to the stratiﬁcation in terms of open Schubert cells, and then p D ≥ admits its usualcharacterization in terms of !-restriction to the open Schubert cells. Proof.

It suﬃces to check after a cover, as pullback is t -exact. This allows us to reduceto the case that G is split. But then it follows from the condition in terms of the hyperboliclocalization. (cid:3) VI.7.1. The Satake category.

We also get the following characterization. The conditionasked here is stronger than perversity.

Proposition

VI.7.7 . Let S be any small v-stack over Div d Y and assume that G is split. Then A ∈ D ULA´et ( H ck G,S/

Div d Y , Λ) has the property that A is a ﬂat perverse sheaf (in the sense that A ⊗ L Λ M is perverse for all Λ -modules M ) if and only if Rπ T ∗ CT B ( A )[deg] ∈ D ´et ( S, Λ) is ´etale locally on S isomorphic to a ﬁnite projective Λ -module in degree . Proof.

The functor Rπ T ∗ CT B ( A )[deg] preserves universally locally acyclic sheaves and hencetakes values in sheaves that are locally constant with perfect ﬁbres. By Proposition VI.7.4 and asany bounded part of Gr T, Div d Y → Div d Y is ﬁnite over the base, the condition A ∈ p D ≤ is equivalent I.7. PERVERSE SHEAVES 217 to Rπ T ∗ CT B ( A )[deg] ∈ D ≤ . The ﬂatness then ensures that this is locally isomorphic to a perfectcomplex of Tor-amplitude [0 , (cid:3) In the following deﬁnition, S → Div d Y is any small v-stack, and G is general. Definition

VI.7.8 . Let

Sat( H ck G,S/

Div d Y , Λ) ⊂ D ´et ( H ck G,S/

Div d Y , Λ) be the full subcategory of all objects that are universally locally acyclic and ﬂat perverse. This deﬁnition has the virtue that it is invariant under switching sw ∗ . Let us give some examplesof objects in the Satake category, when d = 1. Assume for simplicity that G is split. For any µ ∈ X ∗ ( T ) + , we get the open Schubert cell j µ : H ck G, Div Y ,µ (cid:44) → H ck G, Div Y of dimension d µ = (cid:104) ρ, µ (cid:105) . The following proposition gives the analogue of highest weight modulesin the Satake category. Proposition

VI.7.9 . The perverse sheaves p j µ ! Λ[ d µ ] = p H ( j µ ! Λ[ d µ ]) , p Rj µ ∗ Λ[ d µ ] = p H ( Rj µ ∗ Λ[ d µ ]) lie in the Satake category Sat( H ck G, Div Y , Λ) . Proof.

This follows from Proposition VI.7.7 and Proposition VI.7.5. (cid:3)

Definition/Proposition

VI.7.10 . The functor Rπ G,S ∗ : Sat( H ck G,S/

Div d Y , Λ) → D ´et ( S, Λ) of pullback to Gr G,S/

Div d Y and pushforward along π G,S : Gr

G,S/

Div d Y → S takes values in complexes C ∈ D ´et ( S, Λ) such that all H i ( C ) are local systems of ﬁnite projective Λ -modules, and each functor H i ( Rπ G,S ∗ ) : Sat( H ck G,S/

Div d Y , Λ) → LocSys( S, Λ) . is exact.Let F G,S = (cid:77) i ∈ Z H i ( Rπ G,S ∗ ) : Sat( H ck G,S/

Div d Y , Λ) → LocSys( S, Λ) . The functor F G,S is exact, faithful, and conservative. Moreover, if f : A → B is a map in Sat( H ck G,S/

Div d Y , Λ) such that ker F G,S ( f ) is a direct summand of F G,S ( A ) , then f admits a kernel in Sat( H ck G,S/

Div d Y , Λ) ;similarly for cokernels. The ﬁnal statement in particular ensures the condition of “existence of coequalizers of F G,S -splitparallel pairs” appearing in the Barr–Beck theorem.

18 VI. GEOMETRIC SATAKE

Remark

VI.7.11 . It is not clear whether there are natural isomorphisms F G,S (sw ∗ A ) ∼ = F G,S ( A ),so this ﬁbre functor is (at least a priori) destroying part of the symmetry. What makes this questionslightly delicate is that it is asking for extra structure, and ideally one would like to produce thisstructure in a clean way; it is conceivable that one can reduce to geometric points and then use theaﬃrmative answers we give later under stronger assumptions on S . Proof.

Localize on S to reduce to the case that G is split, and ﬁx a Borel B ⊂ G withtorus T . Using the stratiﬁcation of Gr G into the strata S ν , we get a ﬁltration on Rπ G,S ∗ A whoseassociated graded is given by (cid:76) ν Rp ν ! A | S ν . Restricting to connected components of Gr G and for A in the Satake category, these are concentrated in degrees of the same parity, so the correspondingspectral sequence necessarily degenerates. Thus, most of this follows from Proposition VI.7.7 andProposition VI.4.2. Faithfulness of F G,S reduces to conservativity and the Barr–Beck type assertion,so it remains to prove the Barr–Beck type assertion. For this, consider the kernel of f in the categoryof all perverse sheaves on H ck G,S/

Div d Y . We need to see that this is still universally locally acyclic,and ﬂat perverse. These properties can be checked after applying hyperbolic localization, shiftby deg, and pushforward to S (using various t -exactness properties), where they follow from theassumption of being a direct summand. (cid:3) The Satake category also carries a Verdier duality functor. Again, it is not clear that thisfunctor commutes naturally with sw ∗ (we will settle it later under stronger assumptions on S ). Proposition

VI.7.12 . The image of the fully faithful functor

Sat( H ck G,S/

Div d Y , Λ) → D ´et (Gr G,S/

Div d Y , Λ) is stable under Verdier duality D Gr G,S/

Div d Y /S . The induced functor D : Sat( H ck G,S/

Div d Y , Λ) op → Sat( H ck G,S/

Div d Y , Λ) is an equivalence, with D = id . Moreover, it makes the diagram Sat( H ck G,S/

Div d Y , Λ) op D (cid:47) (cid:47) F G,S (cid:15) (cid:15)

Sat( H ck G,S/

Div d Y , Λ) F G,S (cid:15) (cid:15)

LocSys( S, Λ) op V (cid:55)→ V ∗ (cid:47) (cid:47) LocSys( S, Λ) commute naturally. Proof.

The Verdier dual D ( A ) ∈ D ´et (Gr G,S/

Div d Y , Λ) can actually be deﬁned already in D ´et ( H ck G,S/

Div d Y , Λ) bd by using Verdier duality along bounded subsets of H ck G → [ ∗ /L + G ]. It follows from Verdier du-ality that it commutes with the passage to cohomology, i.e. the functor F G,S , and from this onecan deduce that it is ﬂat perverse and hence lies in the Satake category. Biduality follows fromCorollary IV.2.25. (cid:3)

Moreover, the formation of the Satake category is compatible with constant term functors. Wedeﬁne a locally constant function deg P : Gr M, Div d Y → Z I.8. CONVOLUTION 219 as the composite of the projection to X ∗ ( M ) considered before and the map X ∗ ( M ) → Z given bypairing with 2 ρ G − ρ M . Proposition

VI.7.13 . Let P ⊂ G be a parabolic with Levi M . Let S → Div d Y be any smallv-stack. Consider the diagram Gr G,S/

Div d Y q S ←− Gr P,S/

Div d Y p S −→ Gr M,S/

Div d Y . Then the functor Rp S ! q ∗ S [deg P ] deﬁnes a functor CT P,S [deg P ] : Sat( H ck G,S/

Div d Y , Λ) → Sat( H ck M,S/

Div d Y , Λ) . These functors are compatible with composition, i.e. if P (cid:48) ⊂ P is a further parabolic with image Q ⊂ M and Levi M (cid:48) , then there is a natural equivalence CT P (cid:48) ,S [deg P (cid:48) ] ∼ = CT Q,S [deg Q ] ◦ CT P,S [deg P ] : Sat( H ck G,S/

Div d Y , Λ) → Sat( H ck M (cid:48) ,S/ Div d Y , Λ) (and for triple compositions, the obvious diagram commutes). Proof.

Let λ : G m → G be a cocharacter such that P = P λ . This induces in particular a Levisplitting M (cid:44) → P as the centralizer of λ . We can then divide the diagramGr G,S/

Div d Y q S ←− Gr P,S/

Div d Y p S −→ Gr M,S/

Div d Y . by L +Div d Y M to see that one can reﬁne Rp S ! q ∗ S into a functor D ´et ( H ck G,S/

Div d Y , Λ) → D ´et ( H ck M,S/

Div d Y , Λ)via ﬁrst pulling back to L +Div d Y M \ Gr G,S/

Div d Y . It is clear that these functors are compatible withcomposition.We want to see that the image is contained in the Satake category. First, by Proposition IV.6.14,we see that the image is universally locally acyclic. Now the claim follows from Proposition VI.7.7and the compatibility with composition (used for the Borel B ⊂ P ), after passing to an ´etale coverto assume that G is split. (cid:3) VI.8. Convolution

For any d and small v-stack S → Div d Y , the category D ´et ( H ck G,S/

Div d Y , Λ)is naturally a monoidal category. Indeed, with all loop groups taken over Div d Y , there is a convolutionmorphism H ck G, Div d Y × Div d Y H ck G, Div d Y a ←− L + G \ LG × L + G LG/L + G b −→ L + G \ LG/L + G = H ck G, Div d Y where the morphism a is an L + G -torsor, and the right morphism is ind-proper (its ﬁbres are theﬁbres of Gr G, Div d Y → Div d Y ). If one denotes by a S and b S the pullbacks along S → Div d Y , one canthen deﬁne the convolution product (cid:63) on D ´et ( H ck S/ Div d Y , Λ) bd20 VI. GEOMETRIC SATAKE via A (cid:63) A = Rb S ∗ a ∗ S ( A (cid:2) A ) for A , A ∈ D ´et ( H ck S/ Div d Y , Λ) bd . It is easy to see that this isassociative by writing out the corresponding convolution diagrams with multiple factors.In fact, modulo the problem that [Div d Y /L +Div d Y G ] → [Div d Y /L Div d Y G ] is not representable in lo-cally spatial diamonds (only ind-representable), the category D ´et ( H ck S/ Div d Y , Λ) bd is precisely thecategory of endomorphisms of [Div d Y /L +Div d Y G ] × Div d Y S in the 2-category C T for T = [Div d Y /L Div d Y G ] × Div d Y S . This problem is corrected by passing to bounded sheaves – one can extend the formalism to thecase of maps that are ind-representable in locally spatial diamonds, with closed immersions in theind-system, using categories of bounded sheaves as morphisms.Convolution interacts nicely with the classes of sheaves we have previously singled out. Inparticular, it preserves (ﬂat) perverse sheaves; this observation goes back to Lusztig [ Lus83 ]. Proposition

VI.8.1 . Let A , A ∈ D ´et ( H ck S/ Div d Y , Λ) bd . (i) If A and A are universally locally acyclic, then A (cid:63) A is universally locally acyclic. (ii) If A and A lie in p D ≤ , then A (cid:63) A ∈ p D ≤ . (iii) If A , A ∈ Sat( H ck G,S/

Div d Y , Λ) , then also A (cid:63) A ∈ Sat( H ck G,S/

Div d Y , Λ) . Proof.

Part (i) follows from Proposition IV.2.11 and Proposition IV.2.26. For part (ii), weﬁrst make some reductions. Namely, the claim can be checked if S = Spa( C, C + ) is strictly localand G split. Moreover, by a d´evissage one can assume that A and A are the !-extensions of theconstant sheaves on open Schubert cells; in particular, these are universally locally acyclic. By theK¨unneth formula one can then reduce to the case d = 1. In that case, we can pass to the universalcase S = Div Y . Over (Div Y ) , we can consider the moduli space (cid:103) H ck G, (Div Y ) of G -bundles E , E , E over B +(Div Y ) together with isomorphisms between E and E over B +(Div Y ) [ I ] and between E and E over B +(Div Y ) [ I ], where I , I ⊂ O Y S are the ideal sheaves parametrizing the two Cartierdivisors. Away from the diagonal, this is isomorphic to H ck G, (Div Y ) (cid:54) = / Div Y , while over the diagonalit is isomorphic to L +Div Y G \ L Div Y G × L +Div1 Y G L Div Y G/L +Div Y G. There are two natural projections p , p : (cid:103) H ck G, (Div Y ) → H ck G, Div Y keeping track of E and E resp. E and E , and a projection m : (cid:103) H ck G, (Div Y ) → H ck G, (Div Y ) / Div Y keeping track of E and E . One can the form B = Rm ∗ ( p ∗ A ⊗ L Λ p ∗ A ). Recall that we reducedto the case that A and A are moreover universally locally acyclic. By Proposition IV.2.11 andProposition IV.2.26, one sees that also B ∈ D ULA´et ( H ck G, (Div Y ) / Div Y , Λ) . Away from the diagonal, this is simply the exterior tensor product of A and A and in particularlies in p D ≤ . Looking at CT B ( B )[deg], we get a universally locally acyclic sheaf on (a bounded I.9. FUSION 221 subset of) Gr T, (Div Y ) / Div Y whose restriction away from the diagonal lies in degrees ≤

0. Thisimplies that the whole sheaf lies in degrees ≤

0: As any bounded subset of Gr T, (Div Y ) / Div Y is ﬁniteover (Div Y ) , it suﬃces to check this for the pushforward to (Div Y ) . But this pushforward islocally constant with perfect ﬁbres, and the complement of the diagonal is dense.Thus, using Proposition VI.7.4, the restriction of B to the diagonal lies in p D ≤ . But thisrestriction is precisely A (cid:63) A , giving the desired result. Finally, part (iii) easily follows from (i),(ii), and the observation that convolution commutes with Verdier duality. (cid:3) VI.8.1. Dualizability.

Next, we observe that all objects are dualizable.

Proposition

VI.8.2 . All objects of the monoidal category

Sat( H ck G,S/

Div d Y , Λ) are (left andright) dualizable. The right dual of A ∈ Sat( H ck G,S/

Div d Y , Λ) is given by sw ∗ D ( A ) where sw : H ck G, Div d Y ∼ = H ck G, Div d Y is the switching isomorphism (induced by inversion on L Div d Y G ). Remark

VI.8.3 . In the classical setting, this is asserted without indication of proof in [

MV07 ,end of Section 11].

Proof.

All objects of D ULA ( H ck G,S/

Div d Y , Λ) are left dualizable, with right dual given bysw ∗ D ( A ): This follows from Proposition IV.2.24 (modulo the technical nuisance that everything isonly ind-representable here; everything adapts to that setting). Here, sw simply arises by swappingsource and target. Also note that the condition of being universally locally acyclic is invariantunder sw ∗ by Proposition VI.6.2, so also using Proposition VI.7.12, the functor sw ∗ D ( A ) preservesthe Satake category. (cid:3) Remark

VI.8.4 . Again, we stress that all results above also hold if G is reductive over E , andwe replace Div d Y with Div dY or Div dX . Indeed, the case of Div dY follows from the case of Div d Y as itis an open subset, at least if G admits a reductive model over O E . In general, this happens ´etalelocally, making it possible to reduce to this case. Then the case of Div dX follows from the case ofDiv dY as any map S → Div dX can locally be lifted to a map S → Div dY in such a way that thecorresponding pullbacks of the local Hecke stacks are isomorphic. VI.9. Fusion

Now let G be a reductive group over E . From now on, we ﬁx the base ﬁeld k = F q and workon the category Perf k . For brevity, we deﬁne for any ﬁnite set I with d = | I | the local Hecke stack H ck IG = H ck G, Div dX × Div dX (Div X ) I and correspondingly Gr IG = Gr G, Div dX × Div dX (Div X ) I . Definition

VI.9.1 . For any ﬁnite set I , the Satake category Sat IG (Λ) is the category Sat( H ck IG , Λ) of all A ∈ D ´et ( H ck IG , Λ) that are universally locally acylic and ﬂat perverse over (Div X ) I .

22 VI. GEOMETRIC SATAKE

By Proposition VI.7.10, we get a (not yet monoidal) ﬁbre functor F I : Sat IG (Λ) → LocSys((Div X ) I , Λ) . The target category LocSys((Div X ) I , Λ) is in fact very explicit.

Proposition

VI.9.2 . The category

LocSys((Div X ) I , Λ) is naturally equivalent to the category Rep W IE (Λ) of continuous representations of W IE on ﬁnite projective Λ -modules. Proof.

This is a consequence of Proposition IV.7.3. (cid:3)

For any map f : I → J of ﬁnite sets, there is a natural monoidal functor Sat IG (Λ) → Sat JG (Λ).Indeed, there is a natural closed immersion H ck JG × (Div X ) J (Div X ) I (cid:44) → H ck IG , so pull-push along H ck IG ← H ck JG × (Div X ) J (Div X ) I → H ck JG deﬁnes the desired functor. It is easy to see that this functor is compatible with composition ofmaps of ﬁnite sets. Moreover, the functors Sat IG (Λ) → Sat JG (Λ) make the diagramSat IG (Λ) (cid:47) (cid:47) F I (cid:15) (cid:15) Sat JG (Λ) F J (cid:15) (cid:15) Rep W IE (Λ) (cid:47) (cid:47) Rep W JE (Λ)commute naturally, where the lower functor is pullback under W JE → W IE .Actually, the functor I (cid:55)→ Sat IG (Λ) has further functoriality, given by the fusion product.Namely, for ﬁnite sets I , . . . , I k with disjoint union I = I (cid:116) . . . (cid:116) I k , there is a natural monoidalfunctor Sat I G (Λ) × . . . × Sat I k G (Λ) → Sat IG (Λ) , functorial in I , . . . , I k and compatible with composition. To construct this, let j : (Div X ) I ; I ,...,I k ⊂ (Div X ) I be the open subset where x i (cid:54) = x i (cid:48) whenever i, i (cid:48) ∈ I = I (cid:116) . . . (cid:116) I k lie in diﬀerent I j ’s, and letSat I ; I ,...,I k G (Λ) ⊂ D ULA´et ( H ck IG × (Div X ) I (Div X ) I ; I ,...,I k , Λ)be deﬁned similarly as Sat IG (Λ). Proposition

VI.9.3 . The restriction functor j ∗ : Sat IG (Λ) → Sat I ; I ,...,I k G (Λ) is fully faithful. Similarly, j ∗ : LocSys((Div X ) I , Λ) → LocSys((Div X ) I ; I ,...,I k , Λ) is fully faithful. I.9. FUSION 223

Proof.

For the ﬁrst part, it suﬃces to prove that for all A ∈ Sat IG (Λ), the natural map A → p H ( Rj ∗ j ∗ A )is an isomorphism. Let i : Z (cid:44) → (Div X ) I be the complementary closed. It suﬃces to see that i ∗ i ! A ∈ p D ≥ . Working locally to reduce to the case G split, and applying the t -exact hyperboliclocalization functor Rπ T ∗ CT B [deg], taking values in local systems of ﬁnite projective Λ-moduleson Sat IG (Λ), this follows from the observation that i ∗ i ! Λ ∈ D ≥ , which follows from the observationthat Z admits a stratiﬁcation (by partial diagonals) with smooth strata of (cid:96) -codimension ≥ X ) I .This ﬁnal argument in fact proves directly the second part. (cid:3) On the other hand, over (Div X ) I ; I ,...,I k , one has H ck IG × (Div X ) I (Div X ) I ; I ,...,I k ∼ = k (cid:89) j =1 H ck I j G × (cid:81) j (Div X ) Ij (Div X ) I ; I ,...,I k , so there is a natural monoidal functorSat I G (Λ) × . . . × Sat I k G (Λ) → Sat I ; I ,...,I k G (Λ)given by exterior product. Actually, recall that when forming symmetric monoidal tensor products,there are implicit sign rules when commuting factors. We change these here by hand. Namely, notethat each H ck IG = ( H ck IG ) even (cid:116) ( H ck IG ) odd decomposes into open and closed subsets given by the even and the odd part; the even partcontains those Schubert varieties for which d µ • = (cid:80) i (cid:104) ρ, µ i (cid:105) is even, while the odd part containsthose for which d µ • is odd. Note that the dominance order can only nontrivially compare elementswith the same parity, so these are really open and closed subsets. Also note that it follows fromProposition VI.7.4 that for sheaves concentrated on the even (resp. odd) part, the functor F I isconcentrated in even (resp. odd) degrees. Now we impose that when forming the above exteriorproduct, we introduce a minus sign whenever we commute two sheaves concentrated on the oddparts. A diﬀerent way to say it is that there is a natural commutative diagramSat I G (Λ) × . . . × Sat I k G (Λ) (cid:47) (cid:47) ( F I ,...,F Ik ) (cid:15) (cid:15) Sat I ; I ,...,I k G (Λ) F I ; I ,...,Ik (cid:15) (cid:15) LocSys((Div X ) I , Λ) × . . . × LocSys((Div X ) I k , Λ) (cid:2) (cid:47) (cid:47) LocSys((Div X ) I ; I ,...,I k , Λ)functorial in I , . . . , I k , and under permutations of the sets I , . . . , I k . Indeed, note that the functors F I invoke a shift by deg, which exactly introduces this sign rule. This in fact pins down this choiceof signs by faithfulness of the functors. Definition/Proposition

VI.9.4 . The image of

Sat I G (Λ) × . . . × Sat I k G (Λ) → Sat I ; I ,...,I k G (Λ) lands in Sat IG (Λ) ⊂ Sat I ; I ,...,I k G (Λ) , deﬁning the fusion product ∗ : Sat I G (Λ) × . . . × Sat I k G (Λ) → Sat IG (Λ) ,

24 VI. GEOMETRIC SATAKE a functor of monoidal categories, functorial in I , . . . , I k . It makes the diagram Sat I G (Λ) × . . . × Sat I k G (Λ) ∗ (cid:47) (cid:47) ( F I ,...,F Ik ) (cid:15) (cid:15) Sat IG (Λ) F I (cid:15) (cid:15) LocSys((Div X ) I , Λ) × . . . × LocSys((Div X ) I k , Λ) (cid:2) (cid:47) (cid:47) LocSys((Div X ) I , Λ) commute functorially in I , . . . , I k and permutations of I , . . . , I k . Proof.

We can deﬁne a convolution local Hecke stack H ck I ; I ,...,I k G → (Div X ) I as follows. It parametrizes G -bundles E , . . . , E k over B +(Div X ) I together with isomorphisms of E j − and E j after inverting I i for all i ∈ I j , for j = 1 , . . . , k . Here I i ⊂ O X S is the ideal deﬁning the i -thCartier divisor. There are natural projections p j : H ck I ; I ,...,I k G → H ck I j G , j = 1 , . . . , k remembering E j − and E j , and m : H ck I ; I ,...,I k G → H ck IG remembering E and E k . Given A j ∈ Sat I j G (Λ), one can then deﬁne B = Rm ∗ ( p ∗ A ⊗ L Λ . . . ⊗ L Λ p ∗ k A k ) ∈ D ´et ( H ck IG , Λ) . This is still universally locally acyclic, by Proposition IV.2.11 and Proposition IV.2.26. Afterpullback to (Div X ) I ; I ,...,I k , the map m is an isomorphism, and we simply get the exterior productof all A i . Moreover, working locally to reduce to the case G is split, we see that Rπ T ∗ CT B ( B )[deg]is a local system of ﬁnite projective Λ-modules, as it is locally constant with perfect ﬁbres, and overthe dense open subset (Div X ) I ; I ,...,I k , the perfect complex is a ﬁnite projective module in degree0. This means that B ∈ Sat IG (Λ), as desired. (cid:3) In particular, for any ﬁnite set I , this structure makes Sat IG (Λ) into an E ∞ -monoid object inmonoidal categories, functorially in I , by using the compositeSat IG (Λ) × . . . × Sat IG (Λ) → Sat I (cid:116) ... (cid:116) IG (Λ) → Sat IG (Λ) , using the functor corresponding to the natural map I (cid:116) . . . (cid:116) I → I . Recall that E ∞ -monoidstructures on monoidal categories are the same as symmetric monoidal category structures reﬁningthe given monoidal category structure. Thus, each Sat IG (Λ) has become naturally a symmetricmonoidal category with the fusion product, reﬁning the monoidal convolution product; and every-thing is functorial in I . Moreover, by the ﬁnal part of Deﬁnition/Proposition VI.9.4, the functor F I : Sat IG (Λ) → LocSys((Div X ) I , Λ) ∼ = Rep W IE (Λ)is a symmetric monoidal functor, functorially in I .A consequence of these symmetric monoidal structures are the following natural isomorphisms. I.9. FUSION 225

Corollary

VI.9.5 . For A ∈ Sat IG (Λ) , there are natural isomorphisms F I (sw ∗ A ) ∼ = F I ( A ) , D (sw ∗ A ) ∼ = sw ∗ D ( A ) . Moreover, D is naturally a symmetric monoidal functor, and D ◦ F I ∼ = ( F I ) ∗ as symmetric monoidalfunctors. Proof.

By Proposition VI.8.2, all A ∈ Sat IG (Λ) are dualizable, with dual sw ∗ D ( A ). In asymmetric monoidal category, this means that there are natural isomorphismssw ∗ D sw ∗ D ( A ) ∼ = A. As both sw ∗ and D are self-inverse, this amounts to the commutation of D and sw ∗ .Also, as F I is symmetric monoidal, it follows that F I (sw ∗ D ( A )) and F I ( A ) are naturally dual.But by Proposition VI.7.12, the dual of F I ( A ) is also F I ( D ( A )). Replacing D ( A ) by A , we ﬁnd anatural isomorphism F I (sw ∗ A ) ∼ = F I ( A ).Finally, it is easy to see that the whole construction of the fusion product is compatible withVerdier duality, making Verdier duality a symmetric monoidal functor, compatibly with the ﬁbrefunctor. (cid:3) Moreover, the constant term functors are compatible with the fusion product. More precisely,given a parabolic P ⊂ G with Levi M , we have the constant term functorsCT IP [deg P ] : Sat IG (Λ) → Sat IM (Λ) . Proposition

VI.9.6 . For any ﬁnite set I decomposed into ﬁnite sets I = I (cid:116) . . . (cid:116) I k , thediagram Sat I G (Λ) × . . . × Sat I k G (Λ) (cid:47) (cid:47) (CT I P [deg P ] ,..., CT IkP [deg P ]) (cid:15) (cid:15) Sat IG (Λ) CT IP [deg P ] (cid:15) (cid:15) Sat I M (Λ) × . . . × Sat I k M (Λ) (cid:47) (cid:47) Sat IM (Λ) commutes functorially in I and permutations of I , . . . , I k . Proof.

After passing to the open subset (Div X ) I ; I ,...,I k , this follows from the K¨unneth formula,so Proposition VI.9.3 gives the result. (cid:3) In particular, the functor CT IP [deg P ] : Sat IG (Λ) → Sat IM (Λ)is naturally symmetric monoidal with respect to the fusion product. Moreover, everything is com-patible with composition, for another parabolic P (cid:48) ⊂ P .

26 VI. GEOMETRIC SATAKE

VI.10. Tannakian reconstruction

Our next goal is to construct a group scheme whose category of representations recovers thesymmetric monoidal category Sat IG (Λ). More precisely, we want to use some relative Tannakaduality over Rep W IE (Λ). To achieve this, we need the following proposition. Given any ﬁniteand Galois-stable subsets W i ⊂ X ∗ ( T ) + , i ∈ I , closed under the dominance order, we have aquasicompact closed substack H ck IG, ( W i ) i ⊂ H ck IG and we get a corresponding full subcategorySat IG, ( W i ) i (Λ) ⊂ Sat IG (Λ) . Proposition

VI.10.1 . The functor F I : Sat IG, ( W i ) i (Λ) → Rep W IE (Λ) admits a left adjoint L I ( W i ) i , satisfying the following properties. (i) There is a natural isomorphism L I ( W i ) i ( V ) ∼ = L I ( W i ) i (1) ⊗ V, V ∈ Rep W IE (Λ) , where ∈ Rep W IE (Λ) is the tensor unit, and we use that Sat IG (Λ) is tensored over Rep W IE (Λ) . (ii) There is a natural isomorphism L I ( W i ) i (1) ∼ = ∗ i ∈ I L { i } W i (1) as the fusion of L { i } W i (1) ∈ Sat { i } G,W i (Λ) . (iii) If I = { i } has one element and W = W i , then the left adjoint is the restriction of the leftadjoint to F (cid:48) = (cid:77) m H m ( Rπ G ∗ ) : Perv( H ck { i } G,W , Λ) → Mod W E (Λ) . Proof.

It is enough to ﬁnd the value L I ( W i ) i (1) satisfying (ii) and (iii). Indeed, then theformula in (i) deﬁnes the left adjoint in general. Assume now that (iii) holds, and let us denote P W i = L { i } W i (1). Then for part (ii) we ﬁrst observe that F (cid:48) I = (cid:77) m H m ( Rπ G ∗ ) : Perv( H ck IG, ( W i ) i , Λ) → Shv ´et ((Div X ) I , Λ)admits a left adjoint, and this left adjoint evaluated on the unit, P ( W i ) i , is generically on (Div X ) I given by an exterior tensor product of the corresponding left adjoints for I being a singleton.Indeed, note that there is a natural map P ( W i ) i → ∗ i ∈ I P W i adjoint to the section of F (cid:48) I ( ∗ i ∈ I P W i ) ∼ = (cid:2) i ∈ I F (cid:48) ( P W i )given by the exterior tensor product of the classes given by (iii). To check that this is an isomorphismgenerically, we can by ´etale descent reduce to the case that G is split. In that case, one can make I.10. TANNAKIAN RECONSTRUCTION 227 the left adjoint explicit in terms of the left adjoint to hyperbolic localization. Writing hyperboliclocalization as a composite of !-pullback and ∗ -pushforward, this left adjoint is then given in termsof ∗ -pullback and !-pushforward, and the perverse p H . As generically, everything decomposesgeometrically into a product, it follows from the K¨unneth formula that the left adjoint commuteswith exterior products. But as any B ∈ Sat

IG, ( W i ) i (Λ) is equal to p H ( Rj ∗ j ∗ B ) as in the discussionof the fusion product, we see that F (cid:48) I ( B ) ∼ = Hom( P ( W i ) i , B ) ∼ = Hom( P ( W i ) i , p H ( Rj ∗ j ∗ B )) ∼ = Hom( P ( W i ) i , Rj ∗ j ∗ B ) ∼ = Hom( j ∗ P ( W i ) i , j ∗ B ) ∼ = Hom( j ∗ ∗ i ∈ I P W i , j ∗ B ) ∼ = Hom( ∗ i ∈ I P W i , p H ( Rj ∗ j ∗ B )) ∼ = Hom( ∗ i ∈ I P W i , B ) . It remains to prove part (iii). We can assume that Λ = Z /(cid:96) c Z , using base change. Note ﬁrstthat F (cid:48) = (cid:77) m H m ( Rπ G ∗ ) : Perv( H ck { i } G,W , Λ) → Mod W E (Λ)admits a left adjoint L (cid:48) W , by the adjoint functor theorem. We need to see that when evaluatedat the unit, P W := L (cid:48) W (1) is universally locally acyclic, and ﬂat perverse. By the characterizationof these properties, it suﬃces to show that F (cid:48) ( P W ) ∈ Mod W E (Λ) is a representation on a ﬁniteprojective Λ-module. This does not depend on the W E -action, so we can check these things afterpullback along Spd C → Div X , where C is a completed algebraic closure of E . In particular, wecan assume that G is split. For any open Schubert cell j µ : H ck G, Spd

C,µ (cid:44) → H ck G, Spd

C,W for µ ∈ W , of dimension d µ = (cid:104) ρ, µ (cid:105) , we can computeHom( P W , p Rj µ ∗ Λ[ d µ ]) = F (cid:48) ( p Rj µ ∗ Λ[ d µ ]) . By Proposition VI.7.9, this is a ﬁnite free Λ-module. Using adjunction, we thus see thatHom( p j ∗ µ P W , Λ[ d µ ])is a ﬁnite free Λ-module for all µ ∈ W . Now p j ∗ µ P W is concentrated on an open Schubert cell H ck G, Spd

C,µ , which is covered by Spd C , and concentrated in degree − d µ . It is thus given by theconstant sheaf M [ d µ ] for some Λ-module M , and we know that Hom( M, Λ) is ﬁnite free over Λ.As we reduced to Λ = Z /(cid:96) m Z , this implies that M is free.Now argue by induction on W , and take a maximal element µ ∈ W ; let W = W \ { µ } . We getan exact sequence 0 → K → p j µ ! j ∗ µ P W → P W → Q → H ck G, Spd C , Λ) supported on W . In fact, we necessarily have Q = P W (as they represent thesame functor), for which we know by induction that F (cid:48) ( Q ) is a ﬁnite free Λ-module. We claim that K = 0. As K lies in the kernel of p j µ ! j ∗ µ P W → p Rj µ ∗ j ∗ µ P W , it follows from Proposition VI.7.5 that (cid:96) a K = 0 for some a independent of Λ. Using functoriality of the construction for Λ (cid:48) = Z /(cid:96) a + c Z → Λ = Z /(cid:96) c Z and that p j µ ! j ∗ µ P W lies in the Satake category (so in particular it is ﬂat over Λ), we seethe image of K (cid:48) in K is equal to 0. On the other hand, as all constructions are compatible with base

28 VI. GEOMETRIC SATAKE change, the map K (cid:48) → K had to be surjective. It follows that K = 0, as desired. (Alternatively,we could have reduced to Z (cid:96) -coeﬃcients, in which case p j µ ! j ∗ µ P W → p Rj µ ∗ j ∗ µ P W is injective (as thekernel is both (cid:96) -torsion free and killed by (cid:96) a ), implying K = 0 directly.) (cid:3) Now we use the following general Tannakian reconstruction result. This is essentially an ax-iomatization of [

MV07 , Proposition 11.1]. Recall that a symmetric monoidal category is rigid ifall of its objects are dualizable.

Proposition

VI.10.2 . Let A be a rigid symmetric monoidal category, and let C be a symmetricmonoidal category with a tensor action of A . Moreover, let F : C → A be a symmetric monoidal A -linear conservative functor, such that C admits and F reﬂects coequaliz-ers of F -split parallel pairs. Assume that C can be written as a ﬁltered union of full subcategories C i ,stable under coequalizers of F -split parallel pairs and the A -action, such that F | C i is representableby some X i ∈ C .Then H = lim −→ i F ( X i ) ∨ ∈ Ind( A ) admits a natural structure as a bialgebra (with commutative multiplication and associative comul-tiplication), and C is naturally equivalent to the symmetric monoidal category of representations of H in A . If C is rigid, then H admits an inverse, i.e. is a Hopf algebra. Here, the symmetric monoidal category of representations of H is the category of comodules over H as a coalgebra, endowed with the symmetric monoidal structure coming from the commutativemultiplication on H . Proof.

Consider F i = F | C i : C i → A . This admits the left adjoint A (cid:55)→ A ⊗ X i , asHom C i ( A ⊗ X i , Y ) ∼ = Hom C i ( X i , A ∨ ⊗ Y ) ∼ = F ( A ∨ ⊗ Y ) ∼ = A ∨ ⊗ F ( Y ) ∼ = Hom( A, F ( Y )) . By the Barr–Beck monadicity theorem, it follows that C i is equivalent to the category of modulesover the monad A (cid:55)→ F ( A ⊗ X i ) ∼ = A ⊗ F ( X i ) . Note that the monad structure here is equivalently turning F ( X i ) into an associative algebra A i ∈ A , and its category of modules is the category of modules over A i . Passing to duals, we notethat F ( X i ) ∨ is a coalgebra, and its category of comodules is equivalent to the category of modulesover A i , i.e. to C i . Now we can take a colimit over i and see that H = lim −→ i F ( X i ) ∨ is naturally a coalgebra whose category of comodules in A is equivalent to C . The functor is thefollowing: Any X ∈ C deﬁnes the object F ( X ) ∈ A and for any i large enough so that X ∈ C i amap F ( X ) ⊗ X i → X (by adjunction), thus a map F ( X ) ⊗ F ( X i ) ∼ = F ( F ( X ) ⊗ X i ) → F ( X ) , and hence dually we get the map F ( X ) → F ( X ) ⊗ F ( X i ) ∨ → F ( X ) ⊗ H . I.10. TANNAKIAN RECONSTRUCTION 229

Moreover, for any i , j there is some k such that C i ⊗ C j ⊂ C k :indeed, C i (resp. C j ) is generated by X i (resp. X j ) under tensors with A and coequalizers of F -splitparallel pairs, so C i ⊗ C j is generated by X i ⊗ X j under these operations. Thus, for any k such that X i ⊗ X j ∈ C k , we actually have C i ⊗ C j ⊂ C k . Let X k ∈ C k represent F | C k ; then we have a naturalmap X k → X i ⊗ X j . Indeed, this is adjoint to a map 1 → F ( X i ⊗ X j ) = F ( X i ) ⊗ F ( X j ), for which we use the tensorproduct of the unit maps 1 → F ( X i ), 1 → F ( X j ). This means that there is a natural map H ⊗ H = lim −→ i,j F ( X i ) ∨ ⊗ F ( X j ) ∨ ∼ = lim −→ i,j F ( X i ⊗ X j ) ∨ → lim −→ k F ( X k ) ∨ = H , which turns H into a commutative algebra, where the unit is induced by the maps X i → F (1) ∈ A (inducing maps 1 = F (1) → F ( X i ) ∨ ).It is a matter of unraveling deﬁnitions that this makes H into a Hopf algebra whose symmetricmonoidal category of representations in A is exactly C . If C is rigid, one also sees that A admitsan inverse. Indeed, one can write F ( X i ) ∨ ∼ = F ( X ∨ i ) ∼ = H om( X i , X ∨ i ) ∼ = H om( X i ⊗ X i , A . Here H om ∈ A denotesthe internal Hom over A . (cid:3) We can apply Proposition VI.10.2 to Sat IG (Λ) to get Hopf algebras H IG (Λ) ∈ Ind(Rep W IE (Λ)) . Proposition

VI.10.3 . The exterior tensor product (cid:2) i ∈ I : (cid:89) i ∈ I Sat { i } G (Λ) → Sat IG (Λ) induces an isomorphism (cid:79) i ∈ I H { i } G (Λ) ∼ = H IG (Λ) . Proof.

This is a consequence of the construction of the Hopf algebras together with Proposi-tion VI.10.1 (ii), noting that F I ( ∗ i ∈ I A i ) ∼ = (cid:79) i ∈ I F ( A i ) . (cid:3) We see that all information about the categories Sat IG (Λ) is in the Hopf algebra H G (Λ) = H {∗} G (Λ) ∈ Ind(Rep W E (Λ)) . Note also that the construction of H G (Λ) is compatible with base change in Λ, so it is enough toconsider the case Λ = Z /n Z with n prime to p . In fact, note that we can formally take the inverselimit over n to deﬁne Sat G ( (cid:98) Z p ) = lim ←− n Sat G ( Z /n Z )

30 VI. GEOMETRIC SATAKE with a ﬁbre functor into Rep cont W E ( (cid:98) Z p ) = lim ←− n Rep W E ( Z /n Z ) , the category of continuous representations of W E on ﬁnite free (cid:98) Z p = lim ←− n Z /n Z -modules, yieldinga Hopf algebra H G ∈ Ind(Rep cont W E ( (cid:98) Z p )) . This can equivalently be thought of as an aﬃne group scheme G (cid:86) over (cid:98) Z p , with an action of W E that is in a suitable sense continuous. VI.11. Identiﬁcation of the dual group

Our goal is to identify G (cid:86) with the Langlands dual group of G . Recall that the universal Cartanof G deﬁnes a cocharacter group X ∗ as an ´etale sheaf on Spec( E ), i.e. equivalently a ﬁnite freeabelian group X ∗ together with an action of the absolute Galois group of E , and in particular of W E . It comes with the W E -stable set of coroots Φ ∨ ⊂ X ∗ , and the subset of positive roots Φ ∨ + .Dually, we have the cocharacters X ∗ and dominant Weyl chamber ( X ∗ ) + ⊂ X ∗ , and the rootsΦ ⊂ X ∗ , containing the positive roots Φ + ⊂ Φ. These data give rise to a pinned Chevalley groupscheme (cid:98) G over (cid:98) Z p (or already over Z , but we will only consider it over (cid:98) Z p ) corresponding to thedual root data ( X ∗ , Φ , X ∗ , Φ ∨ ). Being pinned, there are distinguished torus and Borel (cid:98) T ⊂ (cid:98) B ⊂ (cid:98) G ,isomorphisms X ∗ ( (cid:98) T ) ∼ = X ∗ under which the positive coroots Φ ∨ + correspond to the weights of (cid:98) T onLie (cid:98) B/ Lie (cid:98) T , so Lie (cid:98) B/ Lie (cid:98) T = (cid:77) a ∈ Φ ∨ + Lie (cid:98) U a for root subgroups (cid:98) U a ⊂ (cid:98) B . Moreover, one has ﬁxed pinnings ψ a : Lie (cid:98) U a ∼ = (cid:98) Z p for all simple roots a . We want to endow (cid:98) G with a W E -action. We already have the W E -action on( X ∗ , Φ , X ∗ , Φ ∨ ), but we need to twist the action on the pinning. More precisely, let us write thepinning instead with a Tate twist as ψ a : Lie (cid:98) U a ∼ = (cid:98) Z p (1) . Then W E acts naturally on the pinning as well, and thereby induces an action of W E on (cid:98) G .We aim to prove the following theorem. Recall that we write G (cid:86) for the Tannaka group arisingfrom the Satake category. Generally, we will denote by − (cid:86) various objects deﬁned via the Satakecategory, while by (cid:98) − we will denote objects formally deﬁned as Langlands duals. Theorem

VI.11.1 . There is a canonical W E -equivariant isomorphism G (cid:86) ∼ = (cid:98) G . We note that the formulation of this theorem is slightly more precise than the formulation in[

MV07 ], where no canonical isomorphism is given. Also, we handle the case of non-split groups.Note that in particular, G (cid:86) only depends on G up to inner automorphisms; this is not clear. I.11. IDENTIFICATION OF THE DUAL GROUP 231

To prove the theorem, we can work over Z (cid:96) for some (cid:96) (cid:54) = p : Indeed, the statement of thetheorem is equivalent to having isomorphisms over Z /n Z for all n prime to p (by the Tannakianperspective), so the reduction follows from the Chinese remainder theorem.We will now ﬁrst prove the theorem when the group G is split; more precisely, if we have ﬁxed asplit torus and Borel T ⊂ B ⊂ G and trivializations of all simple root groups U a ⊂ B . Afterwards,we will verify that the isomorphism does not depend on this pinning (essentially, as pinnings varyalgebraically, while automorphisms of (cid:98) G/ Z (cid:96) form an (cid:96) -adic group), and ﬁnally use Galois descentto deduce the result in general.Note ﬁrst that if G = T is a torus, then Gr T, Div X ∼ = X ∗ ( T ) × Div X , and it is clear that Sat T is just the category of X ∗ ( T )-graded objects in Rep cont W E ( Z (cid:96) ). This implies that T (cid:86) ∼ = (cid:98) T is the dualtorus with X ∗ ( (cid:98) T ) = X ∗ ( T ).We have the symmetric monoidal constant term functorCT B [deg] : Sat G → Sat T , and it commutes with the ﬁbre functors by the identity (cid:76) i H i ( Rπ G ∗ ) ∼ = H ( Rπ T ∗ CT B [deg]). Thisgives rise to a W E -equivariant map (cid:98) T = T (cid:86) → G (cid:86) . Using the objects A µ = p j µ ! Z (cid:96) , whose µ -weightspace is 1-dimensional, we see that the map T (cid:86) → G (cid:86) must be a closed immersion.We have the following information about the generic ﬁbre G (cid:86) Q (cid:96) , following [ MV07 , Section 7].First, it follows from Proposition VI.7.5 that its category of representations Sat G ( Q (cid:96) ) is given bySat G ( Q (cid:96) ) ∼ = (cid:77) µ Rep cont W E ( Q (cid:96) ) ⊗ A µ . (Here Sat G ( Q (cid:96) ) = Sat G ( Z (cid:96) )[ (cid:96) ] , where Sat G ( Z (cid:96) ) = lim ←− m Sat G ( Z /(cid:96) m Z ).) The category of representations of G (cid:86) Q (cid:96) as an abstract groupscheme is then given bySat G ( Q (cid:96) ) ⊗ Rep cont WE ( Q (cid:96) ) Vect( Q (cid:96) ) ∼ = (cid:77) µ Vect( Q (cid:96) ) ⊗ A µ , and in particular is semisimple. As A µ ∗ A µ (cid:48) contains A µ + µ (cid:48) as a direct summand and X + ∗ is ﬁnitelygenerated as a monoid, we see that Sat G ( Q (cid:96) ) has a ﬁnite number of tensor generators. This impliesthat G (cid:86) Q (cid:96) is of ﬁnite type by [ DM82 , Proposition 2.20]. Moreover, it is connected as Sat G ( Q (cid:96) )does not have nontrivial ﬁnite tensor subcategories (as for any A µ with µ (cid:54) = 0, the tensor categorygenerated by A µ contains all A nµ ), cf. [ DM82 , Corollary 2.22]. As Sat G ( Q (cid:96) ) is semisimple, we evenknow that G (cid:86) Q (cid:96) is reductive by [ DM82 , Proposition 2.23]. For any simple object A µ , the weightsof A µ on (cid:98) T Q (cid:96) → G (cid:86) Q (cid:96) are contained in the set of all λ ∈ X ∗ = X ∗ ( (cid:98) T ) such that the dominantrepresentative of λ is bounded by µ in the dominance order, and contains µ (with weight 1). Thisimplies that T (cid:86) Q (cid:96) → G (cid:86) Q (cid:96) is a maximal torus of G (cid:86) Q (cid:96) . We can also deﬁne a subgroup B (cid:86) ⊂ G (cid:86) as thestabilizer of the ﬁltration associated to the cohomological grading of F (stabilizing the ﬁltration (cid:76) m ≤ i R m π G ∗ on the ﬁbre functor F = (cid:76) m R m π G ∗ ); then B (cid:86) Q (cid:96) ⊂ G (cid:86) Q (cid:96) is a Borel.

32 VI. GEOMETRIC SATAKE

Now we analyze the case G = PGL . In that case, we have the minuscule cocharacter µ : G m → G giving rise to the minuscule Schubert cell Gr G, Div X ,µ ∼ = P X . Then F ( A µ ) = H ( P ) ⊕ H ( P ) = Z (cid:96) ⊕ Z (cid:96) ( − W E -representation. This is a representation of G (cid:86) , giving a natural map G (cid:86) → GL( Z (cid:96) ⊕ Z (cid:96) ( − Z (cid:96) ⊕ Z (cid:96) ( − T (cid:86) actson Z (cid:96) ⊕ Z (cid:96) ( −

1) with weight ±

1, and in particular lands inside SL( Z (cid:96) ⊕ Z (cid:96) ( − G (cid:86) Q (cid:96) is reductiveof rank 1, it necessarily follows that G (cid:86) Q (cid:96) → SL( Q (cid:96) ⊕ Q (cid:96) ( − G (cid:86) → SL( Z (cid:96) ⊕ Z (cid:96) ( − G (cid:86) F (cid:96) → SL( F (cid:96) ⊕ F (cid:96) ( − H ⊂ SL( F (cid:96) ⊕ F (cid:96) ( − G (cid:86) F (cid:96) . Notethat the irreducible representations of G (cid:86) F (cid:96) (as an abstract group) are in bijection with dominantcocharacters, corresponding to the simple objects B µ = j µ ! ∗ F (cid:96) on Gr G, Spd C ; each B µ has a highestweight vector given by weight µ . It follows that H satisﬁes the hypothesis of the next lemma. Lemma

VI.11.2 . Let H be a closed subgroup of SL / F (cid:96) containing the diagonal torus such thatits set of irreducible representations injects into Z ≥ via consideration of highest weight vectors.Then H = SL . Proof.

Using a power of Frobenius, one can assume that H is reduced, and thus smooth. By[ DM82 , Corollary 2.22] and consideration of highest weight vectors, one also sees that H mustbe connected. Then H is either the torus, or a Borel, or SL . The ﬁrst cases lead to too manyirreducible representations. (cid:3) Thus, the map G (cid:86) F (cid:96) → SL( F (cid:96) ⊕ F (cid:96) ( − G (cid:86) → SL( Z (cid:96) ⊕ Z (cid:96) ( − Lemma

VI.11.3 . Let f : M → N be a map of ﬂat Z (cid:96) -modules such that M/(cid:96) → N/(cid:96) is injectiveand M [ (cid:96) ] → N [ (cid:96) ] is an isomorphism. Then f is an isomorphism. Proof. As M is ﬂat, f : M → N is injective; moreover, for any x ∈ N there is some minimal k such that (cid:96) k n = f ( m ) lies in the image of M . But if k >

0, then m lies in the kernel of M/(cid:96) → N/(cid:96) ,a contradiction. (cid:3)

The subgroup B (cid:86) ⊂ G (cid:86) is then given by the Borel stabilizing the line Z (cid:96) ⊂ Z (cid:96) ⊕ Z (cid:96) ( − Z (cid:96) ( − → Z (cid:96) , which is canonically isomorphic to Z (cid:96) (1). Thisﬁnishes the proof of the theorem for G = PGL .If now G is of rank 1, we get the map G → G ad ∼ = PGL , where the isomorphism G ad ∼ = PGL is uniquely determined by our choice of pinning. The mapGr G, Div X → Gr G ad , Div X is an isomorphism when restricted to each connected component, inducing an isomorphismGr G, Div X ∼ = π ( G ) × π ( G ad ) Gr G ad , Div X . I.11. IDENTIFICATION OF THE DUAL GROUP 233

Here of course π ( G ad ) ∼ = Z / Z . This implies that Sat G can be equivalently described as thecategory of A ∈ Sat G ad together with a reﬁnement of the Z / Z -grading to a π ( G )-grading. Thisimplies that G (cid:86) = G ad (cid:86) × µ Z (cid:86) where Z (cid:86) is the split torus with character group π ( G ). Thus, one getsan isomorphism G (cid:86) ∼ = (cid:98) G also in this case, including the isomorphism ψ a on the root group.Coming back to a general split group G , let a be any simple coroot. We now look at thecorresponding minimal Levi subgroups M a ⊂ G properly containing T , with parabolic P a ⊂ B . Wehave the symmetric monoidal constant term functorCT P a [deg P a ] : Sat G → Sat M a , commuting with the functors to Sat T . This induces a map M a (cid:86) → G (cid:86) , commuting with the inclusionof T (cid:86) into both. In particular, passing to Lie algebras, we see that a ∈ X ∗ = X ∗ ( T (cid:86) ) is a root of G (cid:86) ,and a ∨ ∈ X ∗ = X ∗ ( T (cid:86) ) is a coroot of G (cid:86) . Moreover, if s a ∈ W is the corresponding simple reﬂectionfor G , we also see that s a ∈ W (cid:86) , the Weyl group of the reductive group G (cid:86) Q (cid:96) . Using this informationfor all a , we see that W ⊂ W (cid:86) , and that under X ∗ = X ∗ ( T (cid:86) ) resp. X ∗ = X ∗ ( T (cid:86) ), we haveΦ ∨ ⊂ Φ( G (cid:86) Q (cid:96) ) , Φ ⊂ Φ ∨ ( G (cid:86) Q (cid:96) ) . Moreover, for any irreducible object A µ ∈ Sat G ( Q (cid:96) ), the weights of A µ are contained in the convexhull of the W -orbit of µ . This implies that these inclusions must be isomorphisms — indeed, thedirections of the edges emanating from µ , for µ regular, correspond to Φ( G (cid:86) Q (cid:96) ). Together with theisomorphisms on simple root groups, we get a unique isomorphism G (cid:86) Q (cid:96) ∼ = (cid:98) G Q (cid:96) . Under this isomorphism, the map (cid:99) M a ∼ = M a (cid:86) → G (cid:86) is compatible with the map (cid:99) M a → (cid:98) G induced byLanglands duality. It follows that G (cid:86) (˘ Z (cid:96) ) ⊂ G (cid:86) ( ˘ Q (cid:96) ) ∼ = (cid:98) G ( ˘ Q (cid:96) )is a subgroup containing all (cid:99) M a (˘ Z (cid:96) ). But these generate (cid:98) G (˘ Z (cid:96) ), so (cid:98) G (˘ Z (cid:96) ) ⊂ G (cid:86) (˘ Z (cid:96) ). Now pick arepresentation G (cid:86) → GL N (given by some object of Sat G ) that is a closed immersion over Q (cid:96) . Bythe inclusion (cid:98) G (˘ Z (cid:96) ) ⊂ G (cid:86) (˘ Z (cid:96) ), we see that the map (cid:98) G Q (cid:96) ∼ = G (cid:86) Q (cid:96) → GL N extends to a map (cid:98) G → GL N .By Lemma VI.11.4, this is necessarily a closed immersion, at least if (cid:96) (cid:54) = 2 or (cid:98) G is simply connected.We can always reduce to the case that (cid:98) G is simply connected by arguing with the adjoint group G ad (whose dual group (cid:100) G ad is simply connected) ﬁrst, as in the discussion of rank-1-groups above. Itthen follows that G (cid:86) → GL N factors over (cid:98) G (cid:44) → GL N , giving a map G (cid:86) → (cid:98) G that is an isomorphismin the generic ﬁbre, and surjective in the special ﬁbre (as any F (cid:96) -point of (cid:98) G lifts to ˘ Z (cid:96) , and then to G (cid:86) (˘ Z (cid:96) )), and hence an isomorphism by Lemma VI.11.3. Lemma

VI.11.4 ([

PY06 , Corollary 5.2]) . Let H be a reductive group over Z (cid:96) , H (cid:48) some aﬃnegroup scheme of ﬁnite type over Z (cid:96) , and let ρ : H → H (cid:48) be a homomorphism that is a closedimmersion in the generic ﬁbre. Assume that (cid:96) (cid:54) = 2 , or that no almost simple factor of the derivedgroup of H Q (cid:96) is isomorphic to SO n +1 (e.g., the derived group of H is simply connected). Then ρ is a closed immersion. This ﬁnishes the proof of Theorem VI.11.1 when G is split, and endowed with a splitting. Nowwe prove independence of the choice of splitting. For this, we note that in fact the cohomological

34 VI. GEOMETRIC SATAKE grading on F alone determines T (cid:86) ⊂ G (cid:86) as its stabilizer, and B (cid:86) ⊂ G (cid:86) as the stabilizer of the associatedﬁltration. It remains to check that the isomorphisms ψ a : Lie (cid:98) U a ∼ = Z (cid:96) (1)are independent of the choices. For this, consider the ﬂag variety F (cid:96) over E , parametrizing Borels B ⊂ G . Each such Borel comes with its torus T , which is the universal Cartan and thus descends to E . Equivalently, note that tori over F (cid:96) are equivalent to ´etale Z -local systems, and as F (cid:96) is simplyconnected all of them come via pullback from E ; this then gives the so-called universal Cartan T over E , which is split as G is split. Let a be a simple coroot of G . At each point of F (cid:96) , we getthe corresponding parabolic P a ⊃ B , with Levi M a . Let (cid:102) F (cid:96) a → F (cid:96) parametrize pinnings of M a ,i.e. isomorphisms of U a with the additive group; this is a G m -torsor. Over (cid:102) F (cid:96) a , the universal group M a is constant, with adjoint group M a, ad ∼ = PGL . Consider S = (cid:102) F (cid:96) ♦ /ϕ Z → Spd

E/ϕ Z = Div X . Applying the constant term functor for P a over S gives a symmetric monoidal functorSat( H ck G, Div X × Div X S, Z (cid:96) ) → Sat( H ck M a , Div X × Div X S, Z (cid:96) );here, being symmetric monoidal is veriﬁed by repeating the construction of the fusion product afterthe smooth pullback S → Div X . Both sides admit ﬁbre functors to LocSys( S, Z (cid:96) ); this containsLocSys(Div X , Z (cid:96) ) = Rep cont W E ( Z (cid:96) ) fully faithfully, and we can consider the symmetric monoidal fullsubcategories on which the ﬁbre functors land in this subcategory. As the constant term functor iscompatible with ﬁbre functors, it induces a symmetric monoidal functor on these full subcategories,which are then easily seen to be equivalent to Sat G and Sat M (reconstructing both starting fromSchubert cells). This shows that the constant term functor Sat G → Sat M a is naturally independentof the choice of Borel, reducing us to the rank 1 case. In the rank 1 case, we can then furtherreduce to PGL , and we have the minuscule Schubert variety, which is the ﬂag variety F (cid:96) ∼ = P of G ∼ = PGL . There are canonical isomorphisms H ( F (cid:96) ) = Z (cid:96) , H ( F (cid:96) ) = Z (cid:96) ( − , and (cid:98) U a is canonically isomorphic to Hom( H ( F (cid:96) ) , H ( F (cid:96) )) ∼ = Z (cid:96) (1).Thus, we have shown that if G is split, the isomorphism G (cid:86) ∼ = (cid:98) G is canonical. Finally, thegeneral case follows by Galois descent from a ﬁnite Galois extension E (cid:48) | E splitting G . VI.12. Cartan involution

Any Chevalley group scheme (cid:98) G comes with the Cartan involution, induced by the map onroot data which on X ∗ is given by µ (cid:55)→ − w ( µ ) where w is the longest Weyl group element.Under the geometric Satake equivalence, this has a geometric interpretation: Namely, it essentiallycorresponds to the switching equivalence sw ∗ . Note that one can upgradesw ∗ : Sat IG (Λ) → Sat IG (Λ)to a symmetric monoidal functor by writing it as the composition of Verdier duality and theduality functor sw ∗ D in Sat IG (Λ); moreover, this symmetric monoidal functor commutes with theﬁbre functor F I (as symmetric monoidal functors), cf. Corollary VI.9.5. Thus, sw ∗ induces anautomorphism of the Tannaka group G (cid:86) , commuting with the W E -action. I.12. CARTAN INVOLUTION 235

Proposition

VI.12.1 . Under the isomorphism G (cid:86) ∼ = (cid:98) G with the dual group, the isomorphism sw ∗ is given by the Cartan involution, up to conjugation by (cid:98) ρ ( − ∈ (cid:98) G ad ( Z (cid:96) ) . Remark

VI.12.2 . There is a diﬀerent construction of the commutativity constraint on Sat G ,not employing the fusion product, that relies on the Cartan involution — this is essentially acategorical version of the classical Gelfand trick to prove commutativity of the Satake algebra.For the Satake category, this construction was ﬁrst proposed by Ginzburg [ Gin90 ], who howeveroverlooked the sign (cid:98) ρ ( − Zhu17 ] of the geometric Satake equivalence for Gr

Witt G used this approach, taking careful control of the signs; these are related to the work of Lusztig–Yun[ LY13 ]. We remark that Zhu gives a diﬀerent construction of the commutation of sw ∗ with the ﬁbrefunctor, using instead that the two actions (on left and right) of H ∗ ([ ∗ /L + G ] , Q (cid:96) ) on H ∗ ( H ck G , A )agree for A ∈ Sat G ( Q (cid:96) ). Proof.

We note that this is really a proposition: The statement only asks about the com-mutation of a certain diagram, not some extra structure. For the statement, we can also forgetabout the W E -action. In particular, enlarging E , we can assume that G is split. As in the proofof Theorem VI.11.1, one can reduce to the case that G is adjoint, so (cid:98) G is semisimple and simplyconnected. We also ﬁx a pinning of G .Now, being pinned, G has its own Cartan involution θ : G → G , and by the functoriality of allconstructions under isomorphisms, the induced automorphism of Sat G corresponds to the Cartaninvolution of (cid:98) G . In other words, we need to see that the automorphism θ ∗ sw ∗ : Sat G → Sat G (whichis symmetric monoidal, and commutes with the ﬁbre functors) induces conjugation by ρ (cid:86) ( −

1) on G (cid:86) .We claim that the natural cohomological grading on the ﬁbre functor F : Sat G (Λ) → Rep W E (Λ)is compatible with sw ∗ . In other words, we need to see that in Corollary VI.9.5, the isomorphism F ( A ) ∼ = F (sw ∗ A ) is compatible with the grading, which follows from its construction. In particular,it follows that the automorphism of G (cid:86) restricts to the identity on the corresponding cocharacter2 ρ (cid:86) : G m ⊂ G (cid:86) . This implies already that it preserves T (cid:86) and the Borel B (cid:86) (as the centralizer anddynamical parabolic). Any such automorphism of G (cid:86) is given by conjugation by some element s ∈ T (cid:86) ad ⊂ G (cid:86) ad . We need to see that s = ρ (cid:86) ( − U (cid:86) a of G (cid:86) .We claim that the symmetric monoidal automorphism θ ∗ sw ∗ : Sat G → Sat G (commuting withthe ﬁbre functor) is compatible with the constant term functors CT P , for any standard parabolic P ⊃ B , and the similar functor on Levi subgroups. Let θ (cid:48) be the composition of θ with conjugationby w . We know, by the proof of Theorem VI.11.1, that any inner automorphism of G induces theidentity on G (cid:86) . Thus, it suﬃces to prove the similar claim for θ (cid:48)∗ sw ∗ : Sat G → Sat G . Let P − ⊂ G be the opposite parabolic of P ; then θ (cid:48) ( P − ) = P , and the induced automorphism of the Levi M is given by the corresponding automorphism θ (cid:48) M deﬁned similarly as θ (cid:48) . Now Proposition IV.6.13and the fusion deﬁnition of the symmetric monoidal structure (along with the deﬁnition of sw ∗ asthe composite of Verdier duality and internal duality) give the claim.These observations reduce us to the case G = PGL . We note that in this case the Cartaninvolution is the identity, so we can ignore θ . We have the minuscule Schubert variety i µ : Gr G,µ = P ⊂ Gr G and the sheaf A = i µ ∗ Z (cid:96) [1]( ) ∈ Sat G (assuming without loss of generality √ q ∈ Λ to

36 VI. GEOMETRIC SATAKE introduce a half-Tate twist), and we know G (cid:86) = SL( F ( A )), where F ( A ) = F ( A ) ⊕ F ( A ) − = H ( P )( ) ⊕ H ( P )( ) . The image of A under sw ∗ is isomorphic to A itself; ﬁx an isomorphism. Then on the one hand F ( A ) ∼ = F (sw ∗ A )as the functor sw ∗ : Sat G → Sat G commutes with the ﬁbre functor F , while on the other hand F (sw ∗ A ) ∼ = F ( A )as the two objects are isomorphic. We need to see that the composite isomorphism is given by thediagonal action of ( u, − u ) for some u ∈ Z × (cid:96) (this claim is independent of the chosen isomorphismbetween A and sw ∗ A ). We already know that the isomorphism is graded, so it is given by diagonalmultiplication by ( u , u ) for some units u , u ∈ Z × (cid:96) .Recall that the ﬁrst isomorphism is constructed as the composite of Verdier duality and internalduality in Sat G . Now the Verdier dual of A is A itself (because of the half-Tate twist), and theVerdier duality pairing F ( A ) ⊗ F ( A ) → Z (cid:96) is the tautological pairing; in particular, restricted to F ( A ) − ⊗ F ( A ) and F ( A ) ⊗ F ( A ) − it is thesame map, up to the natural commutativity constraint on Z (cid:96) -modules. It follows that the internaldual A ∨ of A is also isomorphic to A , and picking such an identiﬁcation we need to understand theinduced pairing F ( A ) ⊗ F ( A ) → F (1) = Z (cid:96) , and show that when restricted to F ( A ) − ⊗ F ( A ) and F ( A ) ⊗ F ( A ) − , the two induced maps diﬀerby a sign (up to the natural commutativity constraint); this claim is again independent of the chosenisomorphism between A ∨ and A . But this is a question purely internal to the symmetric monoidalcategory Sat G ∼ = Rep(SL ) with its ﬁbre functor. In there, we have the tautological representation V = Z (cid:96) = Z (cid:96) e ⊕ Z (cid:96) e , and it has the determinant pairing V ⊗ V → Z (cid:96) as SL -representation,realizing the internal duality. The determinant pairing is alternating, so takes opposite signs on( e , e ) and ( e , e ), as desired. (cid:3) HAPTER VII D (cid:4) ( X ) In order to deal with smooth representations of G ( E ) on Q (cid:96) -vector spaces (not Banach spaces),we extend (a modiﬁed form of) the 6-functor formalism from [ Sch17a ] to the larger class of solidpro-´etale sheaves. The results in this chapter were obtained in discussions with Clausen, and Mannhas obtained analogues of some of these results in the case of schemes. (Strangely enough, in somerespects the formalism actually works better for diamonds than for schemes.)More precisely, we want to ﬁnd a “good” category of Q (cid:96) -sheaves on [ ∗ /G ( E )] that correspondsto smooth representations of G ( E ) with values in Q (cid:96) -vector spaces, and extends to a category of Q (cid:96) -sheaves on Bun G with a good formalism of six operations that allows us to extend the precedingresults for ´etale torsion coeﬃcients. The ﬁrst idea would be to take pro-systems of ´etale torsionsheaves as Z (cid:96) -coeﬃcients and invert formally (cid:96) ; this formalism is easy to construct, see [ Sch17a ,Section 26]. This would give rise to continuous representations of G ( E ) in Q (cid:96) -Banach spaces, andwe do not want that: • supercuspidal representations of G ( E ) in Q (cid:96) -vector spaces are deﬁned over a ﬁnite degree exten-sion of Q (cid:96) , and after twist admit an invariant lattice that allows us to complete them (cid:96) -adically,but we do not want to make such a choice. • we want to construct semi-simple Langlands parameters using the Bernstein center and not some (cid:96) -adic completion of it. • in usual discussions of the cohomology of the Lubin–Tate tower, or more general Rapoport–Zinkspaces, it is possible to use Q (cid:96) -coeﬃcients while talking about usual smooth representations. Wewant to be able to achieve the same on the level of Bun G .We could take Q (cid:96) -pro-´etale sheaves. This would give rise to representations of G ( E ) (seen asa condensed group) with values in condensed Q (cid:96) -vector spaces. This category is too big; there isno hope to obtain a formalism of six operations in this too general context. We need to ask forsome “completeness” of the sheaves, for which we take inspiration from the theory of solid abeliangroups developed in [ CS ].The idea is the following. We deﬁne a category of solid pro-´etale Q (cid:96) -sheaves on Bun G with agood formalism of (a modiﬁed form of) six operations. More precisely, for any small v-stack, wedeﬁne a full subcategory D (cid:4) ( X, Z (cid:96) ) ⊂ D ( X v , Z (cid:96) ) , compatible with pullback, and equipped with a symmetric monoidal tensor product (for whichpullback is symmetric monoidal). A complex is solid if and only if each cohomology sheaf is solid,and this can be checked v-locally. The subcategory D (cid:4) ( X, Z (cid:96) ) is stable under all (derived) limits andcolimits, and the inclusion into D ( X v , Z (cid:96) ) admits a left adjoint. If X is a diamond, then D (cid:4) ( X, Z (cid:96) ) D (cid:4) ( X ) is also a full subcategory of D ( X qpro´et , Z (cid:96) ). If X is a spatial diamond, then on the abelian level, thecategory of solid Z (cid:96) -sheaves is the Ind-category of the Pro-category of constructible ´etale sheaveskilled by some power of (cid:96) . In this way, one can bootstrap many results from the usual ´etale case.For any map f : Y → X of small v-stacks, the pullback functor f ∗ admits a right adjoint Rf ∗ : D (cid:4) ( Y, Z (cid:96) ) → D (cid:4) ( X, Z (cid:96) ) that in fact commutes with any base change, see Proposition VII.2.4.Similarly, the formation of R H om commutes with any base change. Both of these operations cana priori be taken in all v-sheaves, but turn out to preserve solid sheaves. This already gives us fouroperations.Unfortunately, Rf ! does not have the same good properties as usual. In particular, if f is proper(and ﬁnite-dimensional), Rf ∗ does not in general satisfy a projection formula. As a remedy, it turnsout that for all f , the functor f ∗ admits a left adjoint f (cid:92) : D (cid:4) ( Y, Z (cid:96) ) → D (cid:4) ( X, Z (cid:96) ) , given by “relative homology”. This is a completely novel feature, and already for closed immer-sions this takes usual ´etale sheaves to complicated solid sheaves. Again f (cid:92) commutes with anybase change, and also satisﬁes the projection formula (which is just a condition here, as there isautomatically a natural map).When f is “proper and smooth”, one can moreover relative f (cid:92) (“homology”) and Rf ∗ (“coho-mology”) in the expected way. One also gets a formula for the dualizing complex of f in termsof such functors. These results even extend to universally locally acyclic complexes. This solid5-functor formalism thus has some excellent formal properties. We are somewhat confused aboutexactly how expressive it is, and whether it is preferable over the standard 6-functor formalism.One advantage is certainly that f (cid:92) is more canonical, and even deﬁned much more generally, than Rf ! (whose construction for stacky maps would require the resolution of very subtle homotopycoherence issues, and also can only be deﬁned for certain (ﬁnite-dimensional) maps). The mainproblem with the solid formalism is that a stratiﬁcation of a stack does not lead to a semi-orthogonaldecomposition on the level of D (cid:4) .On the other hand, for our concerns here, D (cid:4) (Bun G , Z (cid:96) ) is much too large. On [ ∗ /G ( E )] thisgives rise to representations of G ( E ) (as a condensed group) with values in solid Z (cid:96) -modules. Thecategory of discrete Q (cid:96) -vector spaces injects into the category of solid Z (cid:96) -modules. In fact, Q (cid:96) -vectorspaces are the same as Ind-ﬁnite dimensional vector spaces. Any ﬁnite dimensional Q (cid:96) -vector spaceis complete and thus “solid”. This means that if V is a Q (cid:96) -vector space then it gives rise to thesolid condensed sheaf V ⊗ Q disc(cid:96) Q (cid:96) whose value on the proﬁnite set S islim −→ W ⊂ V ﬁnite dim. Cont(

S, W ) . The category of smooth representations of G ( E ) with values in discrete Q (cid:96) -vector spaces injectsin the category of solid Q (cid:96) -pro-´etale shaves on [ ∗ /G ( E )]. In fact, since (cid:96) (cid:54) = p and G ( E ) is locallypro- p , representations of the condensed group G ( E ) on the condensed Q (cid:96) -vector space V ⊗ Q disc(cid:96) Q (cid:96) are the same as smooth representations of G ( E ) on V . II.1. SOLID SHEAVES 239

We then cut out a subcategory D lis (Bun G , Q (cid:96) ) of D (cid:4) (Bun G , Z (cid:96) ) that gives back the categoryof smooth representations of G ( E ) in Q (cid:96) -vector spaces when we look at [ ∗ /G ( E )]. (Of course, wecan also stick with Z (cid:96) -coeﬃcients.) VII.1. Solid sheaves

In the following, (cid:98) Z always denotes the pro-´etale sheaf (cid:98) Z = lim ←− n Z /n Z where n runs over nonzerointegers. We will quickly restrict attention to (cid:98) Z p = lim ←− ( n,p )=1 Z /n Z , allowing only n prime to p .Let X be a spatial diamond. For any quasi-pro-´etale j : U → X that can be written as acoﬁltered inverse limit of qcqs ´etale j i : U i → X , we let j (cid:92) (cid:98) Z = lim ←− i j i ! (cid:98) Z ;as the pro-system of the U i is unique, this is well-deﬁned. Note that there is a tautological sectionof j (cid:92) (cid:98) Z over U . Equivalently, if one writes (cid:98) Z [ U ] for the free pro-´etale sheaf of (cid:98) Z -modules generatedby U (noting that j ∗ admits a left adjoint on pro-´etale sheaves, being a slice), there is a naturalmap (cid:98) Z [ U ] → lim ←− i (cid:98) Z [ U i ] = j (cid:92) (cid:98) Z .When X = Spa( C ), j (cid:92) (cid:98) Z is the sheaf denoted (cid:98) Z [ U ] (cid:4) in [ CS ], and the same notation will beappropriate here in general. Definition

VII.1.1 . Let F be a pro-´etale sheaf of (cid:98) Z -modules on X . Then F is solid if for all j : U → X as above, the map Hom( j (cid:92) (cid:98) Z , F ) → F ( U ) is an isomorphism. Let us begin with the following basic example. We note ν : X qpro´et → X ´et the projection to the´etale site. Proposition

VII.1.2 . For any ´etale sheaf F of Z /n Z -modules on X ´et , ν ∗ F is solid. Proof.

This is a consequence of [

Sch17a , Proposition 14.9]. (cid:3)

The notion of solid sheaf is well-behaved:

Theorem

VII.1.3 . The category of solid (cid:98) Z -sheaves on X is an abelian subcategory of all pro-´etale (cid:98) Z -sheaves on X , stable under all limits, colimits, and extensions. It is generated by the ﬁnitelypresented objects j (cid:92) (cid:98) Z for quasi-pro-´etale j : U → X as above, and the inclusion admits a left adjoint F (cid:55)→ F (cid:4) that commutes with all colimits.Let F be a pro-´etale (cid:98) Z -sheaf on X . The following conditions are equivalent.(1) The (cid:98) Z -sheaf F is ﬁnitely presented in the category of all pro-´etale (cid:98) Z -sheaves, and is solid.(2) The (cid:98) Z -sheaf F is solid, and ﬁnitely presented in the category of solid (cid:98) Z -sheaves.(3) The (cid:98) Z -sheaf F can be written as a coﬁltered inverse limit of torsion constructible ´etale sheaves.

40 VII. D (cid:4) ( X ) For any such F , the underlying pro-´etale sheaf is representable by a spatial diamond. The categoryof F satisfying (1) – (3) is stable under kernels, cokernels, and extensions, in particular an abeliancategory, and is equivalent to the Pro -category of torsion constructible ´etale sheaves.Moreover, the category of all solid (cid:98) Z -sheaves on X is equivalent to the Ind -category of the

Pro -category of torsion constructible ´etale sheaves.

Question

VII.1.4 . If F is a pro-´etale (cid:98) Z -sheaf on X whose underlying pro-´etale sheaf is repre-sentable by a spatial diamond, or even is just qcqs, is F necessarily solid? If so, these conditionswould also be equivalent to (1) – (3). Let us remark the following lemma.

Lemma

VII.1.5 . Any torsion constructible ´etale sheaf on the spatial diamond X is representedby a spatial diamond. Proof.

Let

F → X be such a sheaf. We can ﬁnd a surjection of ´etale sheaves F (cid:48) = (cid:76) i j i ! Z /n i Z → F for some quasicompact separated ´etale maps j i : U i → X and nonzero inte-gers n i (where the direct sum is ﬁnite). Then F (cid:48) is quasicompact separated ´etale over X , andthus a spatial diamond; and the surjective map F (cid:48) → F is also quasicompact separated ´etale, inparticular universally open, and so also F is a spatial diamond. (cid:3) Before starting the proof, we record a key proposition. Its proof is a rare instance that requiresw-contractible objects — in most proofs, strictly totally disconnected objects suﬃce.

Proposition

VII.1.6 . Let X be a spatial diamond and let F i , i ∈ I , be a coﬁltered system oftorsion constructible ´etale sheaves. Then for all j > the higher inverse limit R j lim ←− i F i = 0 , taken in the category of pro-´etale sheaves on X , vanishes. Proof.

The pro-´etale site of X has a basis given by the w-contractible Y → X , that is strictlytotally disconnected perfectoid spaces Y such that the closed points in | Y | are a closed subset and π Y is an extremally disconnected proﬁnite set; equivalently, any pro-´etale cover (cid:101) Y → Y splits,cf. [ BS15 , Section 2.4] for a discussion of w-contractibility. Thus, it suﬃces to check sections on Y . In other words, we may assume that X is w-contractible, and prove that R j lim ←− i F i ( X ) = 0for all j >

0. As X is strictly totally disconnected, sheaves on X ´et are equivalent to sheaves on | X | .Moreover, we have the closed immersion f : π X → | X | given by the closed points, and pullbackalong this map induces isomorphisms F i ( X ) ∼ = ( f ∗ F i )( π X ). Let S = π X be the extremallydisconnected proﬁnite set.Then any torsion constructible sheaf on S is locally constant with ﬁnite ﬁbres. In particular, f ∗ F i maps isomorphically to H om( H om( f ∗ F i , S ) , S ) where S = R / Z is the sheaf of continuous II.1. SOLID SHEAVES 241 maps to the circle. (We could for the moment also use Q / Z , but it will become critical that S iscompact.) It follows that R j lim ←− i F i ( X ) = Ext jS (lim −→ i H om( f ∗ F i , S ) , S ) . Thus, the result follows from the injectivity of S as stated in the next lemma. (cid:3) Lemma

VII.1.7 . Let S be an extremally disconnected proﬁnite set. Then the abelian sheaf R / Z on S is injective. Proof.

First, we note that R / Z is ﬂasque in a strong sense. Namely, if U ⊂ S is any opensubset with closure U ⊂ S , then U is the Stone- ˇCech compactiﬁcation of U (as U ⊂ S is open byone deﬁnition of extremally disconnected spaces, and then βU (cid:116) ( S \ U ) → S admits a splitting,which in particular gives a splitting of the surjection βU → U that is the identity on U , thusimplying that βU ∼ = U ) and hence any section of R / Z over U extends uniquely to U as R / Z iscompact Hausdorﬀ. Also, all sections over U extend to S , as U is open and closed in S .Let F (cid:44) → G be an injection of sheaves on S with a map F → R / Z . Using Zorn’s lemma,choose a maximal subsheaf of G containing F with an extension of the map to R / Z . Replacing F by this maximal subsheaf, we can assume that F is maximal already. If F → G is not anisomorphism, then it is not an isomorphism on global sections (any local section not in the imagecan be extended by zero to form a global section not in the image), so we can ﬁnd a map Z → G such that F (cid:48) = F × G Z ⊂ Z is a proper subsheaf, and we can replace G by Z and assume that F isa proper subsheaf of Z .For each integer n , we can look at the open subset j n : U n ⊂ S where n ∈ Z lies in F . On thisopen subset, we have a map n Z → R / Z , and by the above this extends uniquely to j n : U n (cid:44) → S .The extension j n ! Z → R / Z necessarily agrees with the restriction of the given map F → R / Z onthe intersection j n ! Z ∩ F ⊂ j n ! Z , as this contains the dense subset j n ! Z and R / Z is separated.Thus, by maximality of F , we see that necessarily all U n are open and closed, hence so areall V n = U n \ (cid:83) m

1, and hence F = Z | V is a direct summand of Z , in which case the possibility of extension is clear. (cid:3) Now we can give the proof of Theorem VII.1.3.

Proof of Theorem VII.1.3.

The Pro-category of torsion constructible ´etale sheaves is anabelian category, and by Proposition VII.1.6 the functor to pro-´etale (cid:98) Z -sheaves is exact. It is alsofully faithful: For this, it suﬃces to see that if F i , i ∈ I , is a coﬁltered inverse system of torsionconstructible ´etale sheaves and G is any ´etale sheaf, thenlim −→ i Hom( F i , G ) → Hom(lim ←− i F i , G )is an isomorphism. But the underlying pro-´etale sheaf of each F i is a spatial diamond over X , soby [ Sch17a , Proposition 14.9] (applied with j = 0) we see that the similar result holds true whenthen taking homomorphisms of pro-´etale sheaves (without the abelian group structure). Enforcing

42 VII. D (cid:4) ( X ) compatibility with addition amounts to a similar diagram for lim ←− i F i × lim ←− i F i to which the sameargument applies.We see that the Pro-category of torsion constructible ´etale sheaves is a full subcategory C of allpro-´etale (cid:98) Z -modules on X , stable under the formation of kernels and cokernels. All of these sheavesare solid: As the condition of being solid is stable under all limits, it suﬃces to see that any ´etalesheaf is solid; this is Proposition VII.1.2.Also, by [ Sch17a , Lemma 11.22] and Lemma VII.1.5 all objects of C have as underlying pro-´etale sheaf a spatial diamond. One also checks that C is stable under extensions, by reduction toconstructible sheaves (cf. proof of Proposition VII.1.12 below).Next, we prove that (3) implies (1), so let F be in C ; in particular, the underlying pro-´etalesheaf is a spatial diamond. Then for any pro-´etale (cid:98) Z -module G , one can describe Hom( F , G ) as themaps F → G of pro-´etale sheaves satisfying additivity and (cid:98) Z -linearity, i.e. certain maps F × F → G resp.

F × (cid:98) Z → G agree. This description commutes with ﬁltered colimits (as for spatial diamonds Y , the functor G (cid:55)→ G ( Y ) commutes with ﬁltered colimits).Now we can describe the full category of solid (cid:98) Z -sheaves. Indeed, using that j (cid:92) (cid:98) Z is ﬁnitelypresented in all pro-´etale (cid:98) Z -modules by the previous paragraph, we see from the deﬁnition thatthe category of solid (cid:98) Z -sheaves is stable under all ﬁltered colimits. In particular, we get an exactfunctor from the Ind-category of C to solid (cid:98) Z -sheaves. This is also fully faithful, as all objectsof C are ﬁnitely presented. Moreover, Ind( C ) is an abelian category for formal reasons. We seethat Ind( C ) is a full subcategory of the category of pro-´etale (cid:98) Z -sheaves stable under kernels andcokernels, and all of its objects are solid. Conversely, any solid (cid:98) Z -sheaf admits a surjection from adirect sum of objects of the form j (cid:92) (cid:98) Z ∈ C , and the kernel of any such surjection is still solid, so wemay write any solid (cid:98) Z -sheaf as the cokernel of a map in Ind( C ). As Ind( C ) is stable under cokernels,we see that Ind( C ) is exactly the category of solid (cid:98) Z -sheaves.As ﬁltered colimits of solid sheaves stay solid, it is now formal that (1) implies (2), and (2)implies (3) as C ⊂

Ind( C ) are the ﬁnitely presented objects (as C is idempotent-complete). Thisﬁnishes the proof of the equivalences.The identiﬁcation with Ind( C ) shows that the category of solid (cid:98) Z -sheaves is stable under kernels,cokernels, and ﬁltered colimits. The latter two imply stability under all colimits, and stability underall limits is clear from the deﬁnition. One also easily checks stability under extensions by reductionto C (again, cf. proof of Proposition VII.1.12). For the existence of the left adjoint, note that itexists on the free pro-´etale (cid:98) Z -modules generated by U , with value j (cid:92) (cid:98) Z , i.e. (cid:98) Z [ U ] (cid:4) = j (cid:92) (cid:98) Z . As these generate all pro-´etale (cid:98) Z -modules, one ﬁnds that the left adjoint F (cid:55)→ F (cid:4) exists in general:one can write any F as a colimit lim −→ α (cid:98) Z [ U α ] and (lim −→ α (cid:98) Z [ U α ]) (cid:4) = lim −→ α (cid:98) Z [ U α ] (cid:4) . (cid:3) We have the following proposition on the functorial behaviour of the notion of solid (cid:98) Z -sheaves. II.1. SOLID SHEAVES 243

Proposition

VII.1.8 . Let f : Y → X be a map of spatial diamonds. Then pullbacks of solid (cid:98) Z -sheaves are solid, and the functor f ∗ commutes with solidiﬁcation. Moreover, if f is surjective,and F is a pro-´etale (cid:98) Z -sheaf on X such that f ∗ F is solid, then F is solid. Proof.

Recall that f ∗ commutes with all limits (and of course colimits) by [ Sch17a , Lemma14.4]. To check that f ∗ commutes with solidiﬁcation, it suﬃces to check on the pro-´etale (cid:98) Z -modules (cid:98) Z [ U ] generated by some quasi-pro-´etale j : U → X , and in that case the claim follows from f ∗ commuting with all limits, f ∗ ( (cid:98) Z [ U ] (cid:4) ) = f ∗ (lim ←− i (cid:98) Z [ U i ]) = lim ←− i f ∗ (cid:98) Z [ U i ] = lim ←− i (cid:98) Z [ f ∗ U i ] = (cid:98) Z [ f ∗ U ] (cid:4) . In particular, applied to solid (cid:98) Z -sheaves on X , this implies that their pullback to Y is already solid.Now assume that f is surjective and F is a pro-´etale (cid:98) Z -sheaf on X such that f ∗ F is solid. Let j : U = lim ←− i U i → X be a quasi-pro-´etale map. We ﬁrst check thatHom( j (cid:92) (cid:98) Z , F ) → F ( U )is injective. Indeed, assume that f : j (cid:92) (cid:98) Z → F lies in the kernel. Then after pullback to Y , thismap vanishes since f ∗ (cid:98) Z [ U ] (cid:4) = (cid:98) Z [ f ∗ U ] (cid:4) . But for any quasi-pro-´etale V → X , the map F ( V ) → ( f ∗ F )( V × X Y ) is injective (using [ Sch17a , Proposition 14.7] one has f ∗ F ( V × X Y ) = ( λ ∗ X F )( V × X Y ) and we conclude since V × X Y → V is a v-cover), so it follows that f = 0.We see that an element of F ( U ) determines at most one map j (cid:92) (cid:98) Z → F , and this assertion staystrue after any pullback. By [ Sch17a , Proposition 14.7], it suﬃces to construct the map v-locally;but it exists after pullback to Y → X , thus proving existence. (cid:3) In particular, it makes sense to make the following deﬁnition.

Definition

VII.1.9 . Let Y be a small v-stack and let F be a v-sheaf of (cid:98) Z -modules on Y . Then F is solid if for all maps f : X → Y from a spatial diamond X , the pullback f ∗ F comes via pullbackfrom a solid (cid:98) Z -sheaf on X qpro´et . Regarding passage to the derived category, we make the following deﬁnition.

Definition

VII.1.10 . Let X be a small v-stack. Let D (cid:4) ( X, (cid:98) Z ) ⊂ D ( X v , (cid:98) Z ) be the full subcate-gory of all A such that each cohomology sheaf H i ( A ) is solid. As being solid is stable under kernels, cokernels, and extensions, this deﬁnes a triangulatedsubcategory.If X is a diamond, one could alternatively deﬁne a full subcategory of D ( X qpro´et , (cid:98) Z ) by the samecondition, and pullback from the quasi-pro-´etale to the v-site deﬁnes a functor. This functor is anequivalence, by repleteness (to handle Postnikov towers, cf. [ BS15 , Section 3]) and the followingproposition that is an amelioration of [

Sch17a , Proposition 14.7] for solid sheaves.

Proposition

VII.1.11 . Let X be a diamond and let F be a sheaf of (cid:98) Z -modules on X qpro´et thatis solid. Let λ : X v → X qpro´et be the map of sites. Then F → Rλ ∗ λ ∗ F is an isomorphism.

44 VII. D (cid:4) ( X ) Proof.

We may assume that X is spatial (or strictly totally disconnected). Then F is a ﬁlteredcolimit of ﬁnitely presented solid (cid:98) Z -sheaves, and the functor Rλ ∗ commutes with ﬁltered colimits in D ≥ . We may thus assume that F is ﬁnitely presented; in that case F is a coﬁltered limit of torsionconstructible ´etale sheaves, and λ ∗ commutes with all limits by [ Sch17a , Lemma 14.4]. Thus, wecan assume that F is an ´etale sheaf, where the claim is [ Sch17a , Proposition 14.7]. (cid:3)

Moreover, solid objects in the derived category satisfy a derived and internal version of Deﬁni-tion VII.1.1.

Proposition

VII.1.12 . Let X be a spatial diamond. For all A ∈ D (cid:4) ( X, (cid:98) Z ) , the map R H om( j (cid:92) (cid:98) Z , A ) → Rj ∗ A | U is an isomorphism for all quasi-pro-´etale j : U → X . Proof.

By taking a Postnikov limit, we can assume that A ∈ D + (cid:4) ( X, (cid:98) Z ), and then one reducesto the case that A = F [0] is concentrated in degree 0. Now by a resolution of Breen [ Bre78 ,Section 3] (appropriately sheaﬁﬁed), there is a resolution of any (cid:98) Z -sheaf G where all terms are ﬁnitedirect sums of sheaves of the form (cid:98) Z [ G i × (cid:98) Z j ]. If G is a spatial diamond, then all G i × (cid:98) Z j are spatialdiamonds, hence R H om( (cid:98) Z [ G i × (cid:98) Z j ] , F )commutes with all ﬁltered colimits. Applied to G = j (cid:92) (cid:98) Z , Breen’s resolution then implies that R H om( G , − ) commutes with all ﬁltered colimits.We may thus assume that F is ﬁnitely presented. But then Theorem VII.1.3 implies that F isa limit of constructible ´etale sheaves, so one can reduce to the case that F is an ´etale sheaf. Butthen Breen’s resolution shows that R H om( j (cid:92) (cid:98) Z , F ) = lim −→ i R H om( j i ! (cid:98) Z , F ) = lim −→ i Rj i ∗ F | U i and [ Sch17a , Proposition 14.9] shows that this identiﬁes with Rj ∗ F | U . (cid:3) Proposition

VII.1.13 . Let X be a spatial diamond. The inclusion D (cid:4) ( X, (cid:98) Z ) ⊂ D ( X qpro´et , (cid:98) Z ) admits a left adjoint A (cid:55)→ A (cid:4) : D ( X qpro´et , (cid:98) Z ) → D (cid:4) ( X, (cid:98) Z ) . Moreover, D (cid:4) ( X, (cid:98) Z ) identiﬁes with the derived category of solid (cid:98) Z -sheaves on X , and A (cid:55)→ A (cid:4) withthe left derived functor of F (cid:55)→ F (cid:4) . The formation of A (cid:55)→ A (cid:4) , for A ∈ D ( X qpro´et , (cid:98) Z ) , commuteswith any base change X (cid:48) → X of spatial diamonds. Proof.

This follows easily from Proposition VII.1.12. (cid:3)

Proposition

VII.1.14 . Let X be a spatial diamond. The kernel of A (cid:55)→ A (cid:4) is a tensor ideal.In particular, there is a unique symmetric monoidal structure − (cid:4) ⊗ L − on D (cid:4) ( X, (cid:98) Z ) making A (cid:55)→ A (cid:4) symmetric monoidal. It is the left derived functor of the induced symmetric monoidal structure onsolid (cid:98) Z -sheaves. This symmetric monoidal structure commutes with all colimits (in each variable)and any pullback. II.1. SOLID SHEAVES 245

Proof.

To check that the kernel is a tensor ideal, take any quasi-pro-´etale j : U → X written asa coﬁltered inverse limit of separated ´etale j i : U i → X , and any further quasi-pro-´etale j (cid:48) : U (cid:48) → X .Then for any solid A ∈ D (cid:4) ( X, (cid:98) Z ), we know by Proposition VII.1.12 that the map R H om( j (cid:92) (cid:98) Z , A ) → Rj ∗ A | U is an isomorphism. Taking sections over U (cid:48) → X , this translates into R Hom( j (cid:92) (cid:98) Z ⊗ L (cid:98) Z (cid:98) Z [ U (cid:48) ] , A ) → R Hom( (cid:98) Z [ U × X U (cid:48) ] , A )being an isomorphism. In other words, taking the tensor product of (cid:98) Z [ U ] → j (cid:92) (cid:98) Z with (cid:98) Z [ U (cid:48) ] stilllies in the kernel, but these generate the tensor ideal generated by the kernel.It is now formal that there is a unique symmetric monoidal structure − (cid:4) ⊗ L − on D (cid:4) ( X, (cid:98) Z )making A (cid:55)→ A (cid:4) symmetric monoidal (given by the solidiﬁcation of the tensor product in all solidpro-´etale sheaves). As solidiﬁcation commutes with all colimits, so does this tensor product. Ongenerators j : U → X , j (cid:48) : U (cid:48) → X as above, it is given by j (cid:92) (cid:98) Z (cid:4) ⊗ L j (cid:48) (cid:92) (cid:98) Z = ( j × X j (cid:48) ) (cid:92) (cid:98) Z , which stillsits in degree 0; this implies that the functor is a left derived functor. Moreover, this descriptioncommutes with any base change. (cid:3) Moreover, the inclusion into all v-sheaves also admits a left adjoint, if X is a diamond. We willlater improve on this proposition when working with (cid:98) Z p -coeﬃcients. Proposition

VII.1.15 . For any diamond X , the fully faithful embedding D (cid:4) ( X, (cid:98) Z ) ⊂ D ( X v , (cid:98) Z ) admits a left adjoint A (cid:55)→ A (cid:4) . The formation of A (cid:4) commutes with quasi-pro-´etale base change X (cid:48) → X . Proof.

Assume ﬁrst that X is strictly totally disconnected. It suﬃces to construct the leftadjoint on a set of generators, such as the pro-´etale sheaves of (cid:98) Z -modules generated by some strictlytotally disconnected Y → X . By [ Sch17a , Lemma 14.5], there is a strictly totally disconnectedaﬃnoid pro-´etale j : Y (cid:48) → X such that Y → X factors over a map Y → Y (cid:48) that is surjective andinduces a bijection of connected components. Then for any B ∈ D (cid:4) ( Y (cid:48) , (cid:98) Z ), the map R Γ( Y (cid:48) , B ) → R Γ( Y, B )is an isomorphism. Indeed, by Postnikov limits this easily reduces to B = F [0] for a solid sheaf of (cid:98) Z -modules, and then to a ﬁnitely presented solid sheaf, and ﬁnally to a constructible ´etale sheaf,for which the result is proved at the end of the proof of [ Sch17a , Lemma 14.4]. This means thatthe left adjoint A (cid:55)→ A (cid:4) when evaluated on (cid:98) Z [ Y ] exists and is given by j (cid:92) (cid:98) Z .The formation of Y (cid:48) → X from Y → X commutes with any quasi-pro-´etale base change ofstrictly totally disconnected X (cid:48) → X . This implies that A (cid:55)→ A (cid:4) commutes with such basechanges. By descent, this implies the existence of the left adjoint in general, and its commutationwith quasi-pro-´etale base change. (cid:3) As usual, we also want to have a theory with coeﬃcients in a ring Λ. As before, we assume thatΛ is constant in the sense that it comes via pullback from the point. In our case, this means that

46 VII. D (cid:4) ( X ) it comes via pullback from the pro-´etale site of a point, i.e. is a condensed ring [ CS ], and we needto assume that it is solid over (cid:98) Z ; in other words, we allow as coeﬃcients any solid (cid:98) Z -algebra Λ.Via pullback, this gives rise to a v-sheaf of (cid:98) Z -algebras on any small v-stack X , and we can consider D ( X v , Λ).

Example

VII.1.16 . We may consider Λ = Z (cid:96) as the solid condensed ring lim −→ L | Q (cid:96) ﬁnite O L . Definition

VII.1.17 . Let D (cid:4) ( X, Λ) ⊂ D ( X v , Λ) be the full subcategory of all A ∈ D ( X v , Λ) such that the image of A in D ( X v , (cid:98) Z ) is solid. On the level of ∞ -categorical enrichments, we thus see that D (cid:4) ( X, Λ) is the category of Λ-modules in D (cid:4) ( X, (cid:98) Z ). It is then formal that the inclusion D (cid:4) ( X, Λ) ⊂ D ( X v , Λ) admits a symmetricmonoidal left adjoint A (cid:55)→ A (cid:4) , compatible with forgetting the Λ-structure. Remark

VII.1.18 . Let us brieﬂy compare the present theory with the one developed in [ CS ].Over a geometric point X = Spa C , D (cid:4) ( X, (cid:98) Z ) is the derived category of solid (cid:98) Z -modules in thesense of [ CS ]. For general Λ, we are now simply considering Λ-modules in D (cid:4) ( X, (cid:98) Z ). This isin general diﬀerent from the theory of Λ (cid:4) -modules, which would ask for a stronger completenessnotion relative to Λ. Our present theory corresponds to the analytic ring structure on Λ inducedfrom (cid:98) Z (cid:4) .One might wonder whether for any analytic ring A in the sense of [ CS ] one can deﬁne a category D ( X, A ) of “ A -complete” pro-´etale sheaves on any spatial diamond X . This does not seem to bethe case; it is certainly not formal. In fact, already for A = Z (cid:4) , problems occur and there iscertainly no abelian category; it is still possible to deﬁne a nice derived category, though. Forgeneral A , deﬁning D ( X, A ) also seems to require extra data beyond the analytic ring structure on A . VII.2. Four functors

Now we discuss some functors on solid sheaves. For this, we assume from now on that we workwith coeﬃcients Λ given by a solid (cid:98) Z p -algebra (so we stay away from p -adic coeﬃcients). For anymap f : Y → X of small v-stacks, we have the pullback functor f ∗ : D (cid:4) ( X, Λ) ⊂ D (cid:4) ( Y, Λ). Thisadmits a right adjoint Rf ∗ ; in fact, one can simply import Rf ∗ from the full D ( Y, Λ):

Proposition

VII.2.1 . Let f : Y → X be a map of small v-stacks and let A ∈ D (cid:4) ( Y, Λ) ⊂ D ( Y, Λ) . Then Rf v ∗ A ∈ D ( X v , Λ) lies in D (cid:4) ( X, Λ) . In particular, Rf v ∗ : D ( Y v , Λ) → D ( X v , Λ) restricts to a functor Rf ∗ : D (cid:4) ( Y, Λ) → D (cid:4) ( X, Λ) that is right adjoint to f ∗ . Proof.

We can formally reduce to the case Λ = (cid:98) Z p . The formation of Rf v ∗ commutes withany pullback (as everything is a slice in the v-site), so using Proposition VII.1.8 we can assumethat X is a spatial diamond. Moreover, taking a simplicial resolution of Y by disjoint unions ofspatial diamonds, and using that D (cid:4) ( X, (cid:98) Z p ) ⊂ D ( X v , (cid:98) Z p ) is stable under all derived limits (as it isstable under all products), we can also assume that Y is a spatial diamond.We may assume A ∈ D + (cid:4) ( Y, (cid:98) Z p ) by a Postnikov limit, then that A = F [0] is concentrated indegree 0, then that F is ﬁnitely presented by writing it as a ﬁltered colimit, and ﬁnally that F is II.2. FOUR FUNCTORS 247 a constructible ´etale sheaf by writing it as a coﬁltered limit. Now the result follows from [

Sch17a ,Proposition 17.6]. (cid:3)

Proposition

VII.2.2 . For any small v-stack X , the inclusion D (cid:4) ( X, Λ) ⊂ D ( X v , Λ) admits a left adjoint A (cid:55)→ A (cid:4) : D ( X v , Λ) → D (cid:4) ( X, Λ) . The functor A (cid:55)→ A (cid:4) commutes with any base change.The kernel of A (cid:55)→ A (cid:4) is a tensor ideal. In particular, there is a unique symmetric monoidalstructure − (cid:4) ⊗ L Λ − on D (cid:4) ( X, Λ) making A (cid:55)→ A (cid:4) a symmetric monoidal functor. The functor − (cid:4) ⊗ L Λ − commutes with all colimits (in each variable) and with all pullbacks f : Y → X . We note that in the case of overlap with previous deﬁnitions of A (cid:55)→ A (cid:4) and − (cid:4) ⊗ L Λ − , thedeﬁnitions agree, by uniqueness of the previous deﬁnitions. Proof.

Again, one can formally reduce to the case Λ = (cid:98) Z p . By descent, we can reduce tothe case that X is strictly totally disconnected. (Note that Y (cid:55)→ D (cid:4) ( Y, (cid:98) Z p ) is a v-sheaf of ∞ -categories — this is clear for D ( Y v , (cid:98) Z p ), and follows for D (cid:4) as being solid can be checked v-locallyby Proposition VII.1.8.) In this case, we already know existence of the left adjoint A (cid:55)→ A (cid:4) byProposition VII.1.15.We check that the left adjoint A (cid:55)→ A (cid:4) commutes with any base change f : Y → X . We alreadyknow that pullbacks of solid objects stay solid, so we have to see that if A ∈ D ( X v , (cid:98) Z p ) satisﬁes A (cid:4) = 0, then also ( f ∗ v A ) (cid:4) = 0. But this statement is adjoint to the statement that Rf v ∗ preserves D (cid:4) , i.e. Proposition VII.2.1.We need to see that the class of all A ∈ D ( X v , (cid:98) Z p ) with A (cid:4) = 0 is a ⊗ -ideal. But we have seenthat for all f : Y → X , also f ∗ v A lies in the corresponding class for Y , and then so does f v(cid:92) f ∗ v A (aspullback preserves D (cid:4) ), where we write f v(cid:92) for the left adjoint of f ∗ v (which exists as it is a slice).But f v(cid:92) f ∗ v A = A ⊗ L (cid:98) Z p f v(cid:92) (cid:98) Z p by the projection formula for slices, so this gives the desired claim. (cid:3) It turns out that for Λ = (cid:98) Z p , the functor − (cid:4) ⊗ L − is actually almost exact. If one would workwith Λ = F (cid:96) -coeﬃcients, it would even be exact. Proposition

VII.2.3 . Let X be a small v-stack and A, B ∈ D (cid:4) ( X, (cid:98) Z p ) be concentrated in degree . Then A (cid:4) ⊗ L B sits in cohomological degrees − and .If X is a spatial diamond and F = lim ←− i F i and G = lim ←− j G j are ﬁnitely presented solid (cid:98) Z p -sheaves written as coﬁltered limits of constructible ´etale sheaves killed by some integer prime to p ,then the natural map F (cid:4) ⊗ L G → R lim ←− i,j F i ⊗ L G j is an isomorphism.

48 VII. D (cid:4) ( X ) Proof.

It suﬃces to prove the ﬁnal assertion, as the statement on A (cid:4) ⊗ L B can be checkedafter pullback to spatial diamonds, and then A and B can be written as ﬁltered colimits of ﬁnitelypresented solid (cid:98) Z p -sheaves (and F i ⊗ L G j sits in degrees − (cid:98) Z p has global dimension 1).Resolving F and G , we can reduce to the case F = j (cid:92) (cid:98) Z p , G = j (cid:48) (cid:92) (cid:98) Z p . But their solid tensor productis indeed given by ( j × X j (cid:48) ) (cid:92) (cid:98) Z p . (cid:3) At this point, we have deﬁned D (cid:4) ( X, Λ) ⊂ D ( X v , Λ) for any small v-stack X , and this sub-category is preserved by pullback and pushforward, and in particular this gives such functors for D (cid:4) ( X, Λ). Moreover, D (cid:4) ( X, Λ) has a natural symmetric monoidal structure − (cid:4) ⊗ L Λ − , commutingwith colimits in both variables, and with pullbacks. Moreover, we have a functor R H om Λ ( − , − ) : D (cid:4) ( X, Λ) op × D (cid:4) ( X, Λ) → D (cid:4) ( X, Λ) , a partial right adjoint to − (cid:4) ⊗ L Λ − as usual. Again, it can be obtained from the correspondingfunctor on D ( X v , Λ) via restriction. In fact, for all A ∈ D ( X v , Λ) and B ∈ D (cid:4) ( X, Λ), one has R H om Λ ( A, B ) ∈ D (cid:4) ( X, Λ). This can be reduced to Λ = (cid:98) Z p and the case A = f (cid:92) (cid:98) Z p for some f : Y → X , and then it amounts to Rf v ∗ f ∗ v B ∈ D (cid:4) ( X, (cid:98) Z p ), which follows from Proposition VII.2.1.There is the following general base change result. We stress the absence of any conditions. Proposition

VII.2.4 . Let Y (cid:48) g (cid:48) (cid:47) (cid:47) f (cid:48) (cid:15) (cid:15) Y f (cid:15) (cid:15) X (cid:48) g (cid:47) (cid:47) X be a cartesian diagram of small v-stacks. For all A ∈ D (cid:4) ( Y, Λ) , the base change map g ∗ Rf ∗ A → Rf (cid:48)∗ g (cid:48)∗ A in D (cid:4) ( X (cid:48) , Λ) is an isomorphism.Similarly, for any map f : Y → X of small v-stacks and all A, B ∈ D (cid:4) ( X, Λ) , the map f ∗ R H om( A, B ) → R H om( f ∗ A, f ∗ B ) in D (cid:4) ( Y, Λ) is an isomorphism. Proof.

The base change is a direct consequence of Proposition VII.2.1, noting that in thev-site, everything is a slice (and hence satisﬁes base change). The statement about R H om followssimilarly from the compatibility with the R H om as formed on the v-site, as was noted above. (cid:3) The projection formula, however, fails to hold.

Warning

VII.2.5 . If f : Y → X is a proper map of small v-stacks that is representable inspatial diamonds with dim . trg f < ∞ , the map A (cid:4) ⊗ L Rf ∗ B → Rf ∗ ( f ∗ A (cid:4) ⊗ L B )may fail to be an isomorphism for A ∈ D (cid:4) ( X, (cid:98) Z p ) and B ∈ D (cid:4) ( Y, (cid:98) Z p ). In fact, already if X = B C isa perfectoid ball and f = j : Y = Spa C → X is the inclusion of a point (which is quasi-pro-´etale), II.2. FOUR FUNCTORS 249 then this fails for A = j (cid:92) (cid:98) Z p and B = (cid:98) Z p . In fact, the map becomes j (cid:92) (cid:98) Z p → Rj ∗ (cid:98) Z p , which is farfrom an isomorphism: For example, on global sections the left-hand side becomes (cid:98) Z p [ − (cid:98) Z p .There is the following result on change of algebraically closed base ﬁeld, an analogue of [ Sch17a ,Theorem 19.5].

Proposition

VII.2.6 . Let X be a small v-stack. (i) Assume that X lives over k , where k is a discrete algebraically closed ﬁeld of characteristic p ,and k (cid:48) /k is an extension of discrete algebraically closed base ﬁelds, X (cid:48) = X × k k (cid:48) . Then the pullbackfunctor D (cid:4) ( X, Λ) → D (cid:4) ( X (cid:48) , Λ) is fully faithful. (ii) Assume that X lives over k , where k is a discrete algebraically closed ﬁeld of characteristic p .Let C/k be an algebraically closed complete nonarchimedean ﬁeld, and X (cid:48) = X × k Spa(

C, C + ) forsome open and bounded valuation subring C + ⊂ C containing k . Then the pullback functor D (cid:4) ( X, Λ) → D (cid:4) ( X (cid:48) , Λ) is fully faithful. (iii) Assume that X lives over Spa(

C, C + ) , where C is an algebraically closed complete nonar-chimedean ﬁeld with an open and bounded valuation subring C + ⊂ C , C (cid:48) /C is an extensionof algebraically closed complete nonarchimedean ﬁelds, and C (cid:48) + ⊂ C (cid:48) an open and bounded val-uation subring containing C + , such that Spa( C (cid:48) , C (cid:48) + ) → Spa(

C, C + ) is surjective. Then for X (cid:48) = X × Spa(

C,C + ) Spa( C (cid:48) , C (cid:48) + ) , the pullback functor D (cid:4) ( X, Λ) → D (cid:4) ( X (cid:48) , Λ) is fully faithful. Proof.

We can assume Λ = (cid:98) Z p . As in [ Sch17a , Theorem 19.5], it suﬃces to prove (iii) andthe restricted case of (ii) where C is the completed algebraic closure of k (( t )) (and hence C + = O C ).Let f : X (cid:48) → X be the map. We have to see that for all A ∈ D (cid:4) ( X, (cid:98) Z p ), the map A → Rf ∗ f ∗ A is an equivalence. This can be checked locally in the v-topology, so we can assume that X =Spa( R, R + ) is an aﬃnoid perfectoid space. By Postnikov limits, we can also assume that A ∈ D + (cid:4) ( X, (cid:98) Z p ), and then that A is concentrated in degree 0. In case (iii), we can now conclude bywriting A as a ﬁltered colimit of ﬁnitely presented solid (cid:98) Z p -modules, and these as coﬁltered limitsof constructible ´etale sheaves, noting that both operations commute with Rf ∗ and f ∗ (as f is qcqsin case (iii)), and hence reducing us to [ Sch17a , Theorem 19.5].It remains to handle case (ii) when C is the completed algebraic closure of k (( t )). In that case X (cid:48) lives over a punctured open unit disc D ∗ X over X , and ﬁxing a pseudouniformizer (cid:36) ∈ R , this can bewritten as the increasing union of quasicompact open subspaces X (cid:48) n = {| t | n ≤ | (cid:36) | ≤ | t | /n } ⊂ X (cid:48) ,with maps f n : X (cid:48) n → X . It suﬃces to prove that for all n , the map A → Rf n ∗ f ∗ n A

50 VII. D (cid:4) ( X ) is an isomorphism. These functors commute again with ﬁltered colimits of sheaves, and hence theprevious reductions apply and reduce the assertion to the ´etale case, which was handled in theproof of [ Sch17a , Theorem 19.5]. (cid:3)

As an application, let us record the following versions of Proposition IV.7.1 and Corollary IV.7.2,where we ﬁx an algebraically closed ﬁeld k | F q and work on Perf k . Corollary

VII.2.7 . For any small v-stack X , the functor ψ ∗ X : D (cid:4) ( X × [ ∗ /W E ] , Λ) → D (cid:4) ( X × Div , Λ) is fully faithful. If the natural pullback functor D (cid:4) ( X, Λ) → D (cid:4) ( X × Spd (cid:98) E, Λ) is an equivalence, then ψ ∗ X is also an equivalence. Proof.

By descent along X → X × [ ∗ /W E ] this reduces to Proposition VII.2.6. (cid:3) Corollary

VII.2.8 . For any small v-stack X and ﬁnite set I , pullback along X × (Div ) I → X × [ ∗ /W IE ] induces a fully faithful functor D (cid:4) ( X × [ ∗ /W IE ] , Λ) → D (cid:4) ( X × (Div ) I , Λ) . Proof.

This follows inductively from Corollary VII.2.7. (cid:3)

We also need a solid analogue of Theorem IV.5.3; we only prove a restricted variant, however.As there, work over Perf k , and let X be a spatial diamond such that X → ∗ is proper, of ﬁnitedim . trg, and take any spatial diamond S . As before, one can introduce the doubly-indexed ind-system { U a,b } ( a,b ) ⊂ X × S , well-deﬁned up to ind-isomorphism; and then U a = (cid:83) b< ∞ U a,b and U b = (cid:83) a> U a,b . Definition

VII.2.9 . The functors Rβ !+ , Rβ ! − : D (cid:4) ( X × S, Λ) → D (cid:4) ( S, Λ) are deﬁned by Rβ !+ C := lim −→ a Rβ ∗ ( j a ! C | U a ) ,Rβ ! − C := lim −→ b Rβ ∗ ( j b ! C | U b ) for C ∈ D (cid:4) ( X × S, Λ) . Here j a ! and j b ! denote the left adjoints to j ∗ a and j ∗ b . Let α : X × S → X be the projection. Theorem

VII.2.10 . Assume that C = α ∗ A for A ∈ D (cid:4) ( X, Λ) , and assume that either A ∈ D + (cid:4) ( X, Λ) , or that X → ∗ is cohomologically smooth. Then Rβ !+ C = 0 = Rβ ! − C. II.3. RELATIVE HOMOLOGY 251

Proof.

We can assume Λ = (cid:98) Z p . All operations commute with any base change; we can thusassume that S = Spa K where K is the complete algebraic closure of k (( t )). We observe that if X → ∗ is cohomologically smooth, then Rβ ∗ : D (cid:4) ( X × S, Λ) → D (cid:4) ( S, Λ) has ﬁnite cohomologicaldimension; this is a statement about sheaves concentrated in degree 0. Any such B can be writtenas the countable limit of Rj a,b, ∗ j ∗ a,b B for the open immersions j a,b : U a,b ⊂ X × S ; it is thus enoughto show that pushforward along U a,b → S has ﬁnite cohomological dimension on solid sheaves.As U a,b → S is qcqs, we can reduce to ﬁnitely presented sheaves; these are coﬁltered limits ofconstructible sheaves. For constructible sheaves, the cohomological dimension is bounded, andeach cohomology group (recall that S = Spa K is a geometric point) is ﬁnite by [ Sch17a , Theorem25.1]. Thus, the coﬁltered limit stays in the same range of degrees.It follows that we can assume that A ∈ D + (cid:4) ( X, (cid:98) Z p ). Arguing as in the proof of Theorem IV.5.3,we can then reduce to the case that X = Spa( R, R + ) is an aﬃnoid perfectoid space with no nonsplitﬁnite ´etale covers, and then to X = Spa K where K is still the completed algebraic closure of k (( t )).In that case, as in the proof of Theorem IV.5.3, one can make a more precise assertion on actualannuli; this statement is compatible with passage to ﬁltered colimits, reducing us to the case that A is a ﬁnitely presented solid sheaf. For A ∈ D (cid:4) (Spa K, (cid:98) Z p ), this means that A is a coﬁltered limitof ﬁnite abelian groups killed by integers prime to p . We can also pull this coﬁltered limit through,reducing us to Theorem IV.5.3. (cid:3) VII.3. Relative homology

A unique feature of the formalism of solid sheaves is the existence of a general left adjoint topullback, with excellent properties. We continue to work with coeﬃcients in a solid (cid:98) Z p -algebra Λ. Proposition

VII.3.1 . Let f : Y → X be any map of small v-stacks. (i) The functor f ∗ : D (cid:4) ( X, Λ) → D (cid:4) ( Y, Λ) admits a left adjoint f (cid:92) : D (cid:4) ( Y, Λ) → D (cid:4) ( X, Λ) . The natural map f (cid:92) ( A (cid:4) ⊗ L Λ f ∗ B ) → f (cid:92) A (cid:4) ⊗ L Λ B is an isomorphism for all A ∈ D (cid:4) ( Y, Λ) and B ∈ D (cid:4) ( X, Λ) . Similarly, the map R H om( f (cid:92) A, B ) → Rf ∗ R H om( A, f ∗ B ) is an isomorphism. (ii) The formation of f (cid:92) commutes with restriction of coeﬃcients along a map Λ (cid:48) → Λ . (iii) For any cartesian diagram Y (cid:48) g (cid:48) (cid:47) (cid:47) f (cid:48) (cid:15) (cid:15) Y f (cid:15) (cid:15) X (cid:48) g (cid:47) (cid:47) X of small v-stacks, the natural map f (cid:48) (cid:92) g (cid:48)∗ A → g ∗ f (cid:92) A is an isomorphism for all A ∈ D (cid:4) ( Y, Λ) .

52 VII. D (cid:4) ( X ) Proof. As f is a slice in the v-site, it is tautological that f ∗ v : D ( X v , Λ) → D ( Y v , Λ) admitsa left adjoint f v(cid:92) : D ( Y v , Λ) → D ( X v , Λ). One can then deﬁne f (cid:92) as the solidiﬁcation of f v(cid:92) . Bygeneral properties of slices, the map f v(cid:92) ( A ⊗ L Λ f ∗ B ) → f v(cid:92) A ⊗ L Λ B is an isomorphism. Passing to solidiﬁcations, using that that this is symmetric monoidal, then givesthat f (cid:92) ( A (cid:4) ⊗ L Λ f ∗ B ) → f (cid:92) A (cid:4) ⊗ L Λ B is an isomorphism. The isomorphism R H om( f (cid:92) A, B ) ∼ = Rf ∗ R H om( A, f ∗ B )then follows by adjointness.For part (ii), we can assume Λ (cid:48) = (cid:98) Z p , and check on generators. These are given by j (cid:92) (cid:98) Z p ⊗ L (cid:98) Z p Λfor j : Y (cid:48) → Y . The claim then follows from the projection formula.Part (iii) is obtained by passing to left adjoints in Proposition VII.2.4. (cid:3) Now if f is “proper and smooth”, we want to relate the left adjoint f (cid:92) (“homology”) and theright adjoint Rf ∗ (“cohomology”). Thus, assume that f : Y → X is a proper map of small v-stacksthat is representable in spatial diamonds with dim . trg f < ∞ , and cohomologically smooth, i.e. (cid:96) -cohomologically smooth for all (cid:96) (cid:54) = p (or just all (cid:96) relevant for Λ). In this case, we want to express Rf ∗ in terms of f (cid:92) . As a ﬁrst step, we show that Rf ∗ has bounded cohomological dimension. Proposition

VII.3.2 . Let f : Y → X be a proper map of small v-stacks that is representablein spatial diamonds with dim . trg f < ∞ , and cohomologically smooth. Then Rf ∗ : D (cid:4) ( Y, Λ) → D (cid:4) ( X, Λ) has bounded cohomological dimension and commutes with arbitrary direct sums. If X isa spatial diamond (thus Y is) and F is a ﬁnitely presented solid (cid:98) Z p -sheaf on Y , then Rf ∗ F is abounded complex all of whose cohomology sheaves are ﬁnitely presented solid (cid:98) Z p -sheaves on X . Proof.

We can assume Λ = (cid:98) Z p . The commutation with arbitrary direct sums follows frombounded cohomological dimension, as one can then reduce to the case of complexes concentratedin degree 0, where Rf ∗ commutes with all direct sums as f is qcqs. For the claim about boundedcohomological dimension, we can argue v-locally, and hence assume that X is a spatial diamond.It suﬃces to prove that for all solid (cid:98) Z p -sheaves F on Y , the complex Rf ∗ F is bounded; thisreduces to the case of ﬁnitely presented solid (cid:98) Z p -sheaves as Rf ∗ commutes with ﬁltered colimits ofsheaves. Now if F is ﬁnitely presented, it is a coﬁltered limit of constructible ´etale sheaves killedby some integer prime to p . As Rf ∗ commutes with this limit, it is now enough to see that Rf ∗ preserves constructible complexes and has bounded amplitude. But this follows from cohomologicalsmoothness, cf. [ Sch17a , Proposition 23.12 (ii)]. (cid:3)

Next, we prove a projection formula for Rf ∗ . Proposition

VII.3.3 . Let f : Y → X be a proper map of small v-stacks that is representablein spatial diamonds with dim . trg f < ∞ , and cohomologically smooth. Then for all A ∈ D (cid:4) ( Y, Λ) and B ∈ D (cid:4) ( X, Λ) , the projection map Rf ∗ A (cid:4) ⊗ L B → Rf ∗ ( A (cid:4) ⊗ L f ∗ B ) II.3. RELATIVE HOMOLOGY 253 is an isomorphism.

Proof.

We can assume Λ = (cid:98) Z p . We note that Rf ∗ and (cid:4) ⊗ L both have bounded cohomologicaldimension, so one easily reduces to the case that A and B are concentrated in degree 0. We canalso assume that X is a spatial diamond (thus Y is, too). Then we can write A and B as ﬁlteredcolimits of ﬁnitely presented solid (cid:98) Z p -sheaves, and reduce to the case that A and B are coﬁlteredlimits of constructible ´etale sheaves killed by some integer prime to p . In that case, it followsfrom Proposition VII.2.3 and Proposition VII.3.2 that all operations commute with these coﬁlteredlimits, and one reduces to the case that A and B are constructible ´etale sheaves killed by someinteger prime to p . Now it follows from [ Sch17a , Proposition 22.11]. (cid:3)

Moreover, the functor Rf ∗ interacts well with g (cid:92) for maps g : X (cid:48) → X . Proposition

VII.3.4 . Let Y (cid:48) g (cid:48) (cid:47) (cid:47) f (cid:48) (cid:15) (cid:15) Y f (cid:15) (cid:15) X (cid:48) g (cid:47) (cid:47) X be a cartesian diagram of small v-stacks, where f : Y → X is proper, representable in spatialdiamonds, with dim . trg f < ∞ and cohomologically smooth. Then the natural transformation g (cid:92) Rf (cid:48)∗ A → Rf ∗ g (cid:48) (cid:92) A is an isomorphism for all A ∈ D (cid:4) ( Y (cid:48) , Λ) . Proof.

We can assume Λ = (cid:98) Z p . By Proposition VII.3.2 both sides commute with Postnikovlimits, so we can assume A ∈ D + , and then reduce to the case that A is concentrated in degree0. We may assume that X is a spatial diamond, and one can also reduce to the case X (cid:48) is aspatial diamond, by writing A as the geometric realization of h (cid:48)• (cid:92) h (cid:48) (cid:92) • A for some simplicial hyper-cover h • : X (cid:48)• → X (cid:48) by disjoint unions of spatial diamonds, and its pullback h (cid:48)• : Y (cid:48)• → Y (cid:48) (andusing Proposition VII.3.2 to commute the geometric realization with pushforward). Under thesecircumstances, one can write A as a ﬁltered colimit of ﬁnitely presented solid (cid:98) Z p -modules, andhence reduce to the case that A is a coﬁltered limit of constructible ´etale sheaves killed by someinteger prime to p . By Proposition VII.3.2 the complex Rf (cid:48)∗ A is then bounded with all cohomologysheaves ﬁnitely presented solid (cid:98) Z p -modules. As g (cid:92) preserves pseudocoherent objects, it follows thatthe map g (cid:92) Rf (cid:48)∗ A → Rf ∗ g (cid:48) (cid:92) A is a map of bounded to the right complexes in D (cid:4) ( X, (cid:98) Z p ) all of whosecohomology sheaves are ﬁnitely presented solid (cid:98) Z p -modules. If the cone of this map is nonzero,then by looking at its ﬁrst nonzero cohomology sheaf, we ﬁnd some nonzero map to a constructible´etale sheaf B on X , killed by some integer prime to p . Note that, using the usual ´etale Rf ! functor,there is a natural adjunction R Hom( Rf ∗ g (cid:48) (cid:92) A, B ) ∼ = R Hom( g (cid:48) (cid:92) A, Rf ! B ) :Indeed, it suﬃces to check this when g (cid:48) (cid:92) A is replaced by a ﬁnitely presented solid (cid:98) Z p -module, by aPostnikov tower (and as all cohomology sheaves of g (cid:48) (cid:92) A are of this form). Writing this as a coﬁltered

54 VII. D (cid:4) ( X ) limit of constructible ´etale sheaves killed by some integer prime to p , both sides turn this coﬁlteredlimit into a ﬁltered colimit, so the claim reduces to the usual ´etale adjunction.Applying R Hom( − , B ) to the map g (cid:92) Rf (cid:48)∗ A → Rf ∗ g (cid:48) (cid:92) A will thus produce R Hom( A, − ) appliedto the base change map Rf (cid:48) ! g ∗ B ← g (cid:48)∗ Rf ! B, which is an isomorphism by [ Sch17a , Proposition 23.12 (iii)]. (cid:3)

Now we can describe the functor Rf ∗ . Indeed, consider the diagram Y ∆ f (cid:47) (cid:47) Y × X Y π (cid:47) (cid:47) π (cid:15) (cid:15) Y f (cid:15) (cid:15) Y f (cid:47) (cid:47) X. Then, under our assumption that f : Y → X is a proper map of small v-stacks that is representablein spatial diamonds with dim . trg f < ∞ and cohomologically smooth, we have Rf ∗ A ∼ = Rf ∗ π (cid:92) ∆ f(cid:92) A ∼ = f (cid:92) Rπ ∗ ∆ f(cid:92) A ∼ = f (cid:92) Rπ ∗ ∆ f(cid:92) ∆ ∗ f π ∗ A ∼ = f (cid:92) Rπ ∗ (∆ f(cid:92) Λ (cid:4) ⊗ L Λ π ∗ A ) ∼ = f (cid:92) ( Rπ ∗ ∆ f(cid:92) Λ (cid:4) ⊗ L Λ A ) . We combine this with the following observation.

Proposition

VII.3.5 . Let f : Y → X be a proper map of small v-stacks that is representablein spatial diamonds with dim . trg f < ∞ and cohomologically smooth. Then Rπ ∗ ∆ f(cid:92) Λ ∈ D (cid:4) ( Y, Λ) is invertible, and its inverse is canonically isomorphic to Rf ! Λ := lim ←− n Rf ! Z /n Z ⊗ L (cid:98) Z p Λ . Thus, there is a canonical isomorphism f (cid:92) A ∼ = Rf ∗ ( A (cid:4) ⊗ L Λ Rf ! Λ) : D (cid:4) ( Y, Λ) → D (cid:4) ( X, Λ) . Thus, we get a somewhat unusual formula for the dualizing complex. We remark that the ﬁbresof Rπ ∗ ∆ f(cid:92) (cid:98) Z p are given by the limit of R Γ c ( U, (cid:98) Z p ) over all ´etale neighborhoods U of the givengeometric point. Remark

VII.3.6 . We see here that an important instance of Rf ! admits an alternative de-scription in terms of g (cid:92) functors. We are a bit confused about exactly how expressive the present5-functor formalism is. So far, we were always able to translate any argument in terms of a 6-functor formalism into this 5-functor formalism, although it is often a nontrivial matter and thereseems to be no completely general recipe. II.4. RELATION TO D ´et Proof.

We can assume Λ = (cid:98) Z p . By the isomorphism Rf ∗ ∼ = f (cid:92) ( Rπ ∗ ∆ f(cid:92) (cid:98) Z p (cid:4) ⊗ L A ), it followsthat Rf ∗ : D (cid:4) ( Y, (cid:98) Z p ) → D (cid:4) ( X, (cid:98) Z p ) admits a right adjoint, given by A (cid:55)→ R H om( Rπ ∗ ∆ f(cid:92) (cid:98) Z p , f ∗ A ) . We claim that this right adjoint maps D ´et ( X, Z /n Z ) into D ´et ( Y, Z /n Z ) for any n prime to p , andthus agrees with the right adjoint Rf ! in that setting. This claim can be checked v-locally, so wecan assume that X is a spatial diamond. Then Rπ ∗ ∆ f(cid:92) (cid:98) Z p ∈ D (cid:4) ( Y, (cid:98) Z p ) is a bounded complexall of whose cohomology sheaves are ﬁnitely presented solid, by Proposition VII.3.2 and as ∆ f is quasi-pro-´etale (so ∆ f(cid:92) (cid:98) Z p is ﬁnitely presented solid). This implies that R H om( Rπ ∗ ∆ f(cid:92) (cid:98) Z p , − )preserves D ´et ( Y, Z /n Z ).Thus, for any A ∈ D ´et ( X, Z /n Z ), there is a natural isomorphism Rf ! A ∼ = R H om( Rπ ∗ ∆ f(cid:92) (cid:98) Z p , f ∗ A ) . Applied with A = Z /n Z , this gives isomorphisms Rf ! Z /n Z ∼ = R H om( Rπ ∗ ∆ f(cid:92) (cid:98) Z p , Z /n Z ) . It remains to see that Rπ ∗ ∆ f(cid:92) (cid:98) Z p is invertible; more precisely, we already get a natural map Rπ ∗ ∆ f(cid:92) (cid:98) Z p → ( Rf ! (cid:98) Z p ) − that we want to prove is an isomorphism. This can again be checked v-locally, so we can assumethat X is a spatial diamond. Then Rπ ∗ ∆ f(cid:92) (cid:98) Z p is a bounded complex all of whose cohomologysheaves are ﬁnitely presented solid; so as in the proof of Proposition VII.3.4, it is enough to checkthat one gets isomorphisms after applying R H om( − , B ) for any B ∈ D ´et ( Y, Z /n Z ). But R H om( Rπ ∗ ∆ f(cid:92) (cid:98) Z p , B ) ∼ = Rπ ∗ R H om(∆ f(cid:92) (cid:98) Z p , Rπ !1 B ) ∼ = ∆ ∗ f Rπ !1 B ∼ = B ⊗ LZ /n Z Rf ! Z /n Z , giving the result.The ﬁnal statement follows formally from the identiﬁcation of Rπ ∗ ∆ f(cid:92) Λ and the discussionleading up to the proposition. (cid:3)

VII.4. Relation to D ´et Assume now that Λ is discrete. In particular, also being a (cid:98) Z p -algebra, we have n Λ = 0 forsome n prime to p . We wish to understand the relation between D ´et ( X, Λ) and D (cid:4) ( X, Λ), and thefunctors deﬁned on them.

VII.4.1. Naive embedding.

For any small v-stack X , we have a fully faithful embedding D ´et ( X, Λ) (cid:44) → D (cid:4) ( X, Λ)as full subcategories of D ( X v , Λ). As usual, the adjoint functor theorem implies that this admits aright adjoint R X ´et : D (cid:4) ( X, Λ) → D ´et ( X, Λ). The full inclusion D ´et ( X, Λ) ⊂ D (cid:4) ( X, Λ) is symmetric

56 VII. D (cid:4) ( X ) monoidal, and compatible with pullback. Moreover, by [ Sch17a , Proposition 17.6], it also com-mutes with Rf ∗ if f : Y → X is qcqs and one restricts to D + ; or in general f is qcqs and of ﬁnitecohomological dimension. Moreover, one always has R X ´et Rf ∗ ∼ = Rf ∗ R Y ´et . Similarly, passing to right adjoints in the commutation with tensor products, we also have R X ´et R H om D ´et ( X, Λ) ( A, B ) ∼ = R H om D (cid:4) ( X, Λ) ( A, R X ´et B )if A ∈ D ´et ( X, Λ) and B ∈ D (cid:4) ( X, Λ). If A is perfect-constructible, then for all B ∈ D ´et ( X, Λ), oneactually has R H om D ´et ( X, Λ) ( A, B ) ∼ = R H om D (cid:4) ( X, Λ) ( A, B ) :by descent, it suﬃces to check this when X is spatial diamond, and then one reduces to A = j ! Λ forsome quasicompact separated ´etale map j : U → X . In that case, it follows from Rj ∗ commutingwith the embedding D ´et ( X, Λ) → D (cid:4) ( X, Λ), as it is qcqs and has cohomological dimension 0.

VII.4.2. Dual embedding.

For a small v-stack X , let D † ´et ( X, Λ) ⊂ D ´et ( X, Λ) be the fullsubcategory of overconvergent objects. Recall that A ∈ D ´et ( X, Λ) is overconvergent if for anystrictly local Spa(

C, C + ) → X , the map R Γ(Spa(

C, C + ) , A ) → R Γ(Spa( C, O C ) , A )is an isomorphism. Proposition

VII.4.1 . Assume that

Λ = Z /n Z with n prime to p . For any overconvergent A ∈ D † ´et ( X, Λ) , let A ∨ = R H om D (cid:4) ( X, Λ) ( A, Λ) ∈ D (cid:4) ( X, Λ) . Then the functor D † ´et ( X, Λ) op → D (cid:4) ( X, Λ) : A (cid:55)→ A ∨ is fully faithful, t -exact (for the standard t -structure), compatible with pullback, and the map A → R H om D (cid:4) ( X, Λ) ( A ∨ , Λ) is an isomorphism. Proof.

As the formation of R H om in the solid context commutes with any base change, allassertions can be proved by v-descent, so we can assume that X is strictly totally disconnected.Then D † ´et ( X, Λ) ∼ = D ( π X, Λ). The heart of the standard t -structure is then an abelian categorywith compact projective generators i ∗ Λ for open and closed subsets i : S ⊂ π X , and the wholecategory is the Ind-category of the constructible complexes of Λ-modules on π X (which are locallyconstant with ﬁnite ﬁbres). Passage to the naive dual is an autoequivalence on constructiblecomplexes (as Λ is selﬁnjective), and thus embeds the whole Ind-category fully faithfully into thePro-category of constructible complexes of Λ-modules on π X , which sits fully faithfully inside thecategory of ﬁnitely presented solid sheaves on X . This already establishes that the functor is fullyfaithful and t -exact, and we already observed at the beginning that it commutes with any pullback.It remains to prove that A → R H om D (cid:4) ( X, Λ) ( A ∨ , Λ) II.4. RELATION TO D ´et is an isomorphism. Again, we can assume Λ = F (cid:96) so that all operations are t -exact. Again, thestatement is clear if A is constructible, and in general it follows from Breen’s resolution that thePro-structure on A ∨ dualizes to a ﬁltered colimit on applying R H om D (cid:4) ( X, Λ) ( − , B ). (cid:3) The functor A (cid:55)→ A ∨ is also close to being symmetric monoidal. Note that it is lax-symmetricmonoidal, i.e. there is a natural functorial map A ∨ (cid:4) ⊗ L Λ B ∨ → ( A ⊗ L Λ B ) ∨ . Proposition

VII.4.2 . Assume that A ∈ D † ´et ( X, Λ) has ﬁnite Tor-amplitude over Λ = Z /n Z ,i.e. for all quotients Λ → F (cid:96) , the complex A ⊗ L Λ F (cid:96) ∈ D † ´et ( X, F (cid:96) ) is bounded. Then for all B ∈ D † ´et ( X, Λ) , the maps A ∨ (cid:4) ⊗ L Λ B ∨ → ( A ⊗ L Λ B ) ∨ , A ⊗ L Λ B → R H om D (cid:4) ( X, Λ) ( A ∨ , B ) are isomorphisms. Proof.

The second follows from the ﬁrst: Using Proposition VII.4.1, R H om D (cid:4) ( X, Λ) ( A ∨ , B ) ∼ = R H om D (cid:4) ( X, Λ) ( A ∨ , R H om D (cid:4) ( X, Λ) ( B ∨ , Λ)) ∼ = R H om D (cid:4) ( X, Λ) ( A ∨ (cid:4) ⊗ L Λ B ∨ , Λ) , which one can further rewrite to A ⊗ L Λ B assuming the ﬁrst isomorphism.We can assume Λ = F (cid:96) , and that A is concentrated in degree 0. Now as functors of B , alloperations are t -exact, so we can reduce to the case that also B is concentrated in degree 0. Wecan assume that X is strictly totally disconnected, and then D † ´et ( X, F (cid:96) ) ∼ = D ( π X, F (cid:96) ). Then A and B are ﬁltered colimits of constructible sheaves on π X , and R H om( − , F (cid:96) ) is a contravariantautoequivalence on constructible F (cid:96) -sheaves on π X . Then the result follows by observing that A (cid:55)→ A ∨ simply exchanges the Ind-category of constructible F (cid:96) -sheaves on π X with its Pro-category. (cid:3) As noted above, the functor A (cid:55)→ A ∨ is compatible with pullback. Regarding pushforward, wehave the following result. Proposition

VII.4.3 . Let f : Y → X be a proper map of small v-stacks that is representablein spatial diamonds with dim . trg f < ∞ . Let A ∈ D † ´et ( Y, Λ) with dual A ∨ ∈ D (cid:4) ( Y, Λ) . Then thereis a natural isomorphism ( Rf ∗ A ) ∨ ∼ = f (cid:92) A ∨ . Note that Rf ∗ A is again overconvergent, by proper base change. Proof.

One has R H om D (cid:4) ( X, Λ) ( f (cid:92) A ∨ , Λ) ∼ = Rf ∗ R H om D (cid:4) ( Y, Λ) ( A ∨ , Λ) ∼ = Rf ∗ A, so by biduality one gets a natural map f (cid:92) A ∨ → ( Rf ∗ A ) ∨ ;we claim that this is an isomorphism. This can be checked v-locally on X , so we can assume that X is w-contractible. One can assume A is bounded above (i.e. A ∈ D − ) as both functors take

58 VII. D (cid:4) ( X ) very coconnective objects to very connective objects; by shifting, we can assume A ∈ D ≤ . Nowusing a Postnikov limit and the assumption dim . trg f < ∞ , we can also assume that A ∈ D + , andhence reduce to A sitting in degree 0. Now we can choose a hypercover of Y by perfectoid spaces Y i that are the canonical compactiﬁcations (relative to X ) of w-contractible spaces. One can thenreplace Y by one of the Y i , so assume that Y is the canonical compactiﬁcation of a w-contractiblespace. In particular, D † ´et ( Y, Λ) ∼ = D ( π Y, Λ), and all operations can be computed on the levelof π f : π Y → π X instead. Here, the result amounts again to the duality between Ind- andPro-objects in the category of constructible sheaves on proﬁnite sets. (cid:3) VII.5. Dualizability

It turns out that most of the results above on Poincar´e duality hold verbatim if the assumptionthat f is cohomologically smooth is relaxed to the assumption that F (cid:96) is f -universally locally acyclicfor all (cid:96) (cid:54) = p . In fact, even more generally, one can obtain certain results comparing twisted formsof f (cid:92) and Rf ∗ for any f -universally locally acyclic complex A .Assume that Λ is a quotient of (cid:98) Z p of the form lim ←− n Z /n Z where n now runs only over someintegers prime to p . If f : X → S is a compactiﬁable map of small v-stacks that is representablein locally spatial diamonds with locally dim . trg f < ∞ , we deﬁne the category D ULA ( X/S,

Λ) of f -universally locally acyclic complexes with coeﬃcients Λ as the limit of the full subcategories D ULA ( X/S, Z /n Z ) ⊂ D ´et ( X, Z /n Z )of f -universally locally acyclic objects in D ´et ( X, Z /n Z ), for n running over the same set of integersprime to p . Equivalently, D ULA ( X/S,

Λ) is the category of all A ∈ D (cid:4) ( X, Λ) such that A n = A ⊗ L Λ Z /n Z lies in D ´et ( X, Z /n Z ) for all such n , is f -universally locally acyclic, and A is the derivedlimit of the A n .Given such an A , in particular all A n are overconvergent, and the functor A (cid:55)→ A ∨ = R H om D (cid:4) ( X, Λ) ( A, Λ) = R lim ←− n A ∨ n ∈ D (cid:4) ( X, Λ)deﬁnes another fully faithful (contravariant) embedding D ULA ( X/S, Λ) op (cid:44) → D (cid:4) ( X, Λ)of f -universally locally acyclic complexes into D (cid:4) ( X, Λ). We can also precompose with Verdierduality D X/S to obtain a covariant fully faithful embedding D ULA ( X/S, Λ) (cid:44) → D (cid:4) ( X, Λ) : A (cid:55)→ D X/S ( A ) ∨ . Example

VII.5.1 . Assume that S = Spa C is a geometric point, and X is the analytiﬁcationof an algebraic variety X alg / Spec C . Then any constructible complex on X alg is universally locallyacyclic over S , yielding a fully faithful embedding D bc ( X alg , Z (cid:96) ) (cid:44) → D ULA ( X/S, Z (cid:96) ) (cid:44) → D (cid:4) ( X, Z (cid:96) ) , embedding the usual bounded derived category of constructible Z (cid:96) -sheaves on X alg into D (cid:4) ( X, Z (cid:96) ).The image lands in bounded complexes with ﬁnitely presented solid cohomology sheaves; in fact,in compact objects. Thus, this fully faithful embedding extends to a fully faithful embeddingInd D bc ( X alg , Z (cid:96) ) (cid:44) → D (cid:4) ( X, Z (cid:96) ) . II.5. DUALIZABILITY 259

The category on the left is the one customarily associated to X alg . This functor takes the sheaf i ∗ Z (cid:96) , for a point i : Spec C → X alg , to the solid sheaf i (cid:92) Z (cid:96) .In many papers in geometric Langlands and related ﬁelds, one often ﬁnds the following con-struction. If Y is a stack on the category of schemes over Spec C , let D ( Y, Z (cid:96) ) := lim ←− X alg → Y Ind D bc ( X alg , Z (cid:96) )where X alg runs over schemes of ﬁnite type over Spec C , and the transition functors are given by Rf ! . This, in fact, embeds naturally into D (cid:4) ( Y ♦ , Z (cid:96) ) via the previous embedding, noting that itintertwines Rf ! with the usual pullback f ∗ on solid sheaves. In fact, D X (cid:48) /S ( Rf ! A ) ∨ ∼ = ( f ∗ D X/S ( A )) ∨ ∼ = f ∗ D X/S ( A ) ∨ for a map f : X (cid:48) alg → X alg of algebraic varieties over Spec C .Now for A ∈ D ULA ( X/S,

Λ), we analyze the functor f (cid:92) ( D X/S ( A ) ∨ (cid:4) ⊗ L Λ − ) : D (cid:4) ( X, Λ) → D (cid:4) ( S, Λ) . We note that from the deﬁnition one sees that this functor commutes with all colimits, the formationof this functor commutes with any base change, and it satisﬁes the projection formula. In fact, thisfunctor extends the functor Rf ! ( A ⊗ L Λ − ). Proposition

VII.5.2 . Assume that A ∈ D ULA ( X/S, Λ) has bounded Tor-amplitude. Let Z /n Z be a discrete quotient of Λ . For B ∈ D ´et ( X, Z /n Z ) , there is a natural equivalence f (cid:92) ( D X/S ( A ) ∨ (cid:4) ⊗ L Λ B ) ∼ = Rf ! ( A n ⊗ LZ /n Z B ) ∈ D ´et ( S, Z /n Z ) . Proof.

We can assume Λ = Z /n Z . Note that for any C ∈ D ´et ( S, Λ), one has R Hom Λ ( f (cid:92) ( D X/S ( A ) ∨ (cid:4) ⊗ L Λ B ) , C ) ∼ = R Hom Λ ( D X/S ( A ) ∨ (cid:4) ⊗ L Λ B, f ∗ C ) ∼ = R Hom Λ ( B, R H om D (cid:4) ( X, Λ) ( D X/S ( A ) ∨ , f ∗ C )) ∼ = R Hom Λ ( B, D X/S ( A ) ⊗ L Λ f ∗ C ) ∼ = R Hom Λ ( B, R H om D ´et ( X, Λ) ( A, Rf ! C )) ∼ = R Hom Λ ( A ⊗ L Λ B, Rf ! C ) ∼ = R Hom Λ ( Rf ! ( A ⊗ L Λ B ) , C ) . Here, we use Proposition VII.4.1 and Proposition IV.2.19. In particular, there is a natural map f (cid:92) ( D X/S ( A ) ∨ (cid:4) ⊗ L Λ B ) → Rf ! ( A ⊗ L Λ B ) . We claim that this is an isomorphism. This can be checked v-locally, so we can assume that S isstrictly totally disconnected. We can assume that X is a spatial diamond by localization. As thefunctor commutes with all colimits in B , we can also assume that B = j ! Λ for some quasicompactseparated ´etale j : V → X . Replacing X by V , we can then even assume B = Λ.Now D X/S ( A ) lies in D + and then again D X/S ( A ) ∨ in D − . It follows that D X/S ( A ) ∨ is acomplex that is bounded above, and ﬁnitely presented solid in each degree. Thus f (cid:92) D X/S ( A ) ∨ is

60 VII. D (cid:4) ( X ) of the same form, and so is the cone Q of f (cid:92) D X/S ( A ) ∨ → Rf ! A . If Q is nonzero, we can look atthe largest i such that H i ( Q ) is nonzero. This is ﬁnitely presented solid, so a coﬁltered limit ofconstructible ´etale sheaves. But R Hom(

Q, C ) = 0 for all C ∈ D ´et ( S, Λ), so it follows that indeed Q = 0. (cid:3) If f is moreover proper, one can also prove the following version of A -twisted Poincar´e duality. Proposition

VII.5.3 . Assume that f : X → S is a proper map of small v-stacks that isrepresentable in spatial diamonds with dim . trg f < ∞ . Let A ∈ D ULA ( X/S, Λ) with boundedTor-amplitude. Then there is a natural equivalence f (cid:92) ( D X/S ( A ) ∨ (cid:4) ⊗ L Λ − ) ∼ = Rf ∗ R H om D (cid:4) ( X, Λ) ( A ∨ , − ) of functors D (cid:4) ( X, Λ) → D (cid:4) ( S, Λ) . Proof.

First, we construct the natural transformation. Let π , π : X × S X → X be the twoprojections. Giving a map f (cid:92) ( D X/S ( A ) ∨ (cid:4) ⊗ L Λ B ) → Rf ∗ R H om D (cid:4) ( X, Λ) ( A ∨ , B )is equivalent to giving a map f ∗ f (cid:92) ( D X/S ( A ) ∨ (cid:4) ⊗ L Λ B ) (cid:4) ⊗ L Λ A ∨ → B. But f ∗ f (cid:92) ( D X/S ( A ) ∨ (cid:4) ⊗ L Λ B ) (cid:4) ⊗ L Λ A ∨ ∼ = π (cid:92) π ∗ ( D X/S ( A ) ∨ (cid:4) ⊗ L Λ B ) (cid:4) ⊗ L Λ A ∨ ∼ = π (cid:92) ( π ∗ D X/S ( A ) ∨ (cid:4) ⊗ L Λ π ∗ B (cid:4) ⊗ L Λ π ∗ A ∨ ) . Thus, it suﬃces to construct a functorial map π ∗ D X/S ( A ) ∨ (cid:4) ⊗ L Λ π ∗ A ∨ (cid:4) ⊗ L Λ π ∗ B → π ∗ B. For this in turn it suﬃces to construct a natural map π ∗ D X/S ( A ) ∨ (cid:4) ⊗ L Λ π ∗ A ∨ → ∆ (cid:92) Λwhere ∆ :

X (cid:44) → X × S X is the diagonal. Here ∆ (cid:92) Λ ∼ = (∆ ∗ Λ) ∨ by Proposition VII.4.3 and π ∗ D X/S ( A ) ∨ (cid:4) ⊗ L Λ π ∗ A ∨ ∼ = ( π ∗ D X/S ( A ) ⊗ L Λ π ∗ A ) ∨ by Proposition VII.4.2. Thus, we have to ﬁnd amap ∆ ∗ Λ → π ∗ D X/S ( A ) ⊗ L Λ π ∗ A or equivalently a section of R ∆ ! ( π ∗ D X/S ( A ) ⊗ L Λ π ∗ A ). But π ∗ D X/S ( A ) ⊗ L Λ π ∗ A ∼ = R H om( π ∗ A, Rπ !1 A )as A is f -universally locally acyclic, and then R ∆ ! ( π ∗ D X/S ( A ) ⊗ L Λ π ∗ A ) ∼ = R ∆ ! R H om( π ∗ A, Rπ !1 A ) ∼ = R H om( A, A ) , where we ﬁnd the identity section.To show that the map is an isomorphism, we can now localize on S , and in particular assumethat S is strictly totally disconnected. By the bounded assumption on A and ﬁnite cohomologicaldimension of f , the functor Rf ∗ R H om Λ ( A ∨ , − ) commutes with all direct sums, and hence wecan assume that B is ﬁnitely presented solid (concentrated in degree 0). Then we can write B II.5. DUALIZABILITY 261 as a coﬁltered limit of constructible ´etale sheaves, and the left-hand side commutes with suchlimits; so we can reduce to B being a constructible ´etale sheaf, where the result follows fromProposition VII.5.2 and Proposition VII.4.1. (cid:3) From the perspective of using sheaves as kernels of induced functors, we have the followingpicture. We can introduce a variant of the category C S introduced above. Namely, for any smallv-stack S , let us consider the 2-category C S, (cid:4) whose objects are relatively 0-truncated small v-stacks X over S , and whose categories of morphismsFun C S, (cid:4) ( X, Y ) = D (cid:4) ( X × S Y, Λ)are given by solid complexes. Again, to any X ∈ C S, (cid:4) , we can associate the triangulated category D (cid:4) ( X, Λ) and to any A ∈ D (cid:4) ( X × S Y, Λ) the functor p (cid:92) ( A (cid:4) ⊗ L Λ p ∗ ) : D (cid:4) ( X, Λ) → D (cid:4) ( Y, Λ)with kernel A . The composition in C S, (cid:4) is deﬁned by the convolution D (cid:4) ( X × S Y, Λ) × D (cid:4) ( Y × S Z, Λ) → D (cid:4) ( X × S Z, Λ) : (

A, B ) (cid:55)→ A (cid:63) B = p (cid:92) ( p ∗ A (cid:4) ⊗ L Λ p ∗ B ) . We wish to compare C S and C S, (cid:4) . Note that the naive embedding D ´et ( X × S Y, Λ) (cid:44) → D (cid:4) ( X × S Y, Λ) is not compatible with the convolution (as one employs Rπ while the other employs π (cid:92) ).On the other hand, we can restrict to the sub-2-category C † S ⊂ C S whose objects are only the proper X/S representable in spatial diamonds of ﬁnite dim . trg, and withFun C † S ( X, Y ) = D † ´et , ftor ( X × S Y, Λ) , where the subscript ftor stands for ﬁnite Tor-dimension over Λ. Then there is a fully faithfulembedding C † S (cid:44) → C co S, (cid:4) where the superscript co means that we change the direction of the arrowswithin each Fun C S, (cid:4) ( X, Y ). Indeed, for any

X, Y ∈ C † S , the functor A (cid:55)→ A ∨ deﬁnes a fully faithfulembedding Fun C † S ( X, Y ) = D † ´et , ftor ( X × S Y, Λ) (cid:44) → D (cid:4) ( X × S Y, Λ) op = Fun C co S, (cid:4) ( X, Y ) . This is compatible with composition by Proposition VII.4.2 and Proposition VII.4.3. This discus-sion leads to another proof of Proposition VII.5.3:

Corollary

VII.5.4 . Let f : X → S be a proper map of small v-stacks that is representable inspatial diamonds with dim . trg f < ∞ . Let A ∈ D ´et ( X, Λ) be f -universally locally acyclic and ofﬁnite Tor-dimension over Λ . Then A ∨ ∈ D (cid:4) ( X, Λ) = Fun C S, (cid:4) ( X, S ) is right adjoint to D X/S ( A ) ∨ ∈ D (cid:4) ( X, Λ) = Fun C S, (cid:4) ( S, X ) . In particular, the functor f (cid:92) ( A ∨ (cid:4) ⊗ L Λ − ) : D (cid:4) ( X, Λ) → D (cid:4) ( S, Λ) is right adjoint to the functor D X/S ( A ) ∨ (cid:4) ⊗ L Λ f ∗ − : D (cid:4) ( S, Λ) → D (cid:4) ( X, Λ) , so f (cid:92) ( A ∨ (cid:4) ⊗ L Λ − ) ∼ = Rf ∗ R H om Λ ( D X/S ( A ) ∨ , − ) : D (cid:4) ( X, Λ) → D (cid:4) ( S, Λ) .

62 VII. D (cid:4) ( X ) Moreover, when applied to the Satake category, we get a fully faithful embedding(Sat IG ) op (cid:44) → D (cid:4) ( H ck IG , (cid:98) Z p ) : A (cid:55)→ A ∨ compatible with the monoidal structure (and functorially in I ), where the right-hand side is givenby Fun C S, (cid:4) ([(Div ) IX /L +(Div X ) I G ] , [(Div ) IX /L +(Div X ) I G ])for S = [(Div X ) I /L (Div X ) I G ]. Precomposing with Verdier duality, we get a covariant fully faithfulembedding Sat IG (cid:44) → D (cid:4) ( H ck IG , (cid:98) Z p ) : A (cid:55)→ D X/S ( A ) ∨ . By Proposition VII.5.2, when one uses objects in the Satake category as kernels to deﬁne Heckeoperators, this fully faithful embedding makes it possible to extend Hecke operators from D ´et to D (cid:4) . VII.6. Lisse sheaves

The category D (cid:4) ( X, Λ) is huge: Already if X is a point and Λ = F (cid:96) , it is the derived categoryof solid F (cid:96) -vector spaces, which is much larger than the category of usual discrete F (cid:96) -vector spaces.When applied to Bun G , we would however really like to study smooth representations on discreteΛ-modules.As coeﬃcients, we will from now on choose a discrete Z (cid:96) -algebra Λ for some (cid:96) (cid:54) = p , or ratherthe corresponding condensed ring Λ := Z (cid:96) ⊗ Z (cid:96), disc Λ disc . (For a technical reason, we have to restrictattention to a particular prime (cid:96) .)It turns out that when X is an Artin v-stack, one can deﬁne a full subcategory D lis ( X, Λ) ⊂ D (cid:4) ( X, Λ) that when specialized to X = Bun G has the desired properties. Here the subscript “lis”is an abbreviation of “lisse” (french smooth), and is not meant to evoke lisse sheaves in the senseof locally constant sheaves, but lisse-´etale sheaves in the sense of Artin stacks [ LMB00 ]. Definition

VII.6.1 . Let X be an Artin v-stack. The full subcategory D lis ( X, Λ) ⊂ D (cid:4) ( X, Λ) is the smallest triangulated subcategory stable under all direct sums that contains f (cid:92) Λ for all maps f : Y → X that are separated, representable in locally spatial diamonds, and (cid:96) -cohomologicallysmooth. In principle, one could give this deﬁnition even when X is any small v-stack, but in that casethere might be very few objects. Proposition

VII.6.2 . Let X be an Artin v-stack. The full subcategory D lis ( X, Λ) ⊂ D (cid:4) ( X, Λ) is stable under − (cid:4) ⊗ L Λ − . Moreover, if f : Y → X is a map of Artin v-stacks, then f ∗ maps D lis ( X, Λ) ⊂ D (cid:4) ( X, Λ) into D lis ( Y, Λ) ⊂ D (cid:4) ( Y, Λ) . Proof.

As tensor products and pullbacks commute with all direct sums, it suﬃces to checkthe claim on the generators g (cid:92) Λ for maps g : Z → X that are separated, representable in locallyspatial diamonds, and (cid:96) -cohomologically smooth. Now the result follows as pullbacks and productsof such maps are of the same form. (cid:3) II.6. LISSE SHEAVES 263

Proposition

VII.6.3 . Let X be an Artin v-stack. The inclusion D lis ( X, Λ) ⊂ D (cid:4) ( X, Λ) admitsa right adjoint A (cid:55)→ A lis : D (cid:4) ( X, Λ) → D lis ( X, Λ) . The kernel of A (cid:55)→ A lis is the class of all A ∈ D (cid:4) ( X, Λ) such that A ( Y ) = 0 for all f : Y → X thatare separated, representable in locally spatial diamonds, and (cid:96) -cohomologically smooth. Proof.

The existence of the right adjoint is formal. We note that the ∞ -category D (cid:4) ( X, Λ) isnot itself presentable, but rather is the large ﬁltered colimit of presentable ∞ -categories D (cid:4) ( X κ , Λ)for uncountable strong limit cardinals κ (restricting the v-site to κ -small perfectoid spaces). Alsonote that D lis ( X, Λ) is contained in D (cid:4) ( X κ , Λ) for some κ : This can be checked when X is a spatialdiamond and for the generators f (cid:92) Λ ∼ = f (cid:92) Z (cid:96) ⊗ LZ (cid:96) Λ when f : Y → X is in addition quasicompact,in which case f (cid:92) Z (cid:96) is the limit of f (cid:92) Z /(cid:96) m Z all of which lie in D ´et ( X, Z /(cid:96) m Z ), so we conclude by[ Sch17a , Remark 17.4]. It follows that the right adjoints to D lis ( X, Λ) → D (cid:4) ( X κ , Λ) for all largeenough κ glue to the desired right adjoint.The description of the kernel is formal. (cid:3) Using Proposition VII.6.3, we can then also deﬁne R H om lis ( A, B ) ∈ D lis ( X, Λ) for

A, B ∈ D lis ( X, Λ) and Rf lis ∗ : D lis ( Y, Λ) → D lis ( X, Λ) for a map f : Y → X of Artin v-stacks, satisfyingthe usual adjunction to the tensor product and pullback.The goal of passing to D lis is to make sheaves “discrete” again. Recall the following result. Proposition

VII.6.4 . For any condensed ring A with underlying ring A ( ∗ ) , the functor M (cid:55)→ M ⊗ A ( ∗ ) A induces a fully faithful functor D ( A ( ∗ )) (cid:44) → D ( A ) from the derived category of usual A ( ∗ ) -modules to the derived category of condensed modules overthe condensed ring A . Proof.

We need to see that for any

M, N ∈ D ( A ( ∗ )), the map R Hom A ( ∗ ) ( M, N ) → R Hom A ( M ⊗ A ( ∗ ) A, N ⊗ A ( ∗ ) A )is an isomorphism. The class of all M for which this happens is triangulated and stable under alldirect sums, so it suﬃces to consider M = A ( ∗ ). Then it amounts to N ( ∗ ) → ( N ⊗ A ( ∗ ) A )( ∗ )being an isomorphism, which follows from evaluation at ∗ being symmetric monoidal. (cid:3) In particular, we have the following result for a geometric point.

Proposition

VII.6.5 . Let X = Spa C for some complete algebraically closed nonarchimedeanﬁeld C . Then D lis ( X, Λ) ∼ = D (Λ) , the derived category of (relatively) discrete Λ -modules. Proof.

We need to see that for all separated (cid:96) -cohomologically smooth maps f : Y → X ofspatial diamonds, one has f (cid:92) Λ ∈ D (Λ). This reduces to Λ = Z (cid:96) . In that case, f (cid:92) Z (cid:96) = lim ←− m f (cid:92) Z /(cid:96) m Z ,where by Proposition VII.5.2 each f (cid:92) Z /(cid:96) m Z ∼ = Rf ! Rf ! Z /(cid:96) m Z ,

64 VII. D (cid:4) ( X ) which is a perfect complex of Z /(cid:96) m Z -modules, in particular discrete. Taking the limit over m , weget a perfect complex of Z (cid:96) -modules, which is in particular (relatively) discrete over Z (cid:96) . (cid:3) When working with torsion coeﬃcients, one recovers D ´et . Proposition

VII.6.6 . Let X be an Artin v-stack, and assume that Λ is killed by a power of (cid:96) . Then D lis ( X, Λ) ⊂ D (cid:4) ( X, Λ) is contained in the image of the naive embedding D ´et ( X, Λ) (cid:44) → D (cid:4) ( X, Λ) . If there is a separated (cid:96) -cohomologically smooth surjection U → X from a locally spatialdiamond U , such that U ´et has a basis with bounded (cid:96) -cohomological dimension, then it induces anequivalence D lis ( X, Λ) ∼ = D ´et ( X, Λ) . Proof. If f : Y → X is separated, representable in locally spatial diamonds, and (cid:96) -cohomologicallysmooth, then f (cid:92) Λ = Rf ! Rf ! Λ lies in D ´et ( X, Λ), hence D lis ( X, Λ) ⊂ D ´et ( X, Λ). To check equality,we can work on an atlas, so by the assumption we can reduce to the case that X is a locallyspatial diamond for which X ´et has a basis with bounded (cid:96) -cohomological dimension. In that case D ´et ( X, Λ) ∼ = D ( X ´et , Λ) by [

Sch17a , Proposition 20.17] (the proof only needs a basis with boundedcohomological dimension), which is generated by j ! Λ for j : U → X quasicompact separated ´etale,which is thus also contained in D lis ( X, Λ). (cid:3)

The most severe problem with the general formalism of solid sheaves is that stratiﬁcationsof a space do not lead to corresponding decompositions of sheaves into pieces on the individualstrata. This problem is somewhat salvaged by D lis ( X, Λ): We expect that it holds true if X and itsstratiﬁcation are suﬃciently nice. Here is a simple instance that will be suﬃcient for our purposes. Proposition

VII.6.7 . Let X be a locally spatial diamond with a closed point x ∈ X , giving acorresponding closed subdiamond i : Z ⊂ X with complement j : U ⊂ X . Assume that Z = Spa C is representable, with C an algebraically closed nonarchimedean ﬁeld. Moreover, assume that Z can be written as a coﬁltered intersection of quasicompact open neighborhoods V ⊂ X such that R Γ( V, F (cid:96) ) ∼ = F (cid:96) .Then one has a semi-orthogonal decomposition of D lis ( X, Λ) into D lis ( U, Λ) and D lis ( Z, Λ) ∼ = D (Λ) . Proof.

We may assume that X is spatial. We analyze the quotient of D lis ( X, Λ) by j ! D lis ( U, Λ).This is equivalently the subcategory of all A ∈ D lis ( X, Λ) with j ∗ A = 0. It is generated bythe images of f (cid:92) Λ for f : Y → X cohomologically smooth separated map of spatial diamonds;under the embedding of the quotient category back into D lis ( X, Λ), this corresponds to the coneof j ! j ∗ f (cid:92) Λ → f (cid:92) Λ. Let M = i ∗ f (cid:92) Λ ∈ D lis ( Z, Λ) ∼ = D (Λ), which in fact is a perfect complex ofΛ-modules (by the proof of Proposition VII.6.5). Then we claim that there is an isomorphismcone( j ! j ∗ f (cid:92) Λ → f (cid:92) Λ) ∼ = cone( j ! M → M ) . To see this, it suﬃces to prove that there is some open neighborhood V of Z such that f (cid:92) Λ | V ∼ = M ,the constant sheaf associated with M . We can reduce to Λ = Z (cid:96) . As f (cid:92) F (cid:96) is constructible, we canﬁnd some such V for which f (cid:92) F (cid:96) | V ∼ = M/(cid:96) . Picking such an isomorphism reducing to the identityat x , and choosing V with the property R Γ( V, F (cid:96) ) ∼ = F (cid:96) , we see that in fact the isomorphism liftsuniquely to Z /(cid:96) m Z for each m , and thus by taking the limit over m to the desired isomorphism f (cid:92) Λ | V ∼ = M . II.7. D lis (Bun G ) 265 Thus, the quotient of D lis ( X, Λ) by j ! D lis ( U, Λ) is generated by the constant sheaf Λ. Moreover,the endomorphisms of Λ in the quotient category are given by the cone of R Γ( X, j ! Λ) → R Γ( X, Λ) . This is equivalently the ﬁltered colimit of R Γ( V, Λ) over all open neighborhoods V of Z ; we canrestrict to those for which R Γ( V, F (cid:96) ) ∼ = F (cid:96) . This implies formally that R Γ( V, Z (cid:96) ) ∼ = Z (cid:96) by passingto limits and then R Γ( V, Λ) ∼ = Λ by passing to ﬁltered colimits. Thus, we get the desired semi-orthogonal decomposition. (cid:3) VII.7. D lis (Bun G )Our goal now is to extend the results of Chapter V to the case of D lis (Bun G , Λ). This willnotably include the case Λ = Q (cid:96) .Thus, let again be E any nonarchimedean local ﬁeld with residue ﬁeld F q and G a reduc-tive group over E . We work with Perf k where k = F q , and ﬁx a complete algebraically closednonarchimedean ﬁeld C/k . Proposition

VII.7.1 . Let b ∈ B ( G ) . The pullback functors D lis (Bun bG , Λ) → D lis ([ ∗ /G b ( E )] , Λ) → D lis ([Spa C/G b ( E )] , Λ) ,D lis (Bun bG , Λ) → D lis (Bun bG × Spa C, Λ) → D lis ([Spa C/G b ( E )] , Λ) are equivalences, and all categories are naturally equivalent (as symmetric monoidal categories) tothe derived category D ( G b ( E ) , Λ) of smooth representations of G b ( E ) on discrete Λ -modules. Proof.

Recall that the map s : [ ∗ /G b ( E )] → Bun bG is cohomologically smooth and surjective; infact, its ﬁbres are successive extensions of positive Banach–Colmez spaces. This implies that s (cid:92) Λ ∼ =Λ. This, in turn, implies by the projection formula for s (cid:92) that s (cid:92) s ∗ A ∼ = A for all A ∈ D (cid:4) (Bun bG , Λ),thus giving fully faithfulness. The same applies after base change to Spa C . Moreover, usingpullback under the projection Bun bG → [ ∗ /G b ( E )], we see that s ∗ is also necessarily essentiallysurjective.It remains to show that the pullback D lis ([ ∗ /G b ( E )] , Λ) → D lis ([Spa C/G b ( E )] , Λ) is an equiv-alence, and identify this symmetric monoidal category with D ( G b ( E ) , Λ). By Proposition VII.2.6,the functor D lis ([ ∗ /G b ( E )] , Λ) → D lis ([Spa C/G b ( E )] , Λ) is fully faithful. One can easily build afunctor D ( G b ( E ) , Λ) → D lis ([ ∗ /G b ( E )] , Λ), and it is enough to see that the composite functor D ( G b ( E ) , Λ) → D lis ([ ∗ /G b ( E )] , Λ) → D lis ([Spa C/G b ( E )] , Λ)is an equivalence. Using that D ( G b ( E ) , Λ) is generated by c -Ind G b ( E ) K Λ for K ⊂ G b ( E ) open pro- p ,one easily sees that the functor is fully faithful, so it remains to prove essential surjectivity. Usingdescent along Spa C → [Spa C/G b ( E )] and the equivalence D lis (Spa C, Λ) ∼ = D (Λ), the target mapsfully faithfully into the derived category of representations of the condensed group G b ( E ) on thefull subcategory of condensed Λ-modules of the form M ⊗ Z (cid:96), disc Z (cid:96) for Λ-modules M . As G b ( E )is locally pro- p , any such action in fact comes from a smooth action on M : For K ⊂ G b ( E ) pro- p , the K -orbit of any m ∈ M lies in some compact submodule, thus in M ⊗ Z (cid:96), disc Z (cid:96) for someﬁnitely generated Z (cid:96) -submodule M (cid:48) ⊂ M . The action of K on m then gives a continuous map

66 VII. D (cid:4) ( X ) K → GL( M (cid:48) ). As the target is locally pro- (cid:96) , this map has ﬁnite image, so that the action of K on m is locally constant. (cid:3) Recall that for any b ∈ B ( G ), we have the cohomologically smooth chart π b : M b → Bun G nearBun bG . This comes with a projection q b : M b → [ ∗ /G b ( E )] which has a natural section, given bythe preimage of Bun bG ⊂ Bun G in M b . Over M b , we have the G b ( E )-torsor (cid:102) M b → M b , and forany complete algebraically closed ﬁeld C over k = F q , the base change (cid:102) M b,C = (cid:102) M b × Spd k Spa C is representable by a locally spatial diamond, endowed with a distinguished point i : Spa C (cid:44) → (cid:102) M b,C .Recall that (cid:102) M b,C is a successive extension of negative Banach–Colmez spaces. Iteratively restrictingto small quasicompact balls inside these negative Banach–Colmez spaces, we see that the closedsubset i : Spa C (cid:44) → (cid:102) M b,C can be written as coﬁltered intersection of quasicompact open subsets V for which R Γ( V, F (cid:96) ) ∼ = F (cid:96) . Proposition

VII.7.2 . For any b ∈ B ( G ) with locally closed immersion i b : Bun bG → Bun G , thefunctor i b ∗ : D lis (Bun G , Λ) → D lis (Bun bG , Λ) ∼ = D lis ([ ∗ /G b ( E )] , Λ) admits a left adjoint, given by π b(cid:92) q ∗ b : D lis ([ ∗ /G b ( E )] , Λ) → D lis (Bun G , Λ) . The unit of the adjunction is given by the equivalence id ∼ = i b ∗ π b(cid:92) q ∗ b arising from base change, andthe identiﬁcation of the pullback of i b along π b with [ ∗ /G b ( E )] ⊂ M b . Proof. As D ([ ∗ /G b ( E )] , Λ) ∼ = D ( G b ( E ) , Λ) is generated by c -Ind G b ( E ) K Λ for open pro- p sub-groups K ⊂ G b ( E ), and as we already determined the unit of the adjunction, it suﬃces to verifythe adjunction on these objects. Let M b,K = (cid:102) M b /K → M b . This comes with a closed immersion i K : [ ∗ /K ] → M b,K . It suﬃces to see that for all A ∈ D lis ( M b,K , Λ), the map R Γ( M b,K , A ) → R Γ([ ∗ /K ] , A )is an isomorphism, where we continue to denote by A any of its pullbacks. Assume ﬁrst that A = j ! A for some A ∈ D lis ( M ◦ b,K , Λ). Then the result follows from Theorem VII.2.10. In generalwe can then replace A by the cone of j ! A → A in the displayed formula. For this statement,we can even base change to Spa C for some complete algebraically closed nonarchimedean ﬁeld C | k , and allow more generally any A ∈ D lis ( M b,K,C , Λ). We can then assume that A = f (cid:92) Z (cid:96) forsome (cid:96) -cohomologically smooth separated qcqs map f : Y → M b,K,C . Then as in the proof ofProposition VII.6.7, A is constant in a neighborhood of [Spa C/K ], which implies the result (asSpa C ⊂ (cid:102) M b,C is a coﬁltered intersection of quasicompact open V ’s with trivial cohomology). (cid:3) Proposition

VII.7.3 . For any quasicompact open substack U ⊂ Bun G , the category D lis ( U, Λ) admits a semi-orthogonal decomposition into the categories D lis (Bun bG , Λ) ∼ = D ( G b ( E ) , Λ) for b ∈| U | ⊂ B ( G ) . Moreover, for any not necessarily quasicompact U , the functor D lis ( U, Λ) → D lis ( U × Spd k Spa C, Λ) is an equivalence. II.7. D lis (Bun G ) 267 Proof.

We argue by induction on | U | , so take some closed element b ∈ | U | ⊂ B ( G ) andlet i : Bun bG → U and j : V → U be the closed and complementary open substacks. We knowthat D lis ( U, Λ) → D lis ( U × Spd k Spa C, Λ) is fully faithful by Proposition VII.2.6, and by induction D lis ( V, Λ) → D lis ( V × Spd k Spa C, Λ) is an equivalence.Now by the previous proposition, i b ∗ admits the left adjoint π b(cid:92) q ∗ b : D lis ([ ∗ /G b ( E )] , Λ) → D lis ( U, Λ) , and in fact the proof of that proposition shows (using our standing induction assumption) that,composed with the embedding into D lis ( U × Spd k Spa C, Λ), it continues to be a left adjoint to i b ∗ : D lis ( U × Spd k Spa C, Λ) → D ([Spa C/G b ( E )] , Λ) ∼ = D ([ ∗ /G b ( E )] , Λ).The unit id → i b ∗ π b(cid:92) q ∗ b of the adjunction is an equivalence. We see that D lis ( U × Spd k Spa C, Λ)has full subcategories given by j ! D lis ( V, Λ) and the essential image of π b(cid:92) q ∗ b (both of which liein D lis ( U, Λ)). To see that one has a semi-orthogonal decomposition, it suﬃces to see that if A ∈ D lis ( U × Spd k Spa C, Λ) with i ∗ A = j ∗ A = 0, then A = 0. This can be checked after pullback to (cid:102) M b,C ,where it follows from Proposition VII.6.7. This also shows that D lis ( U, Λ) → D lis ( U × Spd k Spa C, Λ)is an equivalence. (cid:3)

Now we also want to analyze the compact objects as well as the universally locally acyclicobjects, and various dualities. We start with the compact objects.

Proposition

VII.7.4 . The category D lis (Bun G , Λ) is compactly generated. An object A ∈ D lis (Bun G , Λ) is compact if and only if it has ﬁnite support and i b ∗ A ∈ D lis (Bun bG , Λ) ∼ = D ( G b ( E ) , Λ) is compact for all b ∈ B ( G ) , i.e. lies in the thick triangulated subcategory generated by c - Ind G b ( E ) K Λ for open pro- p subgroups K ⊂ G b ( E ) .Moreover, for each b and K ⊂ G b ( E ) pro- p , letting f K : M b,K → Bun G be the natural map, the object A bK = f K(cid:92) Λ ∈ D lis (Bun G , Λ) is compact, and these generate D lis (Bun G , Λ) . Proof.

By Proposition VII.7.3, the left adjoints π b(cid:92) q ∗ b to i b ∗ generate D lis (Bun G , Λ); as i b ∗ commutes with colimits, these left adjoints also preserve compact objects. As each D ( G b ( E ) , Λ) iscompactly generated, it follows that D lis (Bun G , Λ) is compactly generated, with compact generators A bK .To see that the given property characterizes compact objects, we argue by induction overquasicompact open substacks U ⊂ Bun G . Pick any closed b ∈ | U | ⊂ B ( G ), and assume the resultfor the complementary open j : V ⊂ U . We ﬁrst show that all of the given compact generators(coming from b (cid:48) ∈ | U | ⊂ B ( G )) have the property that all of their stalks are compact. Thisis clear by induction if b (cid:48) ∈ | V | , so we can assume b (cid:48) = b . Then we need to see that j ∗ π b(cid:92) q ∗ b preserves compact objects. But this follows from Lemma VII.7.5 below. Using the semi-orthogonaldecomposition structure, it now follows that conversely, all A with compact stalks are compact. (cid:3) Lemma

VII.7.5 . For K ⊂ G b ( E ) an open pro- p subgroup, the functor R Γ( M ◦ b,K , − ) : D (cid:4) ( M ◦ b,K , Λ) → D (Λ)

68 VII. D (cid:4) ( X ) has ﬁnite cohomological dimension and commutes with all direct sums. Proof. As M ◦ b,K = (cid:102) M ◦ b /K where (cid:102) M ◦ b is a spatial diamond, it suﬃces to prove that the functorhas ﬁnite cohomological dimension. It suﬃces to prove this for (cid:102) M ◦ b (as taking K -invariants is exact).One can formally reduce to Λ = Z (cid:96) and then to ﬁnitely presented solid Z (cid:96) -sheaves F on (cid:102) M ◦ b . Nowthese can be written as coﬁltered inverse limits of constructible F i . The R Γ( (cid:102) M ◦ b , F i ) are uniformlybounded; to see that their derived limit is also bounded, it is then suﬃcient to see that each H j ( (cid:102) M ◦ b , F i ) is ﬁnite. By Theorem IV.5.3, this is isomorphic to H j +1 c ( (cid:102) M ◦ b , F i ). But R Γ c ( (cid:102) M ◦ b , − )preserves compact objects as its right adjoint commutes with all colimits (as (cid:102) M ◦ b is cohomologicallysmooth over Spd k , being open in a successive extension of negative Banach–Colmez spaces). (cid:3) Next, we study Bernstein–Zelevinsky duality. Denoting π : Bun G → ∗ the projection, thepullback π ∗ has a left adjoint π (cid:92) : D lis (Bun G , Λ) → D lis ( ∗ , Λ) ∼ = D (Λ) . This induces a pairing D lis (Bun G , Λ) × D lis (Bun G , Λ) → D (Λ) : ( A, B ) (cid:55)→ π (cid:92) ( A (cid:4) ⊗ L Λ B ) . Proposition

VII.7.6 . For any compact object A ∈ D lis (Bun G , Λ) , there is a unique compactobject D BZ ( A ) ∈ D lis (Bun G , Λ) with a functorial identiﬁcation R Hom( D BZ ( A ) , B ) ∼ = π (cid:92) ( A (cid:4) ⊗ L Λ B ) for B ∈ D lis (Bun G , Λ) . Moreover, the functor D BZ is a contravariant autoequivalence of D lis (Bun G , Λ) ω ,and D is naturally isomorphic to the identity.If U ⊂ Bun G is an open substack and A is concentrated on U , then so is D BZ ( A ) . In particular, D BZ restricts to an autoequivalence of the compact objects in D lis (Bun bG , Λ) ∼ = D ( G b ( E ) , Λ) for b ∈ B ( G ) basic, and in that setting it is the usual Bernstein–Zelevinsky involution. Proof.

The existence of D BZ follows as in Theorem V.5.1, using the left adjoint given byProposition VII.7.2; this construction also shows that D BZ preserves D lis ( U, Λ), and for basic b itrecovers the usual Bernstein–Zelevinsky involution by the same argument as in Theorem V.5.1.We also formally get a morphism D ( A ) → A by adjunctions. We need to see that this is anisomorphism. It suﬃces to check on generators. The Bernstein–Zelevinsky dual of i b ! c -Ind G b ( E ) K Λ is A bK = f K(cid:92)

Λ, for f K : M b,K → Bun G the natural map. Its restriction to Bun bG is again i b ! c -Ind G b ( E ) K Λ,so one easily checks that the map D ( A ) → A is an isomorphism over Bun bG . To see that it isan isomorphism everywhere, one needs to see that if B = Rj ∗ B (cid:48) , B (cid:48) ∈ D lis ( U, Λ) for some opensubstack j : U ⊂ Bun G not containing Bun bG , then π (cid:92) ( A bK (cid:4) ⊗ L Λ B ) = 0 . Twisting a few things away and using the deﬁnition of A bK = f K(cid:92)

Λ, this follows from the assertionthat for all A (cid:48) ∈ D lis ( M ◦ b,K , Λ), with j K : M ◦ b,K (cid:44) → M b,K the open immersion, one has R Γ c ( M b,K , Rj K ∗ A (cid:48) ) = 0 . II.7. D lis (Bun G ) 269 Using the trace map for (cid:102) M b → M b,K , this follows from Theorem VII.2.10. (cid:3) As in Theorem V.6.1, this has the following consequence for Verdier duality.

Proposition

VII.7.7 . Let j : V (cid:44) → U be an open immersion of open substacks of Bun G . Forany A ∈ D lis ( V, Λ) , the natural map j ! R H om lis ( A, Λ) → R H om lis ( Rj lis ∗ A, Λ) is an isomorphism in D lis ( U, Λ) . Proof.

The proof is identical to the proof of Theorem V.6.1. (cid:3)

Using this, one can characterize the reﬂexive objects as in Theorem V.6.2; we omit it here.Finally, one can also characterize the universally locally acyclic A ∈ D lis (Bun G , Λ). Note that wehave not deﬁned a notion of universal local acyclicity for lisse sheaves, but in our present situation wecan simply import the characterization from Proposition IV.2.32 and make the following deﬁnition.

Definition

VII.7.8 . A complex A ∈ D lis (Bun G , Λ) is universally locally acyclic (with respectto Bun G → ∗ ) if the natural map p ∗ R H om lis ( A, Λ) (cid:4) ⊗ L Λ p ∗ A → R H om lis ( p ∗ A, p ∗ A ) is an isomorphism, where p , p : Bun G × Bun G → Bun G are the two projections. We get the following version of Theorem V.7.1.

Proposition

VII.7.9 . Let A ∈ D lis (Bun G , Λ) . Then A is universally locally acyclic if and onlyif for all b ∈ B ( G ) , the pullback i b ∗ A to i b : Bun bG (cid:44) → Bun G corresponds under D lis (Bun bG , Λ) ∼ = D ( G b ( E ) , Λ) to a complex M b of smooth G b ( E ) -representations for which M K is a perfect complexof Λ -modules for all open pro- p subgroups K ⊂ G b ( E ) . The proof is identical to the proof of Theorem V.7.1, and proceeds by proving ﬁrst the followingproposition.

Proposition

VII.7.10 . Let G and G be two reductive groups over E , and let G = G × G .Consider the exterior tensor product − (cid:2) − : D lis (Bun G , Λ) × D lis (Bun G , Λ) → D lis (Bun G , Λ) . For all compact objects A i ∈ D lis (Bun G i , Λ) , i = 1 , , the exterior tensor product A (cid:2) A ∈ D ´et (Bun G , Λ) is compact, these objects form a class of compact generators, and for all furtherobjects B i ∈ D lis (Bun G i , Λ) , i = 1 , , the natural map R Hom( A , B ) ⊗ L Λ R Hom( A , B ) → R Hom( A (cid:2) A , B (cid:2) B ) is an isomorphism. Proof.

The proof is identical to the proof of Proposition V.7.2. (cid:3)

HAPTER VIII L -parameter It is time to understand the other side of the correspondence: In this chapter, we deﬁne, andstudy basic properties of, the stack of L -parameters. These results have recently been obtainedby Dat–Helm–Kurinczuk–Moss [ DHKM20 ], and also Zhu [

Zhu20 ]; previous work in a relateddirection includes [

Hel16 ], [

HH20 ], [

BG19 ], [

BP19 ], [

LTX + , Appendix E].In this chapter, we ﬁx again a nonarchimedean local ﬁeld E with residue ﬁeld F q of characteristic p , and a reductive group G over E , as well as a prime (cid:96) (cid:54) = p . We get the dual group (cid:98) G/ Z (cid:96) , which weendow with its usual “algebraic” action by W E ; the action thus factors over a ﬁnite quotient Q of W E , and we ﬁx such a quotient Q of W E . (The diﬀerence to the cyclotomically twisted W E -actiondisappears after base change to Z (cid:96) [ √ q ], and we could thus obtain analogues of all results below forthis other action by a simple descent along Z (cid:96) [ √ q ] / Z (cid:96) .) We deﬁne a scheme whose Λ-valued points,for a Z (cid:96) -algebra Λ, are the condensed 1-cocycles ϕ : W E → (cid:98) G (Λ) , where Λ = Λ disc ⊗ Z (cid:96), disc Z (cid:96) is regarded as a relatively discrete condensed Z (cid:96) -module. Theorem

VIII.0.1 (Theorem VIII.1.3) . There is a scheme Z ( W E , (cid:98) G ) over Z (cid:96) whose Λ -valuedpoints, for a Z (cid:96) -algebra Λ , are the condensed -cocycles ϕ : W E → (cid:98) G (Λ) . The scheme Z ( W E , (cid:98) G ) is a union of open and closed aﬃne subschemes Z ( W E /P, (cid:98) G ) as P runsthrough open subgroups of the wild inertia subgroup of W E , and each Z ( W E /P, (cid:98) G ) is a ﬂat localcomplete intersection over Z (cid:96) of dimension dim G . To prove the theorem, following [

DHKM20 ] and [

Zhu20 ] we deﬁne discrete dense subgroups W ⊂ W E /P by discretizing the tame inertia, and the restriction Z ( W E /P, (cid:98) G ) → Z ( W, (cid:98) G ) is anisomorphism, where the latter is clearly an aﬃne scheme.We can also prove further results about the (cid:98) G -action on Z ( W E , (cid:98) G ), or more precisely each Z ( W E /P, (cid:98) G ). For this result, we need to exclude some small primes, but if G = GL n , all primes (cid:96) are allowed; for classical groups, all (cid:96) (cid:54) = 2 are allowed. More precisely, we say that (cid:96) is very goodfor (cid:98) G if the following conditions are satisﬁed.(i) The (algebraic) action of W E on (cid:98) G factors over a ﬁnite quotient Q of order prime to (cid:96) .(ii) The order of the fundamental group of the derived group of (cid:98) G is prime to (cid:96) (equivalently, π Z ( G )is of order prime to (cid:96) ). L -PARAMETER (iii) If G has factors of type B , C , or D , then (cid:96) (cid:54) = 2; if it has factors of type E , F , or G , then (cid:96) (cid:54) = 2 ,

3; and if it has factors of type E , then (cid:96) (cid:54) = 2 , , Theorem

VIII.0.2 (Theorem VIII.5.1) . Assume that (cid:96) is a very good prime for (cid:98) G . Then H i ( (cid:98) G, O ( Z ( W E /P, (cid:98) G ))) = 0 for i > and the formation of the invariants O ( Z ( W E /P, (cid:98) G )) (cid:98) G commutes with any base change. The algebra O ( Z ( W E /P, (cid:98) G )) (cid:98) G admits an explicit presentation interms of excursion operators, O ( Z ( W E /P, (cid:98) G )) (cid:98) G = colim ( n,F n → W ) O ( Z ( F n , (cid:98) G )) (cid:98) G where the colimit runs over all maps from a free group F n to W ⊂ W E /P , and Z ( F n , (cid:98) G ) ∼ = (cid:98) G n with the simultaneous twisted (cid:98) G -conjugation.Moreover, the ∞ -category Perf( Z ( W E /P, (cid:98) G ) / (cid:98) G ) is generated under cones and retracts by theimage of Rep( (cid:98) G ) → Perf( Z ( W E /P, (cid:98) G ) / (cid:98) G ) , and Ind Perf( Z ( W E /P, (cid:98) G )) is equivalent to the ∞ -category of modules over O ( Z ( W E /P, (cid:98) G )) in Ind Perf( B (cid:98) G ) .All of these results also hold with Q (cid:96) -coeﬃcients, without any assumption on (cid:96) . With Q (cid:96) -coeﬃcients, these results are simple, as the representation theory of (cid:98) G is semisimple.However, with Z (cid:96) -coeﬃcients, these results are quite subtle, and we need to dive into modularrepresentation theory of reductive groups. Very roughly, the proof of the theorem proceeds byanalyzing the closed (cid:98) G -orbits in the stack of L -parameters ﬁrst, and then use a deformation to thenormal cone to understand the behaviour near any (cid:98) G -orbit. We make critical use of some resultsof Touz´e–van der Kallen [ TvdK10 ]. Let us make some further remarks about the closed (cid:98) G -orbits.First, the closed (cid:98) G -orbits in Z ( W E /P, (cid:98) G ) L , for any algebraically closed ﬁeld L over Z (cid:96) , corre-spond to the semisimple L -parameter ϕ : W E → (cid:98) G ( L ), and also biject to the geometric points ofSpec O ( Z ( W E /P, (cid:98) G )) (cid:98) G . Here, an L -parameter is semisimple if, whenever it factors over a para-bolic (cid:98) P ⊂ (cid:98) G , it also factors over a Levi (cid:99) M ⊂ (cid:98) P . Any semisimple parameter is in fact continuousfor the discrete topology on L , i.e. trivial on an open subgroup of I E . If L = F (cid:96) and Q is of orderprime to (cid:96) , then ϕ is semisimple if and only if it factors over a ﬁnite quotient of order prime to (cid:96) .Its orbit in Z ( W E /P, (cid:98) G ) L is then given by (cid:98) G/S ϕ where S ϕ ⊂ (cid:98) G is the centralizer of ϕ , which isin that case the ﬁxed points under the action of a ﬁnite solvable group F of automorphisms of (cid:98) G ,where the order of F is prime to (cid:96) . We need the following result. Theorem

VIII.0.3 (Theorem VIII.5.14) . Let H be a reductive group over an algebraically closedﬁeld L of characteristic (cid:96) . Let F be a ﬁnite group of order prime to (cid:96) acting on H . Then H F isa smooth linear algebraic group whose connected component is reductive, and with π H F of orderprime to (cid:96) . If F is solvable, the image of Perf( BH ) → Perf( BH F ) generates the whole categoryunder cones and retracts. The last part of this theorem is proved by a very explicit (and exhausting) analysis of allpossible cases. It would be desirable to have a more enlightening argument, possibly also removingthe assumption that F is solvable. In fact, we would expect similar results to hold true in thecase where W is replaced by the fundamental group of a Riemann surface. Our arguments are notgeneral enough to handle that case. III.1. THE STACK OF L -PARAMETERS 273 Remark

VIII.0.4 . While the hypotheses imposed on (cid:96) are surely not optimal, we are quitesure that some hypothesis on (cid:96) is required in Theorem I.8.2. For example, if (cid:98) G = PGL and (cid:96) = 2,we expect problems to arise. For example, one can show that for X = (cid:98) G with the conjugationaction by (cid:98) G , the ∞ -category Perf( X/ (cid:98) G ) is not generated under cones and retracts by the imageof Rep( (cid:98) G ). Our guess would be that the condition that (cid:96) does not divide the order of π ( (cid:98) G der ) isessential.On the other hand, we expect that, for example by the use of z-embeddings [ Kal14 , Section 5],one can reduce all relevant questions (like the general construction of maps on Bernstein centersdiscussed below, or the construction of the spectral action) to the case where (cid:96) does not divide theorder of π ( (cid:98) G der ). We are not taking this up here. VIII.1. The stack of L -parametersVIII.1.1. Deﬁnition and representability. Recall that for a reductive group G over anonarchimedean local ﬁeld E , we have the dual group (cid:98) G over Z (cid:96) , equipped with an action of theWeil group W E . In this chapter, we use the standard action.Now let Λ be any Z (cid:96) -algebra. As in the last chapter, we regard it as a condensed Z (cid:96) -algebra, asΛ disc ⊗ Z (cid:96), disc Z (cid:96) . Its value on a proﬁnite set S is the ring of maps S → Λ that take values in a sub- Z (cid:96) -module of ﬁnite type and are continuous. For example, if Λ = Q (cid:96) then Λ( S ) = lim −→ L ⊂ Q (cid:96) Cont(

S, L )with L | Q (cid:96) ﬁnite. Definition

VIII.1.1 . An L -parameter for G , with coeﬃcients in Λ , is a section ϕ : W E → (cid:98) G (Λ) (cid:111) W E of the natural map of condensed groups (cid:98) G (Λ) (cid:111) W E → W E . Equivalently, an L -parameter for G with coeﬃcients in Λ is a (condensed) -cocycle ϕ : W E → (cid:98) G (Λ) for the given W E -action on (cid:98) G . More concretely, an L -parameter with values in Λ is a 1-cocycle ϕ : W E → (cid:98) G (Λ) such that if (cid:98) G (cid:44) → GL N , the associated map W E → GL N (Λ) is continuous. The preceding means the matrixcoeﬃcients of its restriction to I E are maps I E → Λ that take values in ﬁnite type Z (cid:96) -modules andare continuous. Remark

VIII.1.2 . The standard action of W E factors over a ﬁnite quotient Q . This meansthat L -parameters are also equivalent to maps W E → (cid:98) G (Λ) (cid:111) Q lifting W E → Q .The ﬁrst main result is the following. Theorem

VIII.1.3 . There is a scheme Z ( W E , (cid:98) G ) over Z (cid:96) parametrizing L -parameters for G ,which is a disjoint union of aﬃne schemes of ﬁnite type over Z (cid:96) . It is ﬂat and a relative completeintersection of dimension dim G = dim (cid:98) G .

74 VIII. L -PARAMETER Proof.

Any condensed 1-cocycle ϕ : W E → (cid:98) G (Λ) is trivial on an open subgroup of the wildinertia subgroup P E ; note also that P E acts on (cid:98) G through a ﬁnite quotient. Moreover, for any γ ∈ P E acting trivially on (cid:98) G , the locus where ϕ ( γ ) = 1 is open and closed: Taking a closedembedding (cid:98) G (cid:44) → GL N , this follows from A = 1 being a connected component of the locus of all A ∈ GL N such that A p r = 1, as can be checked by observing that the tangent space at A = 1 istrivial. It follows that the moduli space of L -parameters decomposes as a disjoint union of openand closed subspaces according to the kernel of ϕ on P E .Thus, ﬁx now some quotient W E → W (cid:48) E by an open subgroup of P E such that the action of W E on (cid:98) G factors over W (cid:48) E . We are interested in the moduli space of condensed 1-cocycles W (cid:48) E → (cid:98) G (Λ).Inside W (cid:48) E , we look at the discrete dense subgroup W ⊂ W (cid:48) E generated by the image of P E , a choiceof generator of the tame inertia τ , and a choice of Frobenius σ . Thus, W sits in an exact sequence0 → I → W → σ Z → I in turn sits in an exact sequence0 → P → I → τ Z [ 1 p ] → P is a ﬁnite p -group. Moreover, in W/P , the elements τ and σ satisfy the commutation σ − τ σ = τ q .Now observe that any condensed 1-cocycle W (cid:48) E → (cid:98) G (Λ) is already determined by its restrictionto the discrete group W , as (cid:98) G (Λ) is quasiseparated and W ⊂ W (cid:48) E is dense. Conversely, we claimthat any 1-cocycle W → (cid:98) G (Λ) extends uniquely to a condensed 1-cocycle W (cid:48) E → (cid:98) G (Λ). To checkthis, we may replace E by a ﬁnite extension; we can thus pass to a setting where the action of W (cid:48) E on (cid:98) G is trivial, and where P = 1. Taking a closed immersion (cid:98) G (cid:44) → GL N , it then suﬃces to see thatany representation of τ Z [ 1 p ] (cid:111) σ Z on a ﬁnite free Λ-module extends uniquely to a representation ofthe condensed group (cid:98) Z p (cid:111) σ Z . For this, in turn, it suﬃces to see that for any A ∈ GL N (Λ) suchthat A is conjugate to A q , the map Z → GL N (Λ) : n (cid:55)→ A n extends uniquely to (cid:98) Z p . The assumption on A implies that all eigenvalues of A at all geometricpoints of Spec Λ are roots of unity of order prime to p ; replacing A by a prime-to- p -power (as wemay) we can thus reduce to the case that A is unipotent, i.e. A − n (cid:55)→ X n extends to a continuous map n (cid:55)→ X n = (1 + ( X − n = (cid:88) i ≥ (cid:18) ni (cid:19) ( X − i , deﬁning a map Z (cid:96) → GL N (Λ) (and hence (cid:98) Z p → Z (cid:96) → GL N (Λ)).Thus, we need to see that the space X = Z ( W, (cid:98) G ) of all 1-cocycles ϕ : W → (cid:98) G (Λ) is an aﬃnescheme of ﬁnite type over Z (cid:96) that is ﬂat and a relative complete intersection of dimension dim (cid:98) G .It is clear that it is an aﬃne scheme of ﬁnite type over Z (cid:96) as W is discrete and ﬁnitely generated.To prove the geometric properties, we ﬁnd it slightly more convenient to argue with the Artinstack [ X/ (cid:98) G ], which we aim to prove is ﬂat and a relative complete intersection of dimension 0 over Z (cid:96) . III.1. THE STACK OF L -PARAMETERS 275 We can understand the deformation theory of [ X/ (cid:98) G ]: If Λ is a ﬁeld, then the obstruction groupis H ( W, (cid:98) g ⊗ Z (cid:96) Λ) (where (cid:98) g is the Lie algebra of (cid:98) G ), the tangent space is H ( W, (cid:98) g ⊗ Z (cid:96) Λ), and theinﬁnitesimal automorphisms are H ( W, (cid:98) g ⊗ Z (cid:96) Λ), where in all cases the action of W is twisted by thelocal 1-cocycle ϕ . Now note that by direct computation the prime-to- p cohomological dimension of W is 2, and the Euler characteristic of any representation is equal to 0. Thus, this analysis showsthat we only have to prove that all ﬁbres of [ X/ (cid:98) G ] → Spec Z (cid:96) are of dimension at most 0.Note that X is actually naturally deﬁned over Z [ p ] (as (cid:98) G is, and the discretization W of W (cid:48) E isindependent of (cid:96) ). It follows that it suﬃces to bound the dimension of the ﬁbre over F (cid:96) (as if wecan do this for all closed points of Spec Z [ p ], it follows over the generic ﬁbre by constructibility ofthe dimension of ﬁbers). To do this, we switch back to the picture of condensed 1-cocycles on W E .From now on, we work over F (cid:96) .The stack [ Z ( W E , (cid:98) G ) F (cid:96) / (cid:98) G ] maps to the similar stack parametrizing 1-cocycles ϕ I (cid:96) : I (cid:96) → (cid:98) G (Λ)of the prime-to- (cid:96) inertia subgroup I (cid:96) , up to conjugation. By deformation theory, that stack issmooth and each connected component is a quotient of Spec F (cid:96) by the centralizer group C ϕ I(cid:96) ⊂ (cid:98) G F (cid:96) ,which is a smooth group, whose identity component is reductive by [ PY02 , Theorem 2.1]. We maythus ﬁx ϕ I (cid:96) : I (cid:96) → (cid:98) G ( F (cid:96) ) and consider the open and closed subscheme X ϕ I(cid:96) ⊂ Z ( W E , (cid:98) G ) F (cid:96) of all1-cocycles ϕ : W E → (cid:98) G (Λ) whose restriction to I (cid:96) is equal to ϕ I (cid:96) . Our goal is to show that X ϕ I(cid:96) is of dimension at most dim C ϕ I(cid:96) .Consider the centralizer (cid:101) C of ϕ I (cid:96) inside (cid:98) G F (cid:96) (cid:111) Q . Then X ϕ I(cid:96) maps with ﬁnite ﬁbres to thespace of maps f : W E /I (cid:96) ∼ = Z (cid:96) (cid:111) σ Z −→ (cid:101) C/ϕ I (cid:96) ( I (cid:96) ) . Note that, by representability of X ϕ I(cid:96) , the universal map f factors over a quotient of the form Z /(cid:96) m Z (cid:111) σ Z . Finally, we have reduced to Lemma VIII.1.4 below. (cid:3) Lemma

VIII.1.4 . Let H be a smooth group scheme over F (cid:96) whose identity component is reduc-tive. Then the aﬃne scheme parametrizing maps of groups Z /(cid:96) m Z (cid:111) σ Z → H, where σ acts on Z /(cid:96) m Z via multiplication by q , is of dimension at most dim H . Proof.

The image of the generator of Z /(cid:96) m Z is a unipotent element of H . By ﬁniteness ofthe number of unipotent conjugacy classes, cf. [ Lus76 ], [

FG12 , Corollary 2.6], we can stratify thescheme according to the conjugacy class of the image of τ . But for each ﬁxed conjugacy class,one has to choose the image of σ so as to conjugate τ into τ q : This bounds the dimension of eachstratum by the dimension of the conjugacy class of τ (giving the choices for τ ) plus the codimensionof the conjugacy class of τ (giving the choices for σ , for any given τ ), which is the dimension of H . (cid:3) We observe the following corollary of the proof.

Proposition

VIII.1.5 . Assume that W E acts on (cid:98) G via a ﬁnite quotient Q of order prime to (cid:96) . Let L = F (cid:96) . Let ϕ : W E → (cid:98) G ( L ) be any L -parameter. Then the (cid:98) G -orbit of ϕ in Z ( W E , (cid:98) G ) L isclosed if and only if the map ϕ factors over a ﬁnite quotient of W E of order prime to (cid:96) .

76 VIII. L -PARAMETER Proof.

In the notation of the proof, the group C ϕ I(cid:96) has connected components of order primeto (cid:96) , see Proposition VIII.5.11 below, so one will apply Lemma VIII.1.4 to a group H with π H of order prime to (cid:96) . This implies that the unipotent element is actually an element of H ◦ , and itsorbit is closed only if it is trivial. Moreover, the image of the Frobenius σ deﬁnes a closed orbit ifand only if it is semisimple, i.e. of order prime to (cid:96) , giving the claim. (cid:3) VIII.2. The singularities of the moduli space

The following proposition was already implicitly noted in the proof of Theorem VIII.1.3.

Proposition

VIII.2.1 . For any parameter ϕ : W E → (cid:98) G (Λ) (cid:111) Q corresponding to x : Spec(Λ) → LocSys (cid:98) G , x ∗ L ∨ Z ( W E , (cid:98) G ) / (cid:98) G = R Γ( W E , ( (cid:98) g ⊗ Z (cid:96) Λ) ϕ )[1] where ( (cid:98) g ⊗ Z (cid:96) Λ) ϕ is (cid:98) g ⊗ Z (cid:96) Λ equipped with the twisted action of W E deduced from ϕ . Proof.

This would be clear if we deﬁned the moduli problem on all animated Z (cid:96) -algebras,by deformation theory. Then the cohomological dimension of W E would imply that this moduliproblem is a derived local complete intersection, of expected dimension 0. However, we proved that Z ( W E , (cid:98) G ) / (cid:98) G is a local complete intersection Artin stack of dimension 0, hence it represents thecorrect moduli problem even on all animated Z (cid:96) -algebras, thus giving the result. (cid:3) Proposition

VIII.2.2 . Let M be a free Λ -module of ﬁnite rank equipped with a condensed actionof W E . Then R Γ( W E , M ) is a perfect complex of Λ -modules and there is a canonical isomorphism R Γ( W E , M ) ∗ ∼ = R Γ( W E , M ∗ (1))[2] . Proof.

This follows from Poincar´e duality applied to Div → ∗ , using Proposition VII.3.5 andthe discussion before. It can also be proved by hand, by comparing the W E -cohomology with the W -cohomology, for a discretization W of W E /P as before. (cid:3) Corollary

VIII.2.3 . For any parameter ϕ : W E → (cid:98) G (Λ) (cid:111) W E corresponding to x : Spec(Λ) → LocSys (cid:98) G , x ∗ L LocSys (cid:98) G/ Z (cid:96) = R Γ( W E , ( (cid:98) g ∗ ⊗ Z (cid:96) Λ) ϕ (1))[1] where ( (cid:98) g ∗ ⊗ Z (cid:96) Λ) ϕ is (cid:98) g ∗ ⊗ Z (cid:96) Λ equipped with the twisted action of W E deduced from ϕ . VIII.2.1. The characteristic zero case.

Fix an isomorphism I E /P E ∼ = (cid:98) Z p . There is a (cid:98) G Q (cid:96) -equivariant “unipotent monodromy” morphism M : Z ( W E , (cid:98) G ) Q (cid:96) −→ N (cid:98) G Q (cid:96) where N (cid:98) G Q (cid:96) is the nilpotent cone inside (cid:98) g ⊗ Q (cid:96) .In fact, one can lift the inclusion Z (cid:96) (cid:44) → (cid:98) Z p ∼ = I E /P E to a morphism Z (cid:96) → I E . Now, if ϕ : W E → (cid:98) G (Λ), with Λ a Q (cid:96) -algebra, is a parameter, then ϕ | Z (cid:96) : Z (cid:96) → (cid:98) G (Λ) is such that for n (cid:29) ϕ | (cid:96) n Z (cid:96) is a morphism of condensed groups satisfying ϕ ( σ m ) ϕ | (cid:96) n Z (cid:96) ϕ ( σ m ) − = ϕ q m | (cid:96) n Z (cid:96) III.2. THE SINGULARITIES OF THE MODULI SPACE 277 for m (cid:29)

0. One deduces, using an embedding of (cid:98) G in GL N , that there is a unique N ∈ N (cid:98) G (Λ)such that for n (cid:29) x ∈ (cid:96) n Z (cid:96) , ϕ ( x ) = exp( xN ) . Using these observations, we get a comparison to Weil–Deligne L -parameters. Definition

VIII.2.4 . For Λ a Q (cid:96) -algebra one deﬁnes Par WD (cid:98) G (Λ) to be the set of pairs ( ϕ , N ) where (i) ϕ : W E → (cid:98) G (Λ disc ) is a condensed -cocycle, (ii) N ∈ N (cid:98) G ⊗ Q (cid:96) satisﬁes Ad( ϕ ( σ )) .σN = q | σ | N for all σ ∈ W E . Then we have the following result, which is essentially Grothendieck’s quasi-unipotence theorem.

Proposition

VIII.2.5 ([

Zhu20 , Lemma 3.1.8]) . There is a (cid:98) G -equivariant isomorphism Z ( W E , (cid:98) G ) ⊗ Q (cid:96) ∼ −→ Par WD (cid:98) G . However, we warn the reader that this isomorphism depends on some auxiliary choices, such asthat of a Frobenius element.

VIII.2.2. The singular support.

VIII.2.2.1.

General construction.

Recall the following construction, see for example [

AG15 ].Let A → B be a ﬂat map of commutative rings. One has the Hochschild cohomology HH • ( B/A ) = Ext • B ⊗ A B ( B, B ) . Note that any M ∈ D ( B ⊗ A B ) induces a functor D ( B ) → D ( B ), via N (cid:55)→ M ⊗ L B N (with the“left” B -module structure). Here, M = B ∈ D ( B ⊗ A B ), via the multiplication B ⊗ A B → B ,induces the identity functor. It follows that there is a natural map HH i ( B/A ) = Ext iB ⊗ A B ( B, B ) → Ext iB ( N, N )for any N ∈ D ( B ). Moreover, Hochschild cohomology is naturally a graded algebra, and this mapis a map of algebras HH • ( B/A ) → Ext • B ( N, N ) . There is an identiﬁcation ([

ML95 , Theorem X.3.1]) HH ( B/A ) = Ext B ( L B/A , B )which itself is nothing else than E xalcom A ( B, B ) ([

Gro64 , Chap.0, Sec. 18.4]). We thus have anidentiﬁcation HH ( B/A ) = H ( L ∨ B/A ) . Suppose now that A → B is syntomic, i.e. ﬂat and a local complete intersection. Let X = Spec B → S = Spec A be the associated map of aﬃne schemes. Definition

VIII.2.6 . The scheme

Sing

X/S −→ X represents the functor T /X (cid:55)→ H − ( L X/S ⊗ L O X O T ) .

78 VIII. L -PARAMETER In fact, locally on X , L X/S is isomorphic to a complex of vector bundles [ E − → E ] and thenSing X/S is the kernel of V ( E − ) → V ( E ). Explicitly, Sing X/S is the aﬃne scheme with O (Sing X/S ) = Sym • B H ( L ∨ B/A ) . This is an X -group scheme equipped with an action of G m . The image of

Sing

X/S \ { } → X isthe closed subset complementary of the smooth locus of X → S . Consider now any N ∈ D bcoh ( X ) , and the graded B -algebra Ext • B ( N, N ). Using the map H ( L ∨ B/A ) = HH ( B/A ) → Ext B ( N, N ) , this is in fact naturally a (graded) O (Sing X/S )-algebra. This deﬁnes a G m -equivariant quasi-coherent sheaf µ End( N )on Sing X/S . Suppose now moreover that S is regular. Theorem

VIII.2.7 ([

Gul74 , Theorem 3.1],[

AG15 , Appendix D]) . For N ∈ D bcoh ( X ) , thequasi-coherent sheaf µ End( N ) on Sing

X/S is coherent.

Definition

VIII.2.8 . The singular support of N , SingSupp( N ) , is the support of µ End( F ) asa closed conical subset of Sing

X/S . Of course, the image of SingSupp( N ) → X is contained in Supp( N ). Theorem

VIII.2.9 ([

AG15 , Theorem 4.2.6]) . The following are equivalent: (i) N is a perfect complex, (ii) SingSupp( N ) is contained in the zero section of Sing

X/S . Proof.

We have to prove that if Ext iB ( N, N ) = 0 for i (cid:29) N is a perfect complex. Thisis for example a consequence of [ Jor08 ]. Since S is regular X is Gorenstein. According to [ Jor08 ],if N is a B -module of ﬁnite type that satisﬁes Ext iB ( N, N ) = 0 for i > n , then pd B N ≤ n . Ingeneral, up to taking a shift of N , we can ﬁnd a map N → N (cid:48) , where N (cid:48) is a ﬁnitely generated B -module concentrated in degree 0, such that the cone C of N → N (cid:48) is perfect. Suppose thatExt iB ( N, N ) = 0 for i (cid:29)

0. In the long exact sequence · · · −→

Ext iB ( C, N ) −→ Ext iB ( N (cid:48) , N ) −→ Ext iB ( N, N ) −→ · · · one has Ext iB ( C, N ) = 0 for i (cid:29) C is perfect and Ext iB ( N, N ) = 0 for i (cid:29) iB ( N (cid:48) , N ) = 0 for i (cid:29)

0. In the long exact sequence · · · −→

Ext iB ( N (cid:48) , N ) −→ Ext iB ( N (cid:48) , N (cid:48) ) −→ Ext iB ( N (cid:48) , C ) −→ · · · we have Ext iB ( N (cid:48) , C ) = 0 for i (cid:29) C is perfect and A has ﬁnite injective dimension overitself since it is Gorenstein. Thus, for i (cid:29)

0, Ext iB ( N (cid:48) , N (cid:48) ) ∼ −→ Ext iB ( N (cid:48) , N ) and this vanishes. Wecan thus apply Jorgensen’s theorem to N (cid:48) to conclude that N (cid:48) , and hence N , is perfect. (cid:3) III.2. THE SINGULARITIES OF THE MODULI SPACE 279

Let us note the following corollary.

Corollary

VIII.2.10 . The image of

SingSupp( N ) \ { } → X is the complementary of thebiggest open subset of X on which N is a perfect complex. VIII.2.2.2.

The case of Z ( W E , (cid:98) G ) . Now we apply the preceding theory in the case A = Z (cid:96) and X = Z ( W E , (cid:98) G ) (which is only a union of aﬃne schemes, but this is not a problem). We can alsopass to the quotient stack Z ( W E , (cid:98) G ) / (cid:98) G . According to Corollary VIII.2.3, there is an embeddingSing [ Z ( W E , (cid:98) G ) / (cid:98) G ] / Z (cid:96) [ (cid:98) g ∗ / (cid:98) G ] × B (cid:98) G [ Z ( W E , (cid:98) G ) / (cid:98) G ][ Z ( W E , (cid:98) G ) / (cid:98) G ]where (cid:98) g = Lie (cid:98) G and [ (cid:98) g ∗ / (cid:98) G ] is seen here as a vector bundle on B (cid:98) G = [Spec Z (cid:96) / (cid:98) G ]. Let N ∗ (cid:98) G ⊂ (cid:98) g ∗ be the nilpotent cone; by this we mean the closed subset of all (cid:98) G -orbits whose closure containsthe origin. (If there is a (cid:98) G -equivariant isomorphism between (cid:98) g ∗ and (cid:98) g , this would identify withthe usual nilpotent cone.) Since this is stable under the adjoint action this deﬁnes a Zariski closedsubstack [ N ∗ (cid:98) G / (cid:98) G ] × B (cid:98) G [ Z ( W E , (cid:98) G ) / (cid:98) G ] [ (cid:98) g ∗ / (cid:98) G ] × B (cid:98) G [ Z ( W E , (cid:98) G ) / (cid:98) G ][ Z ( W E , (cid:98) G ) / (cid:98) G ] . Proposition

VIII.2.11 . For a Z (cid:96) -ﬁeld L and a point x : Spec( L ) → LocSys (cid:98) G we have x ∗ Sing [ Z ( W E , (cid:98) G ) / (cid:98) G ] / Z (cid:96) ⊂ N ∗ (cid:98) G ⊗ Z (cid:96) L in the following two cases: (i) L | Q (cid:96) , (ii) (cid:96) does not divide the order of the fundamental group of the adjoint form of G , and if n = f E (cid:48) /E with W E (cid:48) = ker( W E → Out( (cid:98) G )) , (cid:96) (cid:54) | q en − for any exponent e of (cid:98) G . Proof. If x corresponds to the parameter ϕ then x ∗ Sing [ Z ( W E , (cid:98) G ) / (cid:98) G ] / Z (cid:96) = H ( W E , (cid:98) g ∗ ⊗ Z (cid:96) L (1))where the W E action on (cid:98) g ∗ ⊗ Z (cid:96) L (1) is twisted by ϕ . For an element v ∈ (cid:98) g ∼ = (cid:98) g ∗ in this subspacewe thus have that σ.v and qv are in the same orbits under the adjoint action of (cid:98) G ( L ) (here σ.v isgiven by the action of W E on (cid:98) g ∗ deﬁning the L-group). We thus obtain that v is conjugated underthe adjoint action to q n v . There is a morphism (cid:98) g −→ (cid:98) g (cid:12) (cid:98) G = (cid:98) t (cid:12) W ∼ = A m Z (cid:96) given by m homogeneous polynomials of degrees the exponents of the root system. This impliesthat the image of v in A m ( L ) is zero and thus v lies in the nilpotent cone. (cid:3)

80 VIII. L -PARAMETER The supremum of the exponents of (cid:98) G is the Coxeter number h of G . The preceding conditionis satisﬁed if for example (cid:96) > q hn −

1. We refer to [

DHKM20 , Section 5.3] for ﬁner deﬁnitions andresults about (cid:98) G -banal primes; we have not tried to optimize the condition above, and it is likelythat with their results one can obtain a much better condition on (cid:96) . Remark

VIII.2.12 . In the non-banal case things become more complicated and the Arinkin–Gaitsgory condition of nilpotent singular support becomes important. This is also the case when in-teresting congruences modulo (cid:96) between smooth irreducible representations of G ( E ) occur, cf. [ DHKM20 ,Section 1.5].

VIII.3. The coarse moduli space

Let us now describe the corresponding coarse moduli space, i.e. we consider the quotient Z ( W E , (cid:98) G ) (cid:12) (cid:98) G taken in the category of schemes. Concretely, for every connected component Spec A ⊂ Z ( W E , (cid:98) G ),we get a corresponding connected component Spec A (cid:98) G ⊂ Z ( W E , (cid:98) G ) (cid:12) (cid:98) G . VIII.3.1. Geometric points.

For any algebraically closed ﬁeld L over Z (cid:96) , the L -valued pointsof Z ( W E , (cid:98) G ) L (cid:12) (cid:98) G are in bijection with the closed (cid:98) G -orbits in Z ( W E , (cid:98) G ) L .We want to describe L -valued points with closed (cid:98) G -orbit as the “semisimple” parameters. Forthis, recall (cf. [ Bor79 ]) that parabolic subgroups of (cid:98) G L (cid:111) W E surjecting onto W E are up to (cid:98) G ( L )-conjugation given by (cid:98) P L (cid:111) W E for a standard parabolic P ⊂ G ∗ of the quasisplit inner form G ∗ of G . A Levi subgroup is given by (cid:99) M L (cid:111) W E where M ⊂ P is the standard Levi. We now call themthe parabolic subgroups of (cid:98) G (cid:111) W E i.e. we always suppose they surject to W E . If (cid:98) ∆ are the simpleroots of (cid:98) G then the standard parabolic subgroups are in bijection with the ﬁnite W E -stable subsetsof (cid:98) ∆. Definition

VIII.3.1 . Let L be an algebraically closed ﬁeld over Z (cid:96) . An L -parameter ϕ : W E → (cid:98) G ( L ) (cid:111) W E is semisimple if whenever the image of ϕ is contained in a parabolic subgroup of (cid:98) G (cid:111) W E then it is contained in a Levi subgroup of this parabolic subgroup. In terms of the standard parabolic subgroups this means that if some (cid:98) G ( L )-conjugate ϕ (cid:48) of ϕ factorizes through (cid:98) P ( L ) (cid:111) W E , then there exists g ∈ (cid:98) P ( L ) such that gϕ (cid:48) g − = pr L M ◦ ϕ (cid:48) , wherepr L M : (cid:98) P ( L ) (cid:111) W E → (cid:99) M ( L ) (cid:111) W E is the projection onto the standard Levi subgroup. Proposition

VIII.3.2 ([

DHKM20 , Proposition 4.13]) . Let L be an algebraically closed ﬁeldover Z (cid:96) and ϕ : W E → (cid:98) G ( L ) (cid:111) W E a parameter. The following are equivalent: (i) The (cid:98) G -orbit of ϕ in Z ( W E , (cid:98) G ) L is closed. (ii) For any conjugate ϕ (cid:48) of ϕ such that ϕ (cid:48) : W E → (cid:98) P ( L ) (cid:111) W E factors over a standard parabolicsubgroup, ϕ is (cid:98) G ( L ) -conjugate to pr L M ◦ ϕ (cid:48) . (iii) ϕ is semi-simple. III.3. THE COARSE MODULI SPACE 281

Proof.

We use the Hilbert–Mumford–Kempf theorem, cf. [

Kem78 , Corollary 3.5]. Let λ : G m → (cid:98) G L . Up to conjugation we can assume λ ∈ X ∗ ( (cid:98) T ) + . For each τ ∈ W E there is a morphismev τ : Z ( W E , (cid:98) G ) L → (cid:98) G given by evaluating a parameter on τ . Thus, if lim t → λ ( t ) · ϕ exists, i.e.the associated morphism G m,L → Z ( W E , (cid:98) G ) L extends to A L , for each τ ∈ W E one has λ τ = λ and ϕ ( τ ) ∈ Q λ ( L ) (cid:111) τ , cf. Lemma VIII.3.3. One thus has Q λ = (cid:98) P for P a standard parabolic subgroupof G ∗ , and ϕ : W E → (cid:98) P (cid:111) W E .For g ∈ (cid:98) P , lim t → λ ( t ) gλ ( t ) − is the projection onto the standard Levi subgroup (cid:99) M . Thus,using the evaluation morphism ev τ for each τ we deduce that lim t → λ ( t ) · ϕ , if it exists, is given bythe composite W E ϕ −→ (cid:98) P ( L ) (cid:111) W E proj −−→ (cid:99) M ( L ) (cid:111) W E . Reciprocally, since the morphism G m × (cid:98) P → (cid:98) P ,given by ( t, g ) (cid:55)→ λ ( t ) gλ ( t ) − extends to A × (cid:98) P with ﬁber over 0 ∈ A given by the projection to (cid:99) M , for any ϕ : W E → (cid:98) P (cid:111) W E , lim t → λ ( t ) · ϕ exists.From this analysis we deduce the equivalence between (1) and (2). It is clear that (3) implies(2). For the proof of (2) implies (3) we use the results of [ BMR05 ] and [

Ric88 ]. For this we seeparameters as morphisms W → (cid:98) G (Λ) (cid:111) Q where W is discrete ﬁnitely generated as in the proofof Theorem VIII.1.3, and Q is a ﬁnite quotient of W . Let ϕ : W → (cid:98) G ( L ) (cid:111) Q satisfying (2). Let H ⊂ (cid:98) G L (cid:111) Q be the Zariski closure of the image of ϕ . Then if ( x , . . . , x n ) ∈ ( (cid:98) G ( L ) × Q ) n are theimages of a set of generators of W , applying the Hilbert–Mumford–Kempf criterion we see thatthe (cid:98) G L -orbit of ( x , . . . , x n ) via the diagonal action is closed, cf. the proof of [ BMR05 , Lemma2.17]. We can then apply [

Ric88 ], cf. [

BMR05 , Proposition 2.16], to deduce that H is stronglyreductive in (cid:98) G L (cid:111) Q and thus (cid:98) G L -completely reducible. Strictly speaking, since we are working ina non-connected situation, we use in fact [ BMR05 , Section 6]. (cid:3)

Lemma

VIII.3.3 . For λ : G m → (cid:98) G L and g (cid:111) τ ∈ (cid:98) G ( L ) (cid:111) W E , the limit lim t → λ ( t ) gλ ( t ) − τ existsif and only if g ∈ Q λ ( L ) , the parabolic subgroup attached to λ , and λ τ = λ . Proof.

Up to conjugation we can suppose λ ∈ X ∗ ( T ) + . Then, Q λ and Q λ τ are standardparabolic subgroups of (cid:98) G . Let us write g = g (cid:48) . wg (cid:48)(cid:48) with g (cid:48) ∈ Q λ ( L ), g (cid:48)(cid:48) ∈ Q λ τ ( L ) and w ∈ (cid:99) W .Then, writing λ ( t ) gλ ( t ) − τ = ( λ ( t ) g (cid:48) λ ( t ) − )( λ ( t ) . wλ ( t ) − τ )( λ ( t ) τ g (cid:48)(cid:48) λ ( t ) − τ ) , one deduces that lim t → λ ( t ) . wλ ( t ) − τ exists. Thus, lim t → ( λ ( λ − τ ) w )( t ) exists and thus λ = ( λ τ ) w .Since λ τ ∈ X ∗ ( (cid:98) T ) + we deduce λ = λ τ and λ w = λ . (cid:3) The proof shows that up to replacing (cid:98) G ( L ) (cid:111) W E by (cid:98) G ( L ) (cid:111) Q for some ﬁnite quotient Q of W E (as we can), semisimplicity of ϕ is equivalent to the Zariski closure of the image of ϕ beingcompletely reducible in the terminology of [ BMR05 , Section 6].

VIII.3.2. A presentation of O ( Z ( W E , (cid:98) G )) . It will be useful to have a presentation of thealgebra O ( Z ( W E , (cid:98) G )), or rather of the ﬁnite type Z (cid:96) -algebras O ( Z ( W E /P, (cid:98) G )) for open subgroups P of the wild inertia (with the property that the action of W E on (cid:98) G factors over W E /P ). Pick adiscrete dense subgroup W ⊂ W E /P as above, so that Z ( W E /P, (cid:98) G ) = Z ( W, (cid:98) G ). For any n ≥ F n → W , we get a (cid:98) G -equivariant map O ( Z ( F n , (cid:98) G )) → O ( Z ( W, (cid:98) G )) ,

82 VIII. L -PARAMETER where the source is isomorphic to O ( (cid:98) G n ) with appropriately twisted diagonal (cid:98) G -conjugation. Con-sider the category { ( n, F n → W ) } consisting of maps from ﬁnite free groups to W , with maps givenby commutative diagrams F n → F m → W ; this is a sifted index category (as it admits coproducts).The map colim ( n,F n → W ) O ( Z ( F n , (cid:98) G )) → O ( Z ( W, (cid:98) G ))is an isomorphism of algebras with (cid:98) G -action (as 1-cocycles from W to (cid:98) G are uniquely speciﬁed bycompatible collections of 1-cocycles F n → (cid:98) G for all F n → W ). By Haboush’s theorem on geometricreductivity [ Hab75 ] it follows that the mapcolim ( n,F n → W ) O ( Z ( F n , (cid:98) G )) (cid:98) G → O ( Z ( W, (cid:98) G )) (cid:98) G on (cid:98) G -invariants is a universal homeomorphism of ﬁnite type Z (cid:96) -algebras, and an isomorphism afterinverting (cid:96) . Definition

VIII.3.4 . The algebra of excursion operators (for Z ( W, (cid:98) G ) ) is Exc( W, (cid:98) G ) = colim ( n,F n → W ) O ( Z ( F n , (cid:98) G )) (cid:98) G . We see in particular that the geometric points of Exc( W, (cid:98) G ) and Z ( W, (cid:98) G ) agree.Actually, the following higher-categorical variant is true. Proposition

VIII.3.5 . Working in the derived ∞ -category D ( Z (cid:96) ) , the map colim ( n,F n → W ) O ( Z ( F n , (cid:98) G )) → O ( Z ( W, (cid:98) G )) is an isomorphism. Proof.

The left-hand side deﬁnes an animated Z (cid:96) -algebra, in fact the universal animated Z (cid:96) -algebra A with a condensed 1-cocycle W → (cid:98) G ( A ), and the right-hand side is given by π A . Nowthe deformation-theoretic arguments from the proof of Theorem VIII.1.3 show that A is a derivedcomplete intersection, but as π A has the correct dimension, we get A = π A . (cid:3) We will later prove an even ﬁner version, incorporating the (cid:98) G -action; we defer the proof toSection VIII.5. Theorem

VIII.3.6 . Assume that (cid:96) is a very good prime for (cid:98) G . Then the map colim ( n,F n → W ) O ( Z ( F n , (cid:98) G )) → O ( Z ( W, (cid:98) G )) is an isomorphism in the presentable stable ∞ -category Ind Perf( B (cid:98) G ) .In particular, the map Exc( W, (cid:98) G ) = colim ( n,F n → W ) O ( Z ( F n , (cid:98) G )) (cid:98) G → O ( Z ( W, (cid:98) G )) (cid:98) G is an isomorphism. In particular, we see that the algebra of excursion operators plays a central role. In the nextsubsection, we analyze it more explicitly.

III.3. THE COARSE MODULI SPACE 283

VIII.3.3. The algebra of excursion operators.

Fix a ﬁnite quotient Q of W E over whichthe W E -action on (cid:98) G factors. Let ( (cid:98) G (cid:111) Q ) n (cid:12) (cid:98) G be the quotient of ( (cid:98) G (cid:111) Q ) n under simultaneousconjugation by (cid:98) G . Proposition

VIII.3.7 . The algebra of excursion operators

Exc( W, (cid:98) G ) is the universal Z (cid:96) -algebra A equipped with maps Θ n : O (( (cid:98) G (cid:111) Q ) n (cid:12) (cid:98) G ) → Map( W n , A ) for n ≥ , linear over O ( Q n ) → Map( W n , A ) , subject to the following relations. If g : { , . . . , m } →{ , . . . , n } is any map, the induced diagram O (( (cid:98) G (cid:111) Q ) m (cid:12) (cid:98) G ) (cid:47) (cid:47) (cid:15) (cid:15) Map( W m , A ) (cid:15) (cid:15) O (( (cid:98) G (cid:111) Q ) n (cid:12) (cid:98) G ) (cid:47) (cid:47) Map( W n , A ) commutes, where both vertical maps are the natural pullback maps. On the other hand, g alsoinduces a map ( (cid:98) G (cid:111) Q ) m → ( (cid:98) G (cid:111) Q ) n , multiplying in every ﬁbre over i = 1 , . . . , n the terms in g − ( i ) (ordered by virtue of their ordering as a subset of { , . . . , m } ). This map is equivariantunder diagonal (cid:98) G -conjugation, and hence descends to the quotient. Similarly, g induces a map W m → W n . Then also the induced diagram O (( (cid:98) G (cid:111) Q ) n (cid:12) (cid:98) G ) (cid:47) (cid:47) (cid:15) (cid:15) Map( W n , A ) (cid:15) (cid:15) O (( (cid:98) G (cid:111) Q ) m (cid:12) (cid:98) G ) (cid:47) (cid:47) Map( W m , A ) commutes.The (cid:96) -torsion free quotient of Exc( W, (cid:98) G ) is also the universal ﬂat Z (cid:96) -algebra A (cid:48) equipped withmaps Θ (cid:48) n : O (( (cid:98) G (cid:111) Q ) n (cid:12) (cid:98) G ) → Map(( W E /P ) n , A (cid:48) ) for n ≥ , linear over O ( Q n ) → Map(( W E /P ) n , A (cid:48) ) , satisfying the same relations as in Propo-sition VIII.3.7, where the right-hand side Map(( W E /P ) n , A (cid:48) ) denotes the maps of condensed sets(where as usual A (cid:48) is considered as relatively discrete over Z (cid:96) ). In particular, the (cid:96) -torsion freequotient of Exc( W, (cid:98) G ) is independent of the discretization W of W E /P . We do not know whether it is necessary to pass to the (cid:96) -torsion free quotient for the ﬁnalassertion. Note that if (cid:96) is very good for (cid:98) G , then Exc( W, (cid:98) G ) ∼ = O ( Z ( W E /P, (cid:98) G )) (cid:98) G is ﬂat over Z (cid:96) .Moreover note that the (cid:96) -torsion in Exc( W, (cid:98) G ) is always nilpotent, so passing to this quotient is auniversal homeomorphism. Proof.

The datum of the Θ n is equivalent the datum of a map of algebras O ( Z ( F n , (cid:98) G )) (cid:98) G → A for each map F n → W . The relations encode the relations arising in the diagram category( n, F n → W ) corresponding to maps F m → F n (over W ) either sending generators to generators, or

84 VIII. L -PARAMETER multiplying subsets of generators. If one would also allow the inversion of elements, then this wouldgenerate all required relations. We leave it as an exercise to see that this relation, correspondingto F n → F n which is the identity on the ﬁrst n − n -th generator, isin fact enforced by the others. (Hint: Look at the part of Θ n +1 corresponding to ( γ , . . . , γ n , γ − n )and use that under multiplication of the last two variables, this maps to ( γ , . . . , γ n − , γ , . . . , γ n − ).)The second description is a priori stronger as Map(( W E /P ) n , A (cid:48) ) injects into Map( W n , A (cid:48) ) as W ⊂ W E /P is dense. The (cid:96) -torsion free quotient of Exc( W, (cid:98) G ) injects into O ( Z ( W E /P, (cid:98) G )) (cid:98) G (aswe have an isomorphism after inverting (cid:96) ), and it is clear that the elements of Map( W n , Exc( W, (cid:98) G ))map to elements of Map(( W E /P ) n , O ( Z ( W E /P, (cid:98) G )) (cid:98) G ). Thus, this already happens on the (cid:96) -torsion free quotient of Exc( W, (cid:98) G ), which thus has the desired universal property. (cid:3) Regarding the passage to W E in place of W E /P , where there is no natural (ﬁnite type) algebraanymore, we still have the following result. Proposition

VIII.3.8 . Let L be an algebraically closed ﬁeld over Z (cid:96) . Then the following arein canonical bijection. (i) Semisimple L -parameters ϕ : W E → (cid:98) G ( L ) (cid:111) W E , up to (cid:98) G ( L ) -conjugation. (ii) L -valued points of Z ( W E , (cid:98) G ) (cid:12) (cid:98) G . (iii)) Collections of maps of Z (cid:96) -algebras Θ n : O (( (cid:98) G (cid:111) Q ) n (cid:12) (cid:98) G ) → Map( W nE , L ) for n ≥ , linear over O ( Q n ) → Map( W nE , L ) , such that for any map g : { , . . . , m } → { , . . . , n } ,the diagrams O (( (cid:98) G (cid:111) Q ) m (cid:12) (cid:98) G ) Θ m (cid:47) (cid:47) (cid:15) (cid:15) Map( W mE , L ) (cid:15) (cid:15) O (( (cid:98) G (cid:111) Q ) n (cid:12) (cid:98) G ) Θ n (cid:47) (cid:47) Map( W nE , L ) induced by pullback, and O (( (cid:98) G (cid:111) Q ) n (cid:12) (cid:98) G ) Θ n (cid:47) (cid:47) (cid:15) (cid:15) Map( W nE , L ) (cid:15) (cid:15) O (( (cid:98) G (cid:111) Q ) m (cid:12) (cid:98) G ) Θ m (cid:47) (cid:47) Map( W mE , L ) induced by multiplication, commute. Proof.

We already know that (i) and (ii) are in natural bijection. The recipe above gives acanonical map from (ii) to (iii). Now take data as in (iii). Forgetting the continuity of all maps,we see that data as in (iii) gives rise to a semisimple 1-cocycles ϕ : W E → (cid:98) G ( L ) (of discretegroups), up to conjugation. We need to see that if the data in (iii) are maps of condensed sets,then ϕ is also a map of condensed sets (this condition does not depend on the representative of itsconjugacy class). Consider the corresponding map ϕ (cid:48) : W E → (cid:98) G ( L ) (cid:111) Q and let Z ⊂ (cid:98) G L (cid:111) Q be III.4. EXCURSION OPERATORS 285 the Zariski closure of its image; by semisimplicity, Z is completely reductive. We can ﬁnd ﬁnitelymany elements γ , . . . , γ n ∈ W E such that Z is also the normalizer of the ϕ (cid:48) ( γ i ). By a theoremof Richardson [ Ric88 ], see [

BMR05 , Corollary 3.7] in the positive characteristic case, it followsthat the orbit of ( ϕ (cid:48) ( γ ) , . . . , ϕ (cid:48) ( γ n )) under simultaneous (cid:98) G -conjugation is closed. Moreover, forany further element γ , the (cid:98) G -orbit of ( ϕ (cid:48) ( γ ) , . . . , ϕ (cid:48) ( γ n ) , ϕ (cid:48) ( γ )) maps isomorphically to the (cid:98) G -orbitof ( ϕ (cid:48) ( γ ) , . . . , ϕ (cid:48) ( γ n )) as both are given by (cid:98) G/Z . It follows that ϕ (cid:48) ( γ ) ∈ (cid:98) G ( L ) (cid:111) Q is determineduniquely by the condition that ( ϕ (cid:48) ( γ ) , . . . , ϕ (cid:48) ( γ n ) , ϕ (cid:48) ( γ )) lies in the correct (cid:98) G -orbit (which is closed).This is determined by the map O (( (cid:98) G (cid:111) Q ) n +1 (cid:12) (cid:98) G ) → Map( W n +1 E , L )when evaluated at ( γ , . . . , γ n , γ ). As we required this to be a map of condensed sets, the resultfollows. (cid:3) VIII.4. Excursion operators

One can use Proposition VIII.3.8 to construct L -parameters in the following general categoricalsituation. In order to avoid topological problems, we work in the setting of the discrete subgroup W ⊂ W E /P ; in fact, we can take here any discrete group W . Let Λ be a discrete Z (cid:96) -algebra andlet C be a Z (cid:96) -linear category. Assume that functorially in ﬁnite sets I , we are given a monoidalRep Z (cid:96) ( Q I )-linear functor Rep Z (cid:96) ( (cid:98) G (cid:111) Q ) I → End( C ) BW I : V (cid:55)→ T V where End( C ) is the category of endomorphisms of C , and End( C ) BW IE is the category of F ∈ End( C )equipped with a map of groups W I → Aut( F ).The goal of this section is to prove the following theorem; this is essentially due to V. Laﬀorgue[ Laf18 ]. Theorem

VIII.4.1 . Given the above categorical data, there is a natural map of algebras

Exc( W, (cid:98) G ) = colim ( n,F n → W ) O ( Z ( F n , (cid:98) G )) (cid:98) G → End(id C ) to the Bernstein center of C (i.e., the algebra of endomorphisms of the identity of C ). To prove Theorem VIII.4.1, we construct explicit “excursion operators”. These are associatedto the following data.

Definition

VIII.4.2 . An excursion datum is a tuple D = ( I, V, α, β, ( γ i ) i ∈ I ) consisting of aﬁnite set I , an object V ∈ Rep Z (cid:96) (( (cid:98) G (cid:111) Q ) I ) with maps α : 1 → V | Rep Z (cid:96) ( (cid:98) G ) , β : V | Rep Z (cid:96) ( (cid:98) G ) → andelements γ i ∈ W , i ∈ I . Here, the restriction Rep Z (cid:96) (( (cid:98) G (cid:111) Q ) I ) → Rep Z (cid:96) ( (cid:98) G ) is the restriction to the diagonal copy of (cid:98) G ⊂ (cid:98) G I ⊂ ( (cid:98) G (cid:111) Q ) I .Now consider excursion data D = ( I, V, α, β, ( γ i ) i ∈ I ). These give rise to an endomorphism ofthe identity functor of C , as follows. S D : id = T T α −→ T V ( γ i ) i ∈ I −−−−→ T V T β −→ T = id .

86 VIII. L -PARAMETER Varying the γ i , this gives a map of groups W I → Aut(id C )to the automorphisms of the identity functor on C .We note that if we have two excursion data D = ( I, V, α, β, ( γ i ) i ∈ I ) and D (cid:48) = ( I, V (cid:48) , α (cid:48) , β (cid:48) , ( γ i ) i ∈ I )with same ﬁnite set I and elements γ i ∈ W , and a map g : V → V (cid:48) taking α to α (cid:48) and β (cid:48) to β (bypost- and pre-composition), then S D = S D (cid:48) . Indeed, the diagram T T α (cid:47) (cid:47) T V ( γ i ) i ∈ I (cid:47) (cid:47) T g (cid:15) (cid:15) T V T β (cid:47) (cid:47) T g (cid:15) (cid:15) T T T α (cid:48) (cid:47) (cid:47) T V (cid:48) ( γ i ) i ∈ I (cid:47) (cid:47) T V (cid:48) T β (cid:48) (cid:47) (cid:47) T commutes. Now note that ( V, α, β ) give rise to an element f = f ( V, α, β ) ∈ O ( (cid:98) G \ ( (cid:98) G (cid:111) Q ) I / (cid:98) G ) , the quotient under diagonal left and right multiplication. Indeed, given any g i ∈ (cid:98) G (cid:111) Q , i ∈ I , onecan form the composite 1 α −→ V ( g i ) i ∈ I −−−−→ V β −→ , giving an element of the base ring; as α and β are equivariant for the diagonal (cid:98) G -action, this indeedgives an element f = f ( V, α, β ) ∈ O ( (cid:98) G \ ( (cid:98) G (cid:111) Q ) I / (cid:98) G ) . Conversely, given f we can look at the ( (cid:98) G (cid:111) Q ) I -representation V = V f ⊂ O (( (cid:98) G (cid:111) Q ) I / (cid:98) G ) generatedby f . This comes with a map α f : 1 → V f | Rep Z (cid:96) ( (cid:98) G ) induced by the element f , and a map β f : V f | Rep Z (cid:96) ( (cid:98) G ) → ∈ ( (cid:98) G (cid:111) Q ) I . If we replace V by the subrepresentationgenerated by α , then there is a natural map V → V f taking α to α f and β f to β . The abovecommutative diagrams then imply that S D depends on ( V, α, β ) only through f , and we get a map(a priori, of Z (cid:96) -modules) Θ I : O ( (cid:98) G \ ( (cid:98) G (cid:111) Q ) I / (cid:98) G ) → Map( W I , End(id C )) . Restricted to O ( Q I ), this is given by the natural map O ( Q I ) → Map( W I , Λ) (and Λ → End(id C )).Also, it follows from the deﬁnitions that for any map g : I → J , the diagram O ( (cid:98) G \ ( (cid:98) G (cid:111) Q ) I / (cid:98) G ) Θ I (cid:47) (cid:47) (cid:15) (cid:15) Map( W I , End(id C )) (cid:15) (cid:15) O ( (cid:98) G \ ( (cid:98) G (cid:111) Q ) J / (cid:98) G ) Θ J (cid:47) (cid:47) Map( W J , End(id C )) , induced by pullback along g , is cartesian.We want to check that Θ I is a map of algebras. For this, we use a version of “convolutionproduct = fusion product” in this situation. Namely, given f , f ∈ O ( (cid:98) G \ ( (cid:98) G (cid:111) Q ) I / (cid:98) G ), we canbuild their exterior product f (cid:2) f ∈ O ( (cid:98) G \ ( (cid:98) G (cid:111) Q ) I (cid:116) I / (cid:98) G ). Then one easily checksΘ I (cid:116) I ( f (cid:2) f )(( γ i , γ (cid:48) i ) i ∈ I ) = Θ I ( f )(( γ i ) i ∈ I )Θ I ( f )(( γ (cid:48) i ) i ∈ I ) . III.5. MODULAR REPRESENTATION THEORY 287

Applying now functoriality for pullback under I (cid:116) I → I , it follows that indeed Θ I ( f f ) =Θ I ( f )Θ I ( f ).For any n ≥

0, we can identify O ( (cid:98) G \ ( (cid:98) G (cid:111) Q ) { ,...,n } / (cid:98) G ) ⊗ O ( Q { ,...,n } ) O ( Q { ,...,n } ) ∼ = O (( (cid:98) G (cid:111) Q ) n (cid:12) (cid:98) G )via pullback under ( g , . . . , g n ) (cid:55)→ (1 , g , . . . , g n ). This translates Θ { ,...,n } into maps of Z (cid:96) -algebrasΘ n : O (( (cid:98) G (cid:111) Q ) n (cid:12) (cid:98) G ) → Map( W n , End(id C ))over O ( Q n ) → Map( W n , Λ), still satisfying compatibility with pullback under maps g : { , . . . , m } →{ , . . . , n } .Arguing also as in [ Laf18 , Lemma 10.1, equation (10.5)] and the resulting [

Laf18 , Proposition10.8 (iii), Deﬁnition-Proposition 11.3 (d)], one sees that the maps Θ n are also compatible with themultiplication maps induced by such maps g , thus ﬁnishing the proof of Theorem VIII.4.1.In particular, using the description of geometric points, Theorem VIII.4.1 implies the followingproposition. Corollary

VIII.4.3 . Assume that

Λ = L is an algebraically closed ﬁeld and X ∈ C is an objectwith End( X ) = L . Then there is, up to (cid:98) G ( L ) -conjugation, a unique semisimple L -parameter ϕ X : W → (cid:98) G ( L ) (cid:111) W such that for all excursion data D = ( I, V, α, β, ( γ i ) i ∈ I ) , the endomorphism S D ( X ) ∈ End( X ) = L , X = T ( X ) α −→ T V ( X ) ( γ i ) i ∈ I −−−−→ T V ( X ) β −→ T ( X ) = X, is given by the composite L α −→ V ( ϕ X ( γ i )) i ∈ I −−−−−−−→ V β −→ L. VIII.5. Modular representation theory

The goal of this section is to give a proof of Theorem VIII.3.6. In fact, we prove a slightreﬁnement of it, concerning perfect complexes, that will be useful in the construction of the spectralaction.

Theorem

VIII.5.1 . Assume that (cid:96) is a very good prime for (cid:98) G . Then the map colim ( n,F n → W ) O ( Z ( F n , (cid:98) G )) → O ( Z ( W, (cid:98) G )) is an isomorphism in the presentable stable ∞ -category Ind Perf( B (cid:98) G ) . Moreover, the ∞ -category Perf( Z ( W, (cid:98) G ) / (cid:98) G ) is generated under cones and retracts by Perf( B (cid:98) G ) , and Ind Perf( Z ( W, (cid:98) G ) / (cid:98) G ) identiﬁes with the ∞ -category of modules over O ( Z ( W, (cid:98) G )) in Ind Perf( B (cid:98) G ) . The diﬃculties in this theorem all arise on the special ﬁbre. Indeed, we will show below thatwe can reduce to the following version in characteristic (cid:96) .

88 VIII. L -PARAMETER Theorem

VIII.5.2 . Assume that (cid:96) is a very good prime for (cid:98) G , and let L = F (cid:96) . Then the map colim ( n,F n → W ) O ( Z ( F n , (cid:98) G ) L ) → O ( Z ( W, (cid:98) G ) L ) is an isomorphism in the presentable stable ∞ -category Ind Perf( B (cid:98) G ) . Moreover, the ∞ -category Perf( Z ( W, (cid:98) G ) L / (cid:98) G ) is generated under cones and retracts by Perf( B (cid:98) G ) . Then we have the following reduction:

Theorem VIII.5.2 implies Theorem VIII.5.1.

For the colimit claim, we need to see thatfor all representations V of (cid:98) G , the mapcolim ( n,F n → W ) R Γ( (cid:98) G, O ( Z ( F n , (cid:98) G )) ⊗ V ) → R Γ( (cid:98) G, O ( Z ( W, (cid:98) G )) ⊗ V )is an isomorphism. It is an isomorphism after inverting (cid:96) , as then the representation theory of (cid:98) G is semisimple, and it is true on underlying complexes by Proposition VIII.3.5. Thus, it suﬃces toshow that it is an isomorphism after inverting (cid:96) , or even after base change to L , which follows fromTheorem VIII.5.2.For the other half, note ﬁrst that if Perf( B (cid:98) G ) generates Perf( Z ( W, (cid:98) G ) / (cid:98) G ), then it followsby Barr–Beck–Lurie [ Lur16 , Theorem 4.7.4.5] that Ind Perf( Z ( W, (cid:98) G ) / (cid:98) G ) is the ∞ -category ofmodules over O ( Z ( W, (cid:98) G )) in Ind Perf( B (cid:98) G ). Now take any V ∈ Perf( Z ( W, (cid:98) G ) / (cid:98) G ). As its lowestcohomology group is ﬁnitely generated, we can ﬁnd some surjection from an induced vector bundleonto it, and by passing to cones reduce the perfect amplitude until V is a (cid:98) G -equivariant vectorbundle on Z ( W, (cid:98) G ). We may then again ﬁnd a representation W of (cid:98) G and a surjection W ⊗O ( Z ( W, (cid:98) G )) → V . This map splits after inverting (cid:96) , showing that V is a retract of an inducedvector bundle up to a power of (cid:96) . Thus, it suﬃces to show that V /(cid:96) lies in this subcategory, andthis follows from Theorem VIII.5.2. (cid:3)

Thus, we concentrate now on Theorem VIII.5.2, which takes place over an algebraically closedbase ﬁeld L of characteristic (cid:96) . For the proof, we need many preparations on the modular represen-tation theory of reductive groups, for (cid:98) G and many of its subgroups. As everything here happens onthe dual side but we do not want to clutter notation, we will change notation, only for this section,and write G for reductive groups over L . VIII.5.1. Good ﬁltrations.

First, we need to recall the notion of good ﬁltrations. Let G be a reductive group over L ; as disconnected groups will appear frequently below, we stress thatreductive groups are always connected for us, so in particular G is assumed to be connected here.Let T ⊂ B ⊂ G be a torus and Borel for G . For any dominant cocharacter λ of T , we have theinduced representation ∇ λ = H ( G/B, O ( λ )) . A representation V of G has a good ﬁltration if it admits an exhaustive ﬁltration0 = V − ⊂ V ⊂ V ⊂ . . . ⊂ V such that each V i /V i − is isomorphic to a direct sum of ∇ λ ’s. If one picks a total ordering 0 = λ , λ , . . . of the dominant cocharacters, compatible with their dominance order, one can choose III.5. MODULAR REPRESENTATION THEORY 289 V i ⊂ V to be the maximal subrepresentation admitting only weights λ j with j ≤ i . In that case, V i /V i − is generated by its weight space W i of weight λ i , and by adjunction there is a map V i /V i − → W i ⊗ ∇ λ i ;then V admits a good ﬁltration if and only if all of these maps are isomorphisms. For this, it isin fact enough that V i → W i ⊗ ∇ λ i is surjective: The kernel is necessarily given by V i − , as it hasonly smaller weights.A key result is that if V and W admit good G -ﬁltrations, then so does V ⊗ W ; this is a theoremof Donkin [ Don85 ] and Mathieu [

Mat90 ] in general. Moreover, if V admits a good ﬁltration, then H i ( G, V ) = 0 for i >

0: This clearly reduces to the case of V = ∇ λ , in which case it follows fromKempf’s vanishing theorem [ Kem76 ]. These results imply the following standard characterizationof modules admitting a good ﬁltration.

Proposition

VIII.5.3 ([

Don81 ]) . A G -representation V admits a good ﬁltration if and onlyif for all λ , one has H i ( G, V ⊗ ∇ λ ) = 0 for i > . Using this, one can deﬁne a well-behaved notion of a “good ﬁltration dimension” of V , referringto the minimal i such that H j ( G, V ⊗ ∇ λ ) = 0 for all λ and j > i . Equivalently, there is aresolution of length i by representations with a good ﬁltration. We will use the following deepresult of Touz´e–van der Kallen [ TvdK10 , Corollary 1.5].

Theorem

VIII.5.4 . Let A be a ﬁnitely generated L -algebra with an action of G . Let M be a G -equivariant ﬁnitely generated A -module. Then H i ( G, M ) is a ﬁnitely generated A G -module forall i ≥ . Moreover, if A has a good ﬁltration, then M has ﬁnite good ﬁltration dimension. Another key result we need is the following.

Theorem

VIII.5.5 ([

Kop84 ], [

Don88 ]) . The G × G -representation O ( G ) (via left and rightmultiplication) admits a good ﬁltration. In particular, we have the following corollary. For any n ≥

0, let F n be the free group on n letters. Corollary

VIII.5.6 . For any map F n → Aut( G ) , the G -representation O ( Z ( F n , G )) admitsa good ﬁltration. Proof.

Note that Z ( F n , G ) = G n , where the G -action is that of simultaenous twisted conju-gation (by the n given automorphisms of G ). But O ( G n ) admits a good ﬁltration as representationof G n , and restricting to G ⊂ G n it remains good by stability under tensor products (and as theinduced representations of G n are tensor products of induced representations of each factor). (cid:3) Another property of relevance to us is the following.

Theorem

VIII.5.7 ([

AJ84 , 4.4]) . Assume that (cid:96) is very good for G . Then all symmetric powers Sym n g ∗ have a good ﬁltration.

90 VIII. L -PARAMETER VIII.5.2. Reductions.

To prove Theorem VIII.5.2, we will use the following criterion.

Theorem

VIII.5.8 . Let X be an aﬃne scheme of ﬁnite type over L equipped with an action ofa reductive group G . Assume the following conditions. (i) The scheme X is a local complete intersection. (ii) For any closed G -orbit Z ∼ = G/H ⊂ X , the algebra O ( G/H ) has a good G -ﬁltration, and theimage of Perf( BG ) → Perf( BH ) generates under cones and retracts. (iii) With Z as in (ii), the G -equivariant L Z/X [ − ∈ D [ − , ( O ( Z )) has the property that all Sym n O ( Z ) ( L Z/X [ − ∈ D [ − n, ( O ( Z )) have good ﬁltration dimension ; i.e., for all induced rep-resentations V , one has H i ( G, Sym n O ( Z ) ( L Z/X [ − ⊗ V ) = 0 for i ≥ .Then O ( X ) has a good G -ﬁltration, and Perf(

X/G ) is generated (under cones and retracts) by Perf( BG ) . Proof.

To see that A = O ( X ) has a good ﬁltration, we need to show that for every G -representation V with a good ﬁltration, H i ( G, V ⊗ O ( X )) = 0 for i >

0. Note that this cohomologygroup is a ﬁnitely generated A G -module by Theorem VIII.5.4. If it is not zero, pick some closedpoint of Spec A G in its support, and let Z ⊂ X be the corresponding closed G -orbit. There isa (derived) ﬁltration I • ⊂ A such that I = A , I is the ideal of Z , and in general I n / L I n +1 ∼ =Sym n O ( Z ) ( L Z/X [ − X was smooth, we could simply take the ﬁltration by powers ofthe ideal I . In general, we may pick a G -equivariant simplicial resolution of A by smooth algebras,and totalize the corresponding ﬁltrations. Consider the G -equivariant animated A -algebra (cid:101) A = (cid:77) n ∈ Z I n where by deﬁnition I n = A for n ≤

0. (Again, this can be deﬁned by passage to G -equivariantsmooth resolutions.) Let t ∈ (cid:101) A be the generator in degree −

1. Then (cid:101) A/ L t = (cid:77) n ≥ I n / L I n +1 = (cid:77) n ≥ Sym n O ( Z ) ( L Z/X [ − ≤

0. We also note that it is concentrated in a bounded range ofdegrees, bounded by the number of generators for the ideal deﬁning the (local) complete intersection A . This implies that for any V with a good G -ﬁltration, H i ( G, V ⊗ (cid:101) A ) are coherent H ( G, (cid:101) A )-modules whose reduction modulo t vanishes for i >

0. Moreover, after inverting t , one has (cid:101) A [ t − ] = A [ t ± ]. Thus, the support of H i ( G, V ⊗ (cid:101) A ) in Spec H ( G, (cid:101) A ) contains Spec O ( Z )[ t ± ]. As it isclosed, it contains the spectrum of (cid:76) n ≤ ( A/I ) G = O ( Z )[ t ], and thus meets the locus t = 0, incontradiction to its vanishing modulo t . We note that it follows now that also (cid:101) A has good ﬁltrationdimension ≤

0, as all H i ( G, V ⊗ (cid:101) A ) vanish after inverting t and after reduction modulo t .Now we prove that Perf( X/G ) is generated by Perf( BG ). Given N ∈ Perf(Spec

A/G ), let i ∈ Z be maximal such that H i ( N ) (cid:54) = 0. We may pick a G -representation V with a map V [ − i ] → N | Perf( BG ) so that A ⊗ Z (cid:96) V [ − i ] → N has the property that the induced map on H i is surjective. III.5. MODULAR REPRESENTATION THEORY 291

Passing to the ﬁbre, we may then induct on the perfect amplitude of N to reduce to the case that N = M [0] is a vector bundle.Now we use the theorem of Touz´e–van der Kallen, Theorem VIII.5.4, which ensures that M has ﬁnite good ﬁltration dimension, i.e. there is some integer i such that for all λ and all j > i one has H j ( G, M ⊗ ∇ λ ) = 0 for j > i . By Proposition VIII.5.9, when we choose V above, we canassume that V ∗ admits a good ﬁltration. Choosing such a V for N = M ∗ [0], we can thus ﬁnd astrict injection M → A ⊗ V ∗ of vector bundles such that V ∗ admits a good ﬁltration. Passing tothe cokernel then decrases the good ﬁltration dimension of M , and we can reduce to the case that M has a good ﬁltration.We have a G -equivariant resolution . . . → A ⊗ A ⊗ M → A ⊗ M → M of G -equivariant A -modules, which is canonically split in Ind Perf( BG ). The class of all M forwhich this is a resolution in Ind Perf( X/G ) is exactly the class of all M generated under conesand retracts by Perf( BG ) – in one direction, it is a resolution in Ind Perf( X/G ) when M arises viapullback from BG (as it can then be split), and thus for all M generated by Perf( BG ). In the otherdirection, if it is a resolution in Ind Perf( X/G ), then it follows that M is generated by Perf( BG )as all other terms arise via pullback from BG . Now note that this is a resolution in Ind Perf( X/G )if and only if for all N ∈ Perf(

X/G ), which we may again assume to be a G -equivariant ﬁniteprojective A -module with a good G -ﬁltration, the complex . . . → ( N ⊗ A ⊗ M ) G → ( N ⊗ M ) G → ( N ⊗ A M ) G → H i ( G, N ⊗ A M ) = 0 for i >

0. (Note that all terms N ⊗ M , N ⊗ A ⊗ M etc. have a good G -ﬁltration, hence there is no higher G -cohomology.) In particular, these observations show thatif M has a good ﬁltration and lies in the full subcategory of Perf( X/G ) generated by Perf( BG ),then for all G -equivariant ﬁnite projective A -modules N with a good ﬁltration, also M ⊗ A N hasa good G -ﬁltration.We note that the previous arguments also apply in case A = O ( Z ) for an orbit Z as in (ii),in which case the assumption of (ii) implies that for two G -equivariant O ( Z )-perfect complexes M , M ∈ Perf(Spec O ( Z ) /G ) with G -good ﬁltration dimension ≤

0, also their tensor product is of G -good ﬁltration dimension ≤ G -equivariant ﬁnite projective A -module M , we can now ﬁnd somelarge enough V ⊂ M with a good ﬁltration inducing a surjection A ⊗ V → M . Moreover, as( M ⊗ L [ G/U ]) G is ﬁnitely generated over ( A ⊗ L [ G/U ]) G (for example, by Theorem VIII.5.4), wecan even ensure that the map stays surjective after tensoring with L [ G/U ] = (cid:76) λ ∇ λ and taking G -invariants. This implies that the kernel M (cid:48) of A ⊗ V → M still has a good ﬁltration.In other words, we can ﬁnd an inﬁnite resolution . . . → A ⊗ V → A ⊗ V → M → V i are G -representations with a good ﬁltration, and the kernel M i of A ⊗ V i → A ⊗ V i − is always a G -equivariant ﬁnite projective A -module that admits a good ﬁltration. In particular, itis a resolution Ind Perf( BG ). It suﬃces to see that this is in fact a resolution in Ind Perf( X/G ). Tosee this, we need to see that for any N ∈ Perf(

X/G ), which we may assume to be a G -equivariant

92 VIII. L -PARAMETER vector bundle admitting a good G -ﬁltration, also the complex . . . → N ⊗ V → N ⊗ V → N ⊗ A M → BG ). We note that this is automatic if N ⊗ A M i has a good G -ﬁltrationfor all i .Now assume that for any closed point x of Spec A G , with localization ( A G ) x at x , that M ⊗ A G A Gx ∈ Perf(Spec( A ⊗ A G A Gx ) /G ) is generated by Perf( BG ). Then in particular, also M i ⊗ A G A Gx isgenerated by Perf( BG ), and hence N ⊗ A M i ⊗ A G A Gx has a good G -ﬁltration. This then impliesthat the A G -module H > ( G, N ⊗ A M i ⊗ V ) vanishes locally on Spec A G for any representation V with good G -ﬁltration, and hence vanishes, thus showing that N ⊗ A M i has a good G -ﬁltration.By the above, this then implies that M is also globally generated by Perf( BG ).We can thus work locally on Spec A G , and we ﬁx a closed point, corresponding to a closed G -orbit Z ⊂ X . With (cid:101) A = (cid:77) n ∈ Z I n as above, which we endow with an action of (cid:101) G = G × G m (where G m acts via the grading), weconsider (cid:101) G -equivariant ﬁnite projective (cid:101) A -module (cid:102) M = M ⊗ A (cid:101) A . By Theorem VIII.5.4, the O ( Z )-module M ⊗ A A/I has ﬁnite good ﬁltration dimension, and as all I n / L I n +1 have good ﬁltrationdimension ≤

0, it follows that (cid:102)

M /t = (cid:76) n ≥ ( M ⊗ A A/I ) ⊗ A/I I n / L I n +1 has ﬁnite good ﬁltrationdimension. By ﬁnite generation arguments on cohomology groups, this then also implies that (cid:102) M has ﬁnite good ﬁltration dimension. Arguing as for the A -module M , we can then ﬁrst arrangethat (cid:102) M has good ﬁltration dimension ≤

0, and then ﬁnd V , V , . . . ∈ Perf [ − , ( BG ) with goodﬁltration dimension ≤ . . . → (cid:101) A ⊗ V → (cid:101) A ⊗ V → (cid:102) M → B (cid:101) G ), with the limit (cid:102) M i of [ (cid:101) A ⊗ V i → (cid:101) A ⊗ V i − → . . . → (cid:102) M ] being a (cid:101) G -equivariant ﬁnite projective (cid:101) A -module with good ﬁltration dimension ≤

0. Reducing modulo t ,the sequence . . . → (cid:101) A/t ⊗ V → (cid:101) A/t ⊗ V → (cid:102) M /t → (cid:101) A/t ) /G ), as in fact (cid:102) M /t = ( M ⊗ A A/I ) ⊗ A/I (cid:101)

A/t lies in the subcate-gory generated by Perf( BG ), as this is true for M ⊗ A A/I ∈ Perf(

Z/G ) = Perf( BH ) by assumption(ii). Indeed, reversing some arguments above, this implies that for all G -equivariant ﬁnite projective (cid:101) A/t -modules N with good ﬁltration dimension ≤

0, also N ⊗ (cid:101) A/t (cid:102) M i /t has good ﬁltration dimension ≤

0, which gives that the displayed resolution is indeed a resolution in Ind Perf(Spec( (cid:101)

A/t ) /G ).Now consider any (cid:101) N ∈ Perf(Spec (cid:101) A/ (cid:101) G ). We are interested to what extent . . . → ( (cid:101) N ⊗ V ) G → ( (cid:101) N ⊗ V ) G → ( (cid:101) N ⊗ (cid:101) A (cid:102) M ) G → H ( G, (cid:101) A )-modules, and theirreduction modulo t vanishes. This implies, as above, that their support is disjoint from the (cid:101) G -orbitSpec O ( Z )[ t ± ] → Spec (cid:101) A (as the closure of this orbit meets the ﬁbre t = 0). III.5. MODULAR REPRESENTATION THEORY 293

We can apply this in particular to (cid:101) N = N ⊗ A (cid:101) A for some G -equivariant ﬁnite projective A -module with good ﬁltration. Inverting t in the above sequence, we get the complex . . . → ( N ⊗ V ) G [ t ± ] → ( N ⊗ V ) G [ t ± ] → ( N ⊗ A M ) G [ t ± ] → O ( Z )[ t ± ]. In other words, weget that . . . → ( N ⊗ V ) G ⊗ A G A Gx → ( N ⊗ V ) G ⊗ A G A Gx → ( N ⊗ A M ) G ⊗ A G A Gx → . . . → A ⊗ A G A Gx ⊗ V → A ⊗ A G A Gx ⊗ V → M ⊗ A G A Gx → A ⊗ A G A Gx ) /G ). (Note that we can repeat the arguments afterreplacing A G by some localization of it, and N there.) In particular, M ⊗ A G A Gx ∈ Perf(Spec( A ⊗ A G A Gx ) /G ) is generated by Perf( BG ). As we discussed above, this result for all closed points x ∈ Spec A G gives the claim. (cid:3) Proposition

VIII.5.9 . Let V be a representation of G . Then there is a surjection W → V such that W ∗ admits a good ﬁltration. Proof.

We can look at the subcategory of Rep( G ) where all weights are within some ﬁniteset (closed under the dominance order). It is well-known that this category admits a projectivegenerator P , and the projective generator has the property that its dual P ∗ has a good ﬁltration;this follows for example from H ( G, P ∗ ⊗ Z (cid:96) ∇ λ ) = Ext ( P, ∇ λ ) = 0 for all relevant ∇ , and thecharacterization of Proposition VIII.5.3. This implies the proposition. (cid:3) To produce examples of algebras satisfying Theorem VIII.5.8, we will use the following propo-sition.

Proposition

VIII.5.10 . Let A • → A be a G -equivariant simplicial resolution of algebras of ﬁ-nite type over L equipped with an action of G , such that A and all A • are local complete intersections.Assume that A i has a good G -ﬁltration for all i , and that for all closed orbits Z ∼ = G/H ⊂ Spec A , O ( Z ) admits a good G -ﬁltration and the image of Perf( BG ) → Perf( BH ) generates under conesand retracts. Moreover, assume that for all i ≥ and n ≥ , Sym n ( L Z/ Spec A i [ − ∈ Perf( O ( Z ) /G ) has good ﬁltration dimension ≤ , and that for varying i , L Z/ Spec A i ∈ Perf( O ( Z ) /G ) = Perf( BH ) involves only ﬁnitely many irreducible representations of H . Then the map colim A • → A is an isomorphism in Ind Perf( BG ) , and for all n ≥ the map colim Sym n ( L Z/ Spec A • [ − → Sym n ( L Z/ Spec A [ − ∈ Perf( O ( Z ) /G ) is an isomorphism, so the right-hand side has good ﬁltration dimension ≤ . Proof.

First, note that the mapcolim L Z/ Spec A • → L Z/ Spec A ,

94 VIII. L -PARAMETER which is necessarily an isomorphism in D ( L ), is also an isomorphism in Ind Perf( O ( Z ) /G ) =Ind Perf( BH ) as it involves only ﬁnitely many irreducible representations of H . Similarly,colim Sym n ( L Z/ Spec A • [ − → Sym n ( L Z/ Spec A [ − O ( Z ) /G ), and in particular in Ind Perf( BG ). As all terms on the lefthave good ﬁltration dimension ≤

0, this implies that also Sym n ( L Z/ Spec A [ − ≤ A has a good G -ﬁltration. It remains to see thatcolim A • → A is an isomorphism in Ind Perf( BG ). Equivalently, for any representation V of G with a goodﬁltration, the map colim( A • ⊗ V ) G → ( A ⊗ V ) G is an isomorphism.Assume ﬁrst that V = L is trivial. Then A G • is a simplicial ﬁnite type L -algebra, and thus itscolimit B is an animated L -algebra that is almost of ﬁnite type; in other words, π B is a ﬁnitetype L -algebra, and π i B is a ﬁnitely generated π B -module for i >

0. Moreover, it follows fromHaboush’s theorem that the map π B → A G is a universal homeomorphism. Thus, if the map B → A G is not an isomorphism, the homotopy groups of its cone are ﬁnitely generated π B -algebras, and we can take some closed point of Spec π B ∼ = Spec A G in its support, correspondingto some closed G -orbit Z ∼ = G/H ⊂ Spec A . We can then form (cid:101) A and (cid:101) A • as in the proof ofTheorem VIII.5.8, and look at the map colim (cid:101) A G • → (cid:101) A G . Again, the homotopy groups of the cone are ﬁnitely generated, and so as in the proof of Theo-rem VIII.5.8 it suﬃces to show that it is an isomorphism after reduction modulo t . But modulo t ,we get (cid:101) A i /t = (cid:76) n ≥ Sym n ( L Z/ Spec A i [ − (cid:101) A/t , and the claim follows from theﬁrst paragraph of the proof.To see that for all V with a good ﬁltration, the mapcolim( A • ⊗ V ) G → ( A ⊗ V ) G is an isomorphism, we note that now both sides are in D ≤ ( A G ) with ﬁnitely generated cohomologygroups. We can then pick Z as before, and reduce to showing thatcolim( (cid:101) A • ⊗ V ) G → ( (cid:101) A ⊗ V ) G is an isomorphism after reduction modulo t , where it follows again from the direct sum decompo-sition of (cid:101) A • /t and (cid:101) A/t . (cid:3) VIII.5.3. Fixed point groups.

We will need to know some properties of the ﬁxed points H = G F of reductive groups G under a (ﬁnite) group F of automorphisms of G . These will ingeneral not stay connected, so to facilitate inductive arguments, we will also not assume that G isconnected here. We will, however, usually make the assumption that π G is of order prime to thecharacteristic (cid:96) of the base ﬁeld, and we will also assume that F is of order prime to (cid:96) .First, we have the following structural result. III.5. MODULAR REPRESENTATION THEORY 295

Proposition

VIII.5.11 . Let L be an algebraically closed ﬁeld of characteristic (cid:96) > , and let G be a smooth linear algebraic group over L such that G ◦ is reductive and π G is of order primeto (cid:96) . Assume that F is a ﬁnite group of order prime to (cid:96) acting on G and let H = G F be the ﬁxedpoints. Then H is a smooth linear algebraic group, H ◦ is reductive, and π H is of order prime to (cid:96) . We note that our proof of the last part of this proposition probably uses unnecessarily heavymachinery. Under the assumption that F is solvable, we will later give a completely explicit proof. Proof.

It is a standard fact that the ﬁxed points of a smooth aﬃne scheme under a ﬁnitegroup of order prime to the characteristic is still aﬃne and smooth. Moreover, by [

PY02 , Theorem2.1], H ◦ is reductive.For the ﬁnal statement, we can ﬁrst assume that G is connected (as π G is of order primeto (cid:96) ), and thus reductive. We consider the action of G on (cid:81) F G , where it acts on the factorenumerated by Θ ∈ F through Θ-twisted conjugation. Note that O ( (cid:81) F G ) has a good G -ﬁltration.By Theorem VIII.5.4, for any G -equivariant ﬁnitely generated O ( (cid:81) F G )-module M , the goodﬁltration dimension of M is ﬁnite, and in particular H i ( G, M ) = 0 for all large enough i .Now G/H is a closed orbit of G acting on (cid:81) F G (the orbit of the identity element). Moreover,if π H has an element of order (cid:96) , then we get a subgroup H (cid:48) ⊂ H with π H (cid:48) ∼ = Z /(cid:96) Z . In that case, R Γ( G, O ( G/H (cid:48) )) ∼ = R Γ( H (cid:48) , L ) ∼ = R Γ( Z /(cid:96) Z , L )has cohomology in all positive degrees, while O ( G/H (cid:48) ) correponds to a G -equivariant coherentsheaf on (cid:81) F G (equipped with its Θ-conjugation), so this contradicts the previous paragraph. (cid:3) We will also need the following result.

Proposition

VIII.5.12 . In the situation of Proposition VIII.5.11, assume that the simply con-nected cover G (cid:48) of G ◦ der has the property that all Sym n g (cid:48)∗ have a good G (cid:48) -ﬁltration. Then O ( G/H ) has a good G -ﬁltration. We recall that the assumption on symmetric powers of g (cid:48)∗ is satisﬁed for all (cid:96) in type A , forall (cid:96) (cid:54) = 2 in classical types, for all (cid:96) (cid:54) = 2 , E , where one needs (cid:96) (cid:54) = 2 , , Remark

VIII.5.13 . The result reduces easily to the case that G is connected, semisimpleand simply connected. In that case, the result is related, but apparently somewhat weaker, thanthe assertion that H ◦ ⊂ G is a Donkin subgroup, meaning that for any representation V of G with a good G -ﬁltration, its restriction to H ◦ has a good H ◦ -ﬁltration; here we only prove that H i ( H ◦ , V ) = 0 for i >

0. This stronger statement that H ◦ ⊂ G is a Donkin subgroup is actuallyknown in most cases, at least when F is solvable so that one can reduce to automorphisms ofprime order. In particular, involutions are handled completely in [ Bru98 , Section 2], [ vdK01 ],and the (few) remaining cases are mostly done in [

HM13 , Theorem 4.3.3]. In particular, assuming F solvable, one can show this way that the conclusion of the proposition is true for all (cid:96) in classicaltypes or G . The missing cases are the centralizers of semisimple elements of order 3 in F , E and E , and of order 3 and 5 in E . These cases are not handled for small (cid:96) in [ HM13 , Theorem 4.3.3].

96 VIII. L -PARAMETER Proof.

We need to see that for all G -representations V with a good ﬁltration, one has H i ( G, O ( G/H ) ⊗ V ) = 0 for i >

0. This translates into H i ( H, V | H ) = 0 for i >

0. The lat-ter can be checked after replacing H by H ◦ der (as π H is of order prime to (cid:96) ), or even a ﬁnite coverof it. From these remarks, one sees that one can assume that G is connected, semisimple and simplyconnected.As in the proof of Proposition VIII.5.11, we can embed G/H into (cid:81) F G . Moreover, F acts inthe following way on (cid:81) F G : It takes a tuple ( g f ) f ∈ F to ( g ff (cid:48) f ( g f (cid:48) ) − ) f . This action commutes withthe simultaneous twisted G -conjugation. Moreover, the ﬁxed points of F are exactly the 1-cocycles Z ( F, G ) ⊂ (cid:81) F G , with their G -conjugation. The map G/H (cid:44) → Z ( F, G ) is the inclusion of aconnected component. Note that L (cid:81) F G | Z ( F,G ) → L Z ( F,G ) is a map of G -equivariant vector bundles on Z ( F, G ), and it moreover carries a commuting F -action, where the target is the F -orbits of the source. As F is of order prime to (cid:96) , it follows thatthe map splits. In particular, restricting to G/H , we see that L (cid:81) F G | G/H → L G/H splits G -equivariantly. Using the transitivity triangle, it follows that if I ⊂ O ( (cid:81) F G ) is the idealsheaf of G/H , then

I/I is a G -equivariant direct summand of L (cid:81) F G | G/H . Note that G -equivariantvector bundles on G/H are equivalent to representations of H , and L (cid:81) F G | G/H corresponds to therepresentation (cid:76) F g ∗ | H . Taking symmetric powers, it follows that for all n , I n / L I n +1 is a G -equivariant direct summand of the G -equivariant vector bundle on G/H corresponding to a directsum of copies of Sym n g ∗ | H .Now let A = O ( (cid:81) F G ) with its G -action and F -action and G - and F -stable ideal I ⊂ A . Weconsider the complex . . . → A ⊗ F → A ⊗ F → A → O ( Z ( F, G )) → F -orbits on A , in the sense of animated L -algebras. This still computes O ( Z ( F, G )),as F is of order prime to (cid:96) and A is smooth. Let A • = A ⊗ F • be the corresponding simplicial L -algebra, with ideal I • ⊂ A • of G/H . Let (cid:101) A • = (cid:77) n ∈ Z (cid:101) I n • , where (cid:101) I n • = A • for n ≤

0. This deﬁnes a simplicial L [ t ]-algebra, where t is the natural degeneratorin degree −

1. Then (cid:101) A • [ t − ] = A • [ t ± ] and (cid:101) A • /t is the symmetric algebra on (cid:101) I • / (cid:101) I • over O ( G/H ).In particular, this is a resolution of O ( G/H ), and in fact (cid:101) A • is a resolution of O ( G/H )[ t ].Note that the good ﬁltration dimension of each (cid:101) A n is bounded by the good ﬁltration dimensionof O ( G/H ) plus 1. To see this, note that (cid:101) A n [ t − ] = A n [ t ± ] has a good ﬁltration, so it is enoughto show that the good ﬁltration dimension of (cid:101) A n /t is bounded by the good ﬁltration dimension of O ( G/H ). This is because all symmetric powers of (cid:101) I n / (cid:101) I n are direct summands of direct sums ofcopies of symmetric powers of g ∗ restricted to H . We checked this for n = 0, but for n > g ∗ (or applies the reasoning with F replaced by F n +1 ). III.5. MODULAR REPRESENTATION THEORY 297

We claim that for all G -representations V , the mapcolim R Γ( G, (cid:101) A • ⊗ V ) → R Γ( G, O ( G/H )[ t ] ⊗ V )is an isomorphism. Consider the simplicial L -algebra (cid:101) A G • , and its colimit B , an animated L -algebra (with π B ﬁnitely generated, and all π i B ﬁnitely generated π B -modules). Then for all i , H i ( G, (cid:101) A • ⊗ V ) deﬁnes a degree-wise ﬁnitely generated (cid:101) A G • -module (by Theorem VIII.5.4), andhence its colimit deﬁnes an animated B -module whose π j is ﬁnitely generated over π B , for all j .By the good ﬁltration bounds, we also know that these cohomology groups vanish for i suﬃcientlylarge. This implies that each cohomology group of the left-hand side is ﬁnitely generated over π B .Moreover, by Haboush’s theorem, the map π B → O ( G/H )[ t ] is a universal homeomorphism, andhence also the right-hand side is ﬁnitely generated over π B . In particular, the cohomology groupsof the cone have support which is a closed G -invariant subset of Spec π B ∼ = Spec O ( G/H )[ t ]. Allin all, it follows that it suﬃces to prove that the map is an isomorphism modulo t . But modulo t ,we consider colim R Γ( G, (cid:101) A • /t ⊗ V ) → R Γ( G, O ( G/H ) ⊗ V )and all (cid:101) A • /t are given by the symmetric algebras on (cid:101) I • / (cid:101) I • over O ( G/H ), and hence everythingdecomposes into a direct sum over n of the degree n parts. In degree 0, it is the constant simplicialobject given by O ( G/H ). In degree 1, the corresponding simplicial G -equivariant O ( G/H )-module (cid:101) I • / (cid:101) I • is contractible, as it comes from some computation of F -homology, and F is of order prime to (cid:96) . Passing to symmetric powers, the simplicial objects stay contractible, and stay so after taking G -cohomology.Inverting t , we ﬁnd thatcolim R Γ( G, A • ⊗ V )[ t ± ] → R Γ( G, O ( G/H ) ⊗ V )[ t ± ]is an isomorphism; we can then omit the variable t . Applying this to V with a good ﬁltration,we note that all A • have a good G -ﬁltration, hence so does A • ⊗ V , and the left-hand side sits incohomological degrees ≤

0. Thus, the same is true for the right-hand side, showing that O ( G/H )has a good G -ﬁltration. (cid:3) We will need to know that in the situation of Proposition VIII.5.11, the image of Perf( BG ) → Perf( BH ) generates (under cones and retracts), at least if F is solvable. It is very likely that thisassumption is superﬂuous. Theorem

VIII.5.14 . Let L be an algebraically closed ﬁeld of characteristic (cid:96) > , and let G bea smooth linear algebraic group over L such that G ◦ is reductive, and π G is of order prime to (cid:96) .Assume that F is a ﬁnite group of order prime to (cid:96) acting on G and let H = G F be the ﬁxed points.Then H is a smooth linear algebraic group such that H ◦ is reductive, π H is of order prime to (cid:96) ,and the image of Perf( BG ) → Perf( BH ) generates under cones and retracts. Remark

VIII.5.15 . The following example shows that the hypothesis that F is of order primeto (cid:96) is important, and cannot even be weakened to for example “quasi-semisimple” automorphisms(preserving a Borel and a torus); also, the example shows that the precise form of the center iscritical in the theorem. If G = (SL × SL ) /µ with the automorphism switching the two factors,then H = PGL × ( µ × µ ) /µ . If we had (cid:96) = 2 – this is excluded by the hypotheses – then

98 VIII. L -PARAMETER one can show that for all objects A ∈ Perf( BH ) in the image of Perf( BG ), the summand A of A with nontrivial central character has the property that the (homotopy) invariants of the Z / Z ⊂ PGL -action on A are a perfect complex. This implies that the nontrivial character of H is not generated by Perf( BG ) under cones and retracts. On the other hand, the conclusion of thetheorem is (trivially) true for the simply connected and adjoint form of G , as then the inclusion H ⊂ G admits a retraction.Before starting the proof of Theorem VIII.5.14, let us recall the following lemma. Lemma

VIII.5.16 . Let C be an idempotent-complete stable ∞ -category, and let D ⊂ C be afull idempotent-complete stable ∞ -subcategory, with idempotent-completed Verdier quotient E . Let S ⊂ C be a collection of objects of C . Then S generates C under cones, shifts, and retracts if andonly if the image of S in E generates E under cones, shifts, and retracts, and all of objects of D are in the full ∞ -subcategory of C generated by S under cones, shifts, and retracts. Proof.

Let C (cid:48) ⊂ C be the full idempotent-complete stable ∞ -category generated by S ; this is,equivalently, the full ∞ -subcategory generated by S under cones, shifts, and retracts. We want tosee that C (cid:48) = C . Passing to Ind-categories, we have a Verdier quotient sequenceInd( D ) → Ind( C ) → Ind( E )and the quotient Ind( C ) → Ind( E ) has a right adjoint R : Ind( E ) → Ind( C ); see, for example [ NS18 ,Theorem I.3.3, Proposition I.3.5]. In particular, for any X ∈ Ind( C ), with image X ∈ Ind( E ), thecone of X → R ( X ) lies in Ind( D ) ⊂ Ind( C ).To see that C (cid:48) → C is essentially surjective, we need to see that for all X ∈ Ind( C ) withHom( X (cid:48) , X ) = 0 for all X (cid:48) ∈ C (cid:48) , one has X = 0. Now by assumption all objects of D lie in C (cid:48) , soin particular Hom( Y, X ) = 0 for all Y ∈ D . As Hom( Y, R ( X )) = Hom( Y , X ) = 0 for all such Y (as Y = 0 ∈ E ), we see that Hom( Y, cone( X → R ( X ))) = 0 for all Y ∈ D . As the cone lies in Ind( D ),this implies that the cone is equal to 0, so X = R ( X ). But then Hom( X (cid:48) , R ( X )) = Hom( X (cid:48) , X ) = 0for all X (cid:48) ∈ S , which implies that X = 0 by the assumption that S generates E under cones, shifts,and retracts. (cid:3) An application of this is the following result.

Lemma

VIII.5.17 . Let G be a connected reductive group and H ⊂ G be a connected reductivesubgroup of the same rank. Then the image of Perf( BG ) → Perf( BH ) generates under cones andretracts. Proof.

Let T ⊂ H be a maximal torus. Then it is also a maximal torus of G . For anysubset W ⊂ X ∗ ( T ) stable under the Weyl group of G and the dominance order, we have thesubcategory Perf W ( BG ) consisting of those representations whose weights lie in T , and similarlythe subcategory Perf W ( BH ); obviously, restriction gives a functor Perf W ( BG ) → Perf W ( BH ). Weclaim by induction on W that the image generates under cones and retracts. We may always write W = W (cid:48) ∪ W (cid:48)(cid:48) where W (cid:48)(cid:48) is a single Weyl group orbit, and W (cid:48) is still stable under the dominanceorder. Then Perf W ( BG ) / Perf W (cid:48) ( BG ) is isomorphic to Perf( L ), and is generated by the highestweight representation V of weight given by W (cid:48)(cid:48) . Note that that all the weight spaces of V withweights in W (cid:48)(cid:48) have multiplicity exactly 1. The quotient Perf W ( BH ) / Perf W (cid:48) ( BH ) is isomorphic III.5. MODULAR REPRESENTATION THEORY 299 to a product of copies of Perf( L ), enumerated by all the orbits of W (cid:48)(cid:48) under the Weyl group W H of H . The induced map on quotients Perf( L ) → (cid:81) W (cid:48)(cid:48) /W H Perf( L ) is the diagonal embedding,by the multiplicity 1 observation. Thus, its image generates under retracts, and we conclude byinduction. (cid:3) A variation of Lemma VIII.5.16 is the following.

Lemma

VIII.5.18 . Let G be a linear algebraic group over L and let G (cid:48) ⊂ G be a normal subgroup,with quotient G = G/G (cid:48) , and assume that the map L → O ( G ) splits as G -representations. Let C ⊂

Perf( BG ) be an idempotent-complete stable ∞ -subcategory such that for all X, Y ∈ C also X ⊗ Y ∈ C . Assume that C contains Perf( BG ) and the image of C in Perf( BG (cid:48) ) generates undercones and retracts. Then C = Perf( BG ) . Proof.

The functor Ind Perf( BG ) → Ind Perf( BG (cid:48) ) admits a right adjoint R : Ind Perf( BG (cid:48) ) → Ind Perf( BG ), given by pushforward along f : BG (cid:48) → BG . Note that these functors preserve thecoconnective part of these ∞ -categories, and Ind Perf ≥ ( BG ) ∼ = D ≥ ( BG ). In particular, this in-duces a monad on Ind Perf( BG ), given by Rf ∗ f ∗ . By the projection formula, this monad is givenby V (cid:55)→ V ⊗ O ( G ). By the assumption that L → O ( G ) splits as G -representations, it follows thatfor all V ∈ Ind Perf( BG ), V is a direct summand of V ⊗ O ( G ).Now take any V ∈ Ind Perf( BG ); we want to see that it lies in Ind( C ). Let V (cid:48) ∈ Ind Perf( BG (cid:48) )be the image of V . The functor Ind( C ) → Ind Perf( BG (cid:48) ) is surjective, so we can write V (cid:48) as theimage of some X ∈ Ind( C ) ⊂ Ind Perf( BG ). Then V is a direct summand of R ( V (cid:48) ) = X ⊗ O ( G ).As O ( G ) ∈ Ind Perf( BG ) lies in Ind( C ) by assumption, and tensor products preserve Ind( C ), weget the result. (cid:3) Proof of Theorem VIII.5.14.

By induction, we can assume that F is cyclic of prime order p (cid:54) = (cid:96) . The following argument actually proves simultaneously that π H is of order prime to (cid:96) , and H ◦ is reductive, but this is already part of Proposition VIII.5.11.For the statement that Perf( BG ) → Perf( BH ) generates, we ﬁrst reduce to the case that G isconnected and semisimple. Indeed, we have an exact sequence1 → G ◦ der → G → D → D ◦ is a torus and π D is of order prime to (cid:96) ; in particular D has semisimple representationtheory and L → O ( D ) splits as D -representations. Intersecting with H , we get a similar exactsequence, and then Lemma VIII.5.18 reduces us to the case of G ◦ der .Now we claim that we can reduce to the case that Θ permutes the almost simple factors of G transitively. Let G → G be the simply connected cover, to which Θ lifts, and let H = G Θ1 be thecorresponding ﬁxed points. Lemma

VIII.5.19 . Let G be a connected semisimple group, and let Θ : G → G be an automor-phism of order p (cid:54) = (cid:96) ; it also induces an automorphism Θ : G → G of the simply connected cover.Let H = G Θ ⊂ G be the ﬁxed points, and similarly H = G Θ1 . Then H is connected, and π H isan elementary abelian p -group which injects naturally into the coinvariants ker( Z ( G ) → Z ( G )) Θ . Proof.

For the claim that H is connected, see [ Ste68 , Theorem 8.1]. We deﬁne a map ψ : H → ker( Z ( G ) → Z ( G )) Θ . Given h ∈ H , lift to (cid:101) h ∈ G . Then (cid:101) h Θ( (cid:101) h ) − must lie in the kernel

00 VIII. L -PARAMETER of G → G , i.e. in the kernel of Z ( G ) → Z ( G ). Taking a diﬀerent choice of (cid:101) h multiplies (cid:101) h Θ( (cid:101) h ) − by z Θ( z ) − , so the image in the coinvariants is well-deﬁned. Moreover, it is clear that the imageis 0 when h ∈ H lifts to H . Conversely, if the image in the coinvariants is 0, then we can ﬁnd achoice of (cid:101) h so that (cid:101) h = Θ( (cid:101) h ), i.e. we can lift h to (cid:101) h ∈ H . This shows that π H = H/ im( H → H )injects into ker( Z ( G ) → Z ( G )) Θ . It is also easy to check that it is a group homomorphism, so π H is an abelian group.Moreover, all elements of π H are of order p . Indeed, consider (cid:101) h Θ( (cid:101) h ) − above. Its image in thecoinvariants agrees with Θ i ( (cid:101) h )Θ i +1 ( (cid:101) h ) − for all i = 0 , . . . , p −

1, so its p -th power is (cid:101) h Θ p ( (cid:101) h ) − = 1,as Θ is of order p . (cid:3) Now assume that the theorem holds when Θ permutes the almost simple factors of G transitively. This implies that it holds for all simply connected groups: Indeed, we can decomposethese into factors. Now for general G , we induct on the order of ker( Z ( G ) → Z ( G )). Assume thatthere is some normal Θ-stable connected semisimple subgroup G (cid:48) ⊂ G such that G = G/G (cid:48) hasalmost simple factors permuted transitively by Θ, and ker( Z ( G ) → Z ( G )) Θ (cid:54) = 0. In that case,let G (cid:48) → G be the cover with ker( Z ( G (cid:48) ) → Z ( G )) = ker( Z ( G ) → Z ( G )) Θ , and let G (cid:48) → G beits pullback. Then Θ also acts on G (cid:48) , and the action on W = ker( Z ( G (cid:48) ) → Z ( G )) is trivial. Inparticular, if H (cid:48) = ( G (cid:48) ) Θ , then H (cid:48) contains W = ker( Z ( G (cid:48) ) → Z ( G )). By induction, we can assumethat Perf( BG (cid:48) ) → Perf( BH (cid:48) ) generates under cones and retracts. Both categories decomposeinto direct sums over the characters of W , so it follows that also Perf( BG ) → Perf( B ( H (cid:48) /W ))generates under cones and retracts. Now H (cid:48) /W ⊂ H is a union of connected components, so byLemma VIII.5.18, it suﬃces to shows that O ( H/H (cid:48) ) is generated by Perf( BG ). But as G (cid:48) = G × G G (cid:48) ,we also similarly have H (cid:48) = H × H H (cid:48) , and hence H/H (cid:48) = H/H (cid:48) , and hence this part follows from O ( H/H (cid:48) ) being generated by Perf( BG ).Thus, we can assume that for all such quotients G , one has ker( Z ( G ) → Z ( G )) Θ = 0. Thisactually implies that ker( Z ( G ) → Z ( G )) is of order prime to p . If this happens for all suchquotients G , then in fact ker( Z ( G ) → Z ( G )) is of order prime to p . By Lemma VIII.5.19, itis then the case that H is connected. If ker( Z ( G ) → Z ( G )) Θ (cid:54) = 0, then we can pass to thecorresponding cover G (cid:48) → G and the previous arguments reduce us to G (cid:48) . Thus, we can furtherassume that ker( Z ( G ) → Z ( G )) Θ = 0. In that case also the Θ-invariants are trivial (as the groupis prime to p ), and the map H → H is an isomorphism. Moreover, we can ﬁnd a maximal quotient G → G (cid:48) such that ker( Z ( G ) → Z ( G (cid:48) )) is prime to p and ker( Z ( G ) → Z ( G (cid:48) )) Θ = 0, and thisdecomposes into factors. This ﬁnishes the reduction to the case when Θ permutes thealmost simple factors of G transitively.Assume that there is more than one simple factor. Then Θ permutes them cyclically,with Θ p acting as the identity, so G = (cid:81) Z /p Z H . There is nothing to do in the simply connectedtypes E , F and G . Consider next the types B , C , D and E where the fundamentalgroup is a -group. Assume ﬁrst that also p > . Then as in the previous paragraph, we canassume that G → G is the maximal Θ-stable quotient such that H = H = G Θ ; then Z ( H ) → Z ( G )is an isomorphism. Let V be any representation of H . Then V splits oﬀ V ⊗ V ∗ ⊗ V , and inductivelyit splits oﬀ V ⊗ m ⊗ V ∗⊗ ( m − for any m ≥

1. In type B , C , D n with n even and E , take n = p : Then V ⊗ ( n − is a representation of the adjoint form, hence generated by Perf( BG ), and V ⊗ p lifts to arepresentation of G , whence lies in the subcategory of Perf( BH ) generated by Perf( BG ), showing III.5. MODULAR REPRESENTATION THEORY 301 that also V lies in there. In type D n with n odd, the same argument works if p ≡ p ≡ m = p + 1: Then V ⊗ m is a representation of the adjoint group, and V ∗⊗ ( m − extends to G . Now consider the case p = 2 in type B , C , D and E , so necessarily (cid:96) (cid:54) = 2 . In type B , C and E , one has Z ( G ) = ( Z / Z ) , and besides the simply connected and adjointform where the claim is clear, there is only one Θ-stable subgroup. In that case G = ( H × H ) /µ ,with µ diagonally embedded, and the ﬁxed points are given by H = H /µ × ( µ × µ ) /µ . ByLemma VIII.5.18, it suﬃces to generate the nontrivial character of ( µ × µ ) /µ . In type B , theexterior tensor product of two copies of the spinor representation deﬁnes a representation W of G of dimension a power of 2 whose restriction to H has nontrivial central character, and admits acharacter as a summand, thus generating the nontrivial character of µ × µ /µ . In type C , onecan argue similarly with two copies of the i -th exterior power of the standard representation ofSp n , with 0 < i < n odd such that (cid:0) ni (cid:1) is not divisible by (cid:96) – for example, take for i the exactpower of (cid:96) dividing n . In type E , there is an embedding into Sp , reducing to type C . In type D , we make a case distinction according to the size of π ( G ), whose size is a power of 2 between 1and 16, where the extreme cases are clear. If it is of size 2, then G = (Spin n × Spin n ) /µ wherethe µ is diagonally embedded, and in each copy is the kernel of Spin n → SO n . In that case H = SO n × ( µ × µ ) /µ , which can be handled similarly to type B . If π ( G ) is of size 8, then G = (SO n × SO n ) /µ and H = PSO n × ( µ × µ ) /µ . This case can be handled like the caseof type C . There remains the case that π ( G ) is of size 4, where either G = SO n × SO n wherethe claim is clear, or else G ∼ = (Spin n × Spin n ) /Z where Z is the center of Spin n , embeddedeither diagonally, or twisted via the outer automorphism of Spin n in one copy. In this case wealso need to distinguish between even and odd n . Assume ﬁrst that n is odd and Z is embeddeddiagonally. Then H = PSO n × ( µ × µ ) /µ . This case can be reduced to the case that π ( G ) is oforder 2 by passing to a cover. If n is odd and Z is embedded twisted via the outer automorphism,then H ∼ = (SO n × µ ) /µ , so H ◦ ∼ = SO n and π H ∼ = Z / Z . The set of connected componentsactually maps isomorphically to the components in the case where π ( G ) is of order 8, so O ( π H )is generated by Perf( BG ) via reduction to that case. Moreover, taking an exterior tensor productof two copies of a spinor representation gives a representation of G whose restriction to H ◦ = SO n generates the part with nontrivial central character. Now assume that n is even, and Z is embeddeddiagonally. Then H = PSO n × ( Z × Z ) /Z , and again one can generate the characters of ( Z × Z ) /Z via restrictions of exterior tensor products of spinor representations. Finally, if n is even and Z isembedded twisted via the outer automorphism, then H = SO n . Taking an exterior tensor productof the two distinct spinor representations deﬁnes a representation of G whose restriction to H generates the part of Perf( BH ) with nontrivial central character. This ﬁnishes types B , C , D and E . Consider next type E , where the fundamental group is Z / Z . Assume ﬁrst p (cid:54) = 3 . As before, one can reduce to the case G → G is the maximal Θ-stable quotient such that H = H = G Θ is the simply connected form of E . In that case, we use again that any V can besplit oﬀ from V ⊗ m ⊗ V ∗⊗ ( m − . If p ≡ m = p , otherwise m = p + 1: In theﬁrst case V ⊗ m extends to G and V ∗⊗ ( m − is a representation of the adjoint group, in the secondcase it is the other way around. Now consider p = 3 , so necessarily (cid:96) (cid:54) = 3 . Then G = E and Z ( G ) = ( Z / Z ) . The order of π ( G ) is then a power of 3 between 1 and 27, with the extremecases being trivial, and in both other cases there is exactly one possibility. For π ( G ) of order 3,we get H = E , ad × Z where Z ∼ = µ is a central subgroup, given as the kernel of the sum map µ → µ quotient by the diagonal copy of µ . Consider the 27-dimensional representation V of E .Then W = V ⊗ V ∗ ⊗ G on which Z acts via a nontrivial character, and

02 VIII. L -PARAMETER the restriction of W to H splits oﬀ this character (as the dimension of W is a power of 3, henceinvertible modulo (cid:96) ). If π ( G ) is of order 9, then H = E , ad × Z (cid:48) , where Z (cid:48) ∼ = µ is the quotient of µ by the kernel of the sum map. In this case, V ⊗ V ⊗ V deﬁnes a representation of G on which Z (cid:48) acts via a nontrivial character, and the same argument applies. It remains to handle type A . Thus, G = ( (cid:81) Z /p Z SL n ) /Z . Let n (cid:48) be the maximal divisorof n that is prime to p . We can decompose Z = Z p × Z p where Z p is of order a power of p , and Z p is prime to p . We can assume that Z p ⊂ µ pn (cid:48) is the set of all elements with product 1 (whichis the maximal Θ-invariant complement to the diagonal copy of µ n (cid:48) ). On the other hand, letting p a be the exact power of p dividing n , Z p is a subgroup of µ pp a ∼ = ( Z /p a Z ) p that is stable under thethe Θ-action. One can then show that π H ∼ = H (Θ , Z p ). We will now inductively reduce to thecase that Z p is trivial, by replacing Z p by a (certain) subgroup Z (cid:48) p ⊂ Z p of index p and arguingthat the result for G (cid:48) = ( (cid:81) Z /p Z SL n ) /Z (cid:48) p Z p implies the result for G . First note that Z (cid:48) p /Z p ⊂ G (cid:48) is necessarily contained in H (cid:48) = ( G (cid:48) ) Θ , so if Perf( BG (cid:48) ) → Perf( BH (cid:48) ) generates under cones andretracts, the same is true for Perf( BG ) → Perf( B (im( H (cid:48) → H ))). By Lemma VIII.5.18, it remainsto generate O ( H/H (cid:48) ). This is given by H (Θ , Z p ) /H (Θ , Z (cid:48) p ), which injects into H (Θ , Z p /Z (cid:48) p ) ∼ = Z /p Z . Moreover, if a (cid:48) is chosen maximal so that Z p ⊂ ( p a (cid:48) Z /p a Z ) p , we can assume that Z (cid:48) p contains( p a (cid:48) +1 Z /p a Z ) p ∩ Z p . With this choice, we can assume that ( p a (cid:48) +1 Z /p a Z ) p ⊂ Z (cid:48) p ⊂ Z p ⊂ ( p a (cid:48) Z /p a Z ) p ,which then determines Z (cid:48) p uniquely (given Z p ), as F pp has a unique Θ-stable ﬁltration. Now choose i , 0 < i < n , with i not divisible by p , such that (cid:0) ni (cid:1) is not divisible by (cid:96) – for example, takefor i the exact power of (cid:96) dividing n , which works as long as n is not itself a power of (cid:96) , butthen Z p = 0 anyway. Then consider the n (cid:48) p a − a (cid:48) − -th tensor power V of the i -th exterior tensorpower of the standard representation; this V is of dimension prime to (cid:96) , and a representation ofSL n /µ n (cid:48) p a − a (cid:48)− whose central character on µ p a − a (cid:48) /µ p a − a (cid:48)− is nontrivial. Taking tensor productsof V , one can then deﬁne a representation of G whose central character factors over a nontrivialcharacter of Z p /Z (cid:48) p , from which one gets the desired result. Thus, in type A , we can assume that Z p = 0. Then G = ( (cid:81) Z /p Z SL n ) /Z p and H = SL n . We show that the standard representation V of H is a retract of a representation of G ; this implies the same for its exterior powers, and thesegenerated Perf( BH ). We can split oﬀ V from any V ⊗ m ⊗ V ∗⊗ ( m − . Choose a positive integer q such that pq ≡ n (cid:48) and let m = pq . Then V ⊗ m = ( V ⊗ q ) ⊗ p extends to a representationof G , while V ∗⊗ ( m − is a representation of PGL n , which also extends to G . This ﬁnishes thereduction to the case that G is almost simple.Thus, from now on assume that G is almost simple. Assume ﬁrst that Θ is aninner automorphism. If G is simply connected, then H ⊂ G is the centralizer of a semisimpleelement, and thus connected and of the same rank as G , so the claim follows from Lemma VIII.5.17.Moreover, for inner automorphisms Lemma VIII.5.19 shows that the connected components of H always inject into the ones for the adjoint form, and moreover the center of G is always containedin H . So for inner automorphisms, it remains to handle the case of adjoint G . Moreover,by Lemma VIII.5.18, it suﬃces to show that O ( π H ) is in the subcategory of Perf( BH ) generatedunder cones and retracts by the image of Perf( BG ) → Perf( BH ). As a general observation, wenote that the adjoint representation g is a representation of G , and it restricts to a representation of H × Z /p Z . We may decompose according to the (semisimple) action of Θ ∈ Z /p Z , and in particular III.5. MODULAR REPRESENTATION THEORY 303 the ﬁxed points h = g Θ are a retract of g | H . This implies that also all (cid:86) i h ∈ Perf( BH ) are retractsof (cid:86) i g | H .Now we go through the list of types. Note that by Lemma VIII.5.19, π H is an elementaryabelian p -group injecting into Z ( G sc ). In particular, for p >

3, only G = PGL pn is relevant.Moreover, the only relevant H is given by GL pn / G m (cid:111) Z /p Z , which occurs for the centralizer of thesemisimple element that is the image of the diagonal element of GL pn with n occurences of each p -throot of unity. Changing the center, we can also look at restriction from GL pn to GL pn (cid:111)Z /p Z . Forany i , consider the representation (cid:86) i (std) of GL pn . Its restriction to GL pn decomposes as a directsum (as the standard representation decomposes, and a wedge power of a direct sum decomposesinto various pieces), in particular a canonical direct summand (cid:76) pj =1 (cid:86) i (std j ) where std j is thestandard representation of the j -th factor of GL pn . In particular, as a representation of H , therestriction of (cid:86) i (std) contains (cid:76) pj =1 (cid:86) i (std j ), which is naturally a representation of H . If we take i = n , then each (cid:86) i (std j ) is just the determinant of the standard representation. Thus, we get arepresentation W of G pm (cid:111) Z /p Z . This group has semisimple representation theory, and W ⊗ W ∗ admits O ( Z /p Z ) as a direct summand, so the claim follows. This handles the case of type A , infact for all p ≥ p = 3, which is E . In that case, H ◦ is of the form SL /µ (for some embedding of µ into the center µ ), and π H = Z / Z permutes the three factors. Inparticular, h | H ◦ decomposes into the direct sum of the adjoint representations of the three factors(each of which is of dimension 8), permuted cyclically by π H . Taking (cid:86) h , it contains a directsummand that restricted to H ◦ is the direct sum of (cid:86) sl ∼ = L for each factor, permuted cyclicallyby π H ; in other words, the regular representation of π H .The remaining types are now B , C , D and E , all for p = 2 (in particular, (cid:96) (cid:54) = 2). In type B , G = SO n +1 , and we are looking at the centralizer of a diagonal element with entries 1 and −

1. This is of the form S(O m × O n +1 − m ) for some m = 1 , . . . , n . This maps naturally to O m ,identifying connected components. The natural representation decomposes into the sum of thenatural representations of O m and O n +1 − m . As before, it follows that all exterior powers of thenatural representation of O m lie in the relevant subcategory of Perf( BH ), but its determinantdeﬁnes a representation of O m that restricts to the nontrivial character of π H .In type C , G = PSp n , and we are looking at the centralizer of an element of GSp n that hasdiagonal entries 1 and −

1. There are two cases where π H is nontrivial. On the one hand, if n = 2 m , one can get H ◦ = Sp m /µ with π H = Z / Z ; this case can be handled like the caseof E , taking (cid:86) d h for d the dimension of Sp m . In the other case, we get H = GL n /µ (cid:111) Z / Z ,where the action is given by A (cid:55)→ t A − . In that case, we again enlarge the center as in type A ,so consider (cid:101) H = (GL n × G m ) (cid:111) Z / Z ⊂ (cid:101) G = GSp n , where ( A, λ ) ∈ GL n × G m embeds GSp n via the block diagonal matrix with entries A and λ t A − , and Z / Z maps to the block matrix (cid:18) I n I n (cid:19) . Restricting the natural representation of GSp n to (cid:101) H , its restriction to (cid:101) H ◦ is a sumof two n -dimensional representations, permuted by π (cid:101) H . Passing to the n -th exterior power, weget as a direct summand a 2-dimensional representation W of the torus quotient G m (cid:111) Z / Z of (cid:101) H ,and W ⊗ W ∗ contains O ( Z / Z ) as a direct summand.

04 VIII. L -PARAMETER In type D , G = PSO n , and we are looking at the centralizer of a diagonal element of (cid:101) G = GSO n with entries ±

1. In the following, we describe the preimage of H ⊂ G in (cid:101) G . Unless the number of1’s and − n ∩ (GL m × GL n − m ) for some even m , 0 < m < n .In that case, H has two connected components, and these two connected components are stillseen when taking the centralizer S(O m × O n − m ) in SO n , so we may replace the adjoint groupby SO n in this example. Then the same argument as for type B works. There remains thecase that the number of 1’s and − (cid:101) H = GSO n ∩ (GL n × GL n (cid:111)Z / Z ). If n is odd, then this intersection agreeswith GSO n ∩ (GL n × GL n ), and the previous argument applies. If n is even, then π (cid:101) H agreeswith Z ( G sc ) = ( Z / Z ) . Restricting (cid:86) n of the standard representation of GSO n to (cid:101) H admits a2-dimensional summand W which on (cid:101) H ◦ factors over the torus quotient G m . Let us identify this W : We can also ﬁrst restrict from GL n to GL n (cid:111)Z / Z , in which case we get a representationof G m (cid:111) Z / Z that on G m is the direct sum of the two characters given by the two projections.In particular, W ⊗ W ∗ already admits O ( Z / Z ) as a direct summand for GL n (cid:111)Z / Z , and hencecontinues to do so after restriction to (cid:101) H . We may then apply Lemma VIII.5.18 to reduce to theintersection of GSO n with GL n , where we can as before reduce to the cover SO n of G = PSO n ,and the argument of type B applies. Finally, there is an inner involution for which (cid:101) H ◦ = GL n × G m with π (cid:101) H = Z / Z . This can be handled like the similar case in type C .For inner automorphisms, it remains to handle the case of E . In that case, there is an em-bedding E → (cid:101) G = Sp mapping the center to the center. In particular, any semisimple elementof E , ad maps to a semisimple element of PSp , and we get an inclusion H ⊂ (cid:101) H of centralizers.Lemma VIII.5.19 implies that π H → π (cid:101) H is injective, so this case reduces to type C . Now we consider the case Θ is an outer automorphism. Let us ﬁrst treat thecase of triality, so p = 3 (and (cid:96) (cid:54) = 3 ). Then G is of type D and either simply connected oradjoint. By Lemma VIII.5.19, the ﬁxed points H are always connected. Moreover, H is unchangedby taking G adjoint, so we can assume G adjoint. There are two conjugacy of order 3 outerautomorphisms of Spin (see e.g. [ Tit59 ]): The diagram automorphism with ﬁxed points G , andan automorphism whose ﬁxed points agree with the ﬁxed points of the order 3 semisimple elementof G with centralizer PGL ⊂ G . The second case thus follows from the ﬁrst. For the ﬁrst, noteﬁrst that any highest weight for G lifts to a highest weight for Spin , so it follows easily fromhighest weight theory that Perf( B Spin ) → Perf( BG ) generates under cones and retracts. Nowfor the adjoint form, there is the problem that there is a fundamental weight for G that does notlift to a highest weight for PSpin ; rather, if would lift to either of the three fundamental weightsfor Spin giving the two spinor representations V , V and the standard representation V (whichare permuted under triality). But then V = V ⊗ V ⊗ V has trivial central character, and uponrestriction to G all three of these are isomorphic and selfdual. Thus V | G splits oﬀ V | G (as ingeneral W splits oﬀ W ⊗ W ∗ ⊗ W ), giving the desired result. It remains to handle the case of outer involutions, so p = 2 (and (cid:96) (cid:54) = 2 ). In con-tinuing the tradition of handling exceptional cases ﬁrst for outer involutions, consider E . ByLemma VIII.5.19, the ﬁxed points are necessarily connected, and moreover are unchanged whengoing to the adjoint form of E . We can thus assume that the group is adjoint. There are twoconjugacy classes of outer automorphisms of E : The diagram automorphism, with ﬁxed points III.5. MODULAR REPRESENTATION THEORY 305 F , and one automorphism with ﬁxed points PSp . Consider ﬁrst the diagram automorphism. If G is simply connected, any highest weight of F or PSp lifts to a highest weight for E , and theclaim is easy. Otherwise, if G is adjoint, let W be an irreducible representation of the simplyconnected form of E with nontrivial central character. Then W | F is selfdual, so W | F splits oﬀ( W ⊗ W ⊗ W ) | F , and W ⊗ W ⊗ W is always a representation of the adjoint form G .In type A , if the simply connected cover is SL n with n odd, there is just one conjugacy classof outer involutions, with ﬁxed points SO n (independently of the center of G ). The worst case isthus G = PGL n . But for any irreducible representation V of SL n , the restriction of V to SO n isselfdual, and hence V | SO n splits oﬀ V ⊗ n | SO n (noting that n is odd), where V ⊗ n is a representationof the adjoint form G . This reduces us back to the simply connected case, where the claim followseasily from highest weights. If the simply connected cover is SL n , there are two conjugacy classesof outer involutions, with ﬁxed points Sp n resp. SO n (for the simply connected form). Arguing asfor SL n with n odd, we can assume that the fundamental group of G is a 2-group (but the 2-part of π ( G ) changes the structure of H ), and so G = SL n /µ a for some a ≤ a divides 2 N .In the simply connected form, a = 0, both cases follow from consideration of highest weights. Ingeneral, we reduce to the simply connected case by induction on a . Unless a is the exact power of2 dividing 2 n (and assuming a > H = H ◦ × µ a +1 /µ a where H ◦ is PSp n resp. PSO n .Moreover, H (cid:48) → H factors over a µ -cover H (cid:48) → H ◦ , so by Lemma VIII.5.18, it suﬃces to generatethe nontrivial character of µ a +1 /µ a . For this, take V the i -th exterior power of the standardrepresentation where i is the exact power of (cid:96) dividing n , so that V is of dimension prime to (cid:96) .Consider V ⊗ a . This is a representation of G such that the central character of µ a +1 /µ a ⊂ G isnontrivial. The restriction of V to H will then split oﬀ the nontrivial character of µ a +1 /µ a , notingthat the restriction of V ⊗ a to PSp n resp. PSO n splits oﬀ the trivial representation (as a > V becomes selfdual). It remains to consider the case that a is the exact power of 2 dividing2 n . In the symplectic case, H is connected, and the reduction to H (cid:48) works immediately. In theorthogonal case, H is given by PO n , and we need to construct the nontrivial character of π H .Here, we observe as above that the subcategory of Perf( BH ) generated by Perf( BG ) under conesand retracts contains all wedge powers of h = so n . Taking the determinant of h , we observe thatit gives the nontrivial character of π H , giving the claim.In type D , we consider ﬁrst type D n with n odd. In the simply connected form H ⊂ Spin n , thecentralizer is the diagonal µ -cover H → SO m × SO n − m for some odd m . This case is again handledby highest weights. For the intermediate form G = SO n , the centralizer is S(O m × O n − m ) =SO m × SO n − m × µ /µ . In that case, let V be a spinor representation of Spin n , and consider V ⊗ . Then V becomes selfdual upon restriction to Spin m × Spin n − m (as there is only one spinorrepresentation in type B ), and hence V ⊗ splits oﬀ the trivial representation as a representationof SO m × SO n − m . On the other hand, the central character is the nontrivial character of µ /µ ,so V ⊗ restricted to H splits oﬀ the nontrivial character of µ /µ and hence Lemma VIII.5.18handles this case again. Finally, if G = PSO n is adjoint, then the centralizer is SO m × SO n − m and connected as long as m (cid:54) = n , when the reduction to the previous case is immediate. If onthe other hand m = n , then the centralizer H has H ◦ = SO n × SO n and π H = Z / Z , switchingthe two factors. Here we use again that the adjoint representation h lies in the subcategory ofPerf( BH ) generated by Perf( BH ) under cones and retracts. This is given by so n × so n , where π H switches the two factors. Passing to the d -th exterior power, where d is the dimension of so n , weget a two-dimensional summand given by the sum of two copies of the determinants of so n , with

06 VIII. L -PARAMETER π H acting by switching. This generates the regular representation of π H , as desired. Finally,consider the case of D n with n even. In that case, the discussion works in exactly the same way(except that µ is replaced by µ × µ , which however causes no diﬀerence), except that as m isstill odd the case m = n cannot occur, so the most diﬃcult case does not arise. (cid:3) VIII.5.4. End of proof.

Finally, we can prove Theorem VIII.5.2. We switch back to thenotation employed there.Note that W can be written (inside animated groups) as a geometric realization of a simplicialgroup which is ﬁnite free in each degree. In particular, the sifted colimitcolim ( n,F n → W ) O ( Z ( F n , (cid:98) G ) L )can be understood to be a geometric realization of a simplicial diagram.We want to use Proposition VIII.5.10 to show thatcolim ( n,F n → W ) O ( Z ( F n , (cid:98) G ) L ) → O ( Z ( W, (cid:98) G ) L )is an isomorphism in Ind Perf( B (cid:98) G ), and that Z ( W, (cid:98) G ) L satisﬁes the hypotheses of Theorem VIII.5.8.To do so, it suﬃces to see that for all closed (cid:98) G -orbits Z ∼ = (cid:98) G/H ⊂ Z ( W, (cid:98) G ) L , the (cid:98) G -representation O ( Z ) has a good ﬁltration, the image of Perf( B (cid:98) G ) → Perf( BH ) generates under cones and retracts,and for any closed immersion Z ∼ = Z ( W, (cid:98) G ) ⊂ Z ( F n , (cid:98) G ) ∼ = (cid:98) G n , the conormal sheaf I/I deﬁnes a representation of H that is a direct summand of a direct sum ofcopies of (cid:98) g ∗ | H .Note that by Proposition VIII.1.5, any such closed orbit Z has the property that ϕ : W E → (cid:98) G ( L )factors over a ﬁnite quotient F of order prime to (cid:96) . Then H = Cent (cid:98) G ( ϕ ) ⊂ (cid:98) G is the centralizer ofa group of automorphisms of (cid:98) G of order prime to (cid:96) . By Proposition VIII.5.12, O ( (cid:98) G/H ) has a good (cid:98) G -ﬁltration, and by Theorem VIII.5.14, the map Perf( B (cid:98) G ) → Perf( BH ) generates under conesand retracts. Finally, the claim about the conormal sheaf for the closed immersion Z ⊂ Z ( F n , (cid:98) G ) ∼ = (cid:98) G n can be reduced to the case that F n → W E has the property that the images of the generators givea set of representatives for the quotient W E → F in W E , where it follows from the argument in theproof of Proposition VIII.5.12.HAPTER IX The Hecke action

The time has come to put everything together. As before, let E be any nonarchimedean localﬁeld with residue ﬁeld F q of residue characteristic p , and let G be a reductive group over E . For any Z (cid:96) -algebra Λ, we have deﬁned D lis (Bun G , Λ), we have the geometric Satake equivalence relating (cid:98) G to perverse sheaves on the Hecke stack, and we have studied the stack of L -parameters.Our ﬁrst task is to use the geometric Satake equivalence to deﬁne the Hecke operators on D lis (Bun G , Λ). As in the last chapter, we work over a Z (cid:96) [ √ q ]-algebra Λ in order to trivialize thecyclotomic twist in the geometric Satake equivalence; let Q be a ﬁnite quotient of W E over whichthe action on (cid:98) G factors. If Λ is killed by a power of (cid:96) , then we can deﬁne Hecke operators in thefollowing standard way. For any ﬁnite set I and V ∈ Rep Λ ( (cid:98) G (cid:111) Q ) I , we get a perverse sheaf S V on H ck IG , which we can pull back to the global Hecke stack Hck IG ; we denote its pullback still by S V .Using the correspondence Hck IGp (cid:123) (cid:123) p (cid:38) (cid:38) Bun G Bun G × (Div ) I we get the Hecke operator T V : D ´et (Bun G , Λ) → D ´et (Bun G × (Div ) I , Λ) : A (cid:55)→ Rp ∗ ( p ∗ A ⊗ L Λ S V ) . By Corollary IV.7.2, the target has D ´et (Bun G × [ ∗ /W IE ] , Λ) as a full subcategory, and we will seebelow that T V will factor over this subcategory. Working ∞ -categorically in order to have descent,and using a little bit of condensed formalism in order to deal with W IE not being discrete, we canin fact rewrite D ´et (Bun G × [ ∗ /W IE ] , Λ) ∼ = D ´et (Bun G , Λ) BW IE as the W IE -equivariant objects of the condensed ∞ -category D ´et (Bun G , Λ); we will discuss thecondensed structure below.The following theorem summarizes the properties of the Hecke operators. In particular, itasserts that these functors are deﬁned even when Λ is not torsion.

Theorem

IX.0.1 (Theorem IX.2.2; Corollary IX.2.4, Proposition IX.5.1) . For any Z (cid:96) [ √ q ] -algebra Λ , any ﬁnite set I , and any V ∈ Rep Λ ( (cid:98) G (cid:111) Q ) I , there is a natural Hecke operator T V : D lis (Bun G , Λ) → D lis (Bun G , Λ) BW IE . (i) Forgetting the W IE -action, i.e. as an endofunctor of D lis (Bun G , Λ) , the functor T V is functorialin V ∈ Rep Λ (cid:98) G I . Moreover, T V commutes with all limits and colimits, and preserves compactobjects and universally locally acyclic objects. Letting sw ∗ : Rep Λ (cid:98) G I → Rep Λ (cid:98) G I be the involutionof Proposition VI.12.1, there are natural isomorphisms D BZ ( T V ( A )) ∼ = T sw ∗ V ∨ ( D BZ ( A )) , R H om lis ( T V ( A ) , Λ) ∼ = T sw ∗ V ∨ R H om lis ( A, Λ) . (ii) As a functor of V , it induces an exact Rep Λ ( Q I ) -linear monoidal functor Rep Λ ( (cid:98) G (cid:111) Q ) I → End Λ ( D lis (Bun G , Λ) ω ) BW IE . Moreover, for any compact object X ∈ D lis (Bun G , Λ) ω , there is some open subgroup P of the wildinertia subgroup of W E such that for all I and V , the P I -action on T V ( X ) is trivial. In particular,one can write D lis (Bun G , Λ) ω as an increasing union of full stable ∞ -subcategories D P lis (Bun G , Λ) ω such that the Hecke action deﬁnes functors Rep Λ ( (cid:98) G (cid:111) Q ) I → End Λ ( D P lis (Bun G , Λ) ω ) B ( W E /P ) I . (iii) Varying also I , the functors of (ii) are functorial in I . In particular, the categories D P lis (Bun G , Λ) ω ﬁt the bill of the discussion of Section VIII.4, soTheorem VIII.4.1 gives a construction of excursion operators. To state the outcome, we make thefollowing deﬁnitions as in the introduction. Definition

IX.0.2 . (i) The Bernstein center of G ( E ) is Z ( G ( E ) , Λ) = π End(id D ( G ( E ) , Λ) ) = lim ←− K ⊂ G ( E ) Z (Λ[ K \ G ( E ) /K ]) where K runs over open pro- p subgroups of G ( E ) , and Λ[ K \ G ( E ) /K ] = End G ( E ) ( c - Ind G ( E ) K Λ) isthe Hecke algebra of level K . (ii) The geometric Bernstein center of G is Z geom ( G, Λ) = π End(id D lis (Bun G , Λ) ) . Inside Z geom ( G, Λ) , we let Z geomHecke ( G, Λ) be the subring of all endomorphisms f : id → id commutingwith Hecke operators, in the sense that for all V ∈ Rep( (cid:98) G I ) and A ∈ D lis (Bun G , Λ) , one has T V ( f ( A )) = f ( T V ( A )) ∈ End( T V ( A )) . (iii) The spectral Bernstein center of G is Z spec ( G, Λ) = O ( Z ( W E , (cid:98) G ) Λ ) (cid:98) G , the ring of global functions on Z ( W E , (cid:98) G ) Λ (cid:12) (cid:98) G . The inclusion D ( G ( E ) , Λ) (cid:44) → D lis (Bun G , Λ) induces a map of algebra Z geom ( G, Λ) → Z ( G ( E ) , Λ).

Corollary

IX.0.3 . Assume that (cid:96) is invertible in Λ , or (cid:96) is a very good prime for (cid:98) G . Thenthere is a canonical map Z spec ( G, Λ) → Z geomHecke ( G, Λ) ⊂ Z geom ( G, Λ) , X. THE HECKE ACTION 309 and in particular a map Ψ G : Z spec ( G, Λ) → Z ( G ( E ) , Λ) . In general, there is such a map, up to replacing Z ( W E , (cid:98) G ) Λ (cid:12) (cid:98) G by a universally homeomorphicscheme. In particular, if Λ = L is an algebraically closed ﬁeld over Z (cid:96) [ √ q ], we get the following con-struction of L -parameters. Definition

IX.0.4 . Let L be an algebraically closed ﬁeld over Z (cid:96) [ √ q ] , and let A ∈ D lis (Bun G , L ) be a Schur-irreducible object, i.e. End( A ) = L as condensed algebras. Then there is a uniquesemisimple L -parameter ϕ A : W E → (cid:98) G ( L ) (cid:111) Q such that for all excursion data ( I, V, α, β, ( γ i ) i ∈ I ) consisting of a ﬁnite set I , V ∈ Rep(( (cid:98) G (cid:111) Q ) I ) , α : 1 → V | (cid:98) G , β : V | (cid:98) G → and γ i ∈ W E for i ∈ I , the endomorphism A = T ( A ) α −→ T V ( A ) ( γ i ) i ∈ I −−−−→ T V ( A ) β −→ T ( A ) = A is given by the scalar L α −→ V ( ϕ A ( γ i )) i ∈ I −−−−−−−→ V β −→ L. We can apply this in particular in the case of irreducible smooth representations π of G ( E ).Concerning the L -parameters we construct, we can prove the following basic results. (In fact, weprove slightly ﬁner results on the level of Bernstein centers.) Theorem

IX.0.5 (Sections IX.6, IX.7) . (i) If G = T is a torus, then π (cid:55)→ ϕ π is the usual Langlands correspondence. (ii) The correspondence π (cid:55)→ ϕ π is compatible with twisting. (iii) The correspondence π (cid:55)→ ϕ π is compatible with central characters. (iv) The correspondence π (cid:55)→ ϕ π is compatible with passage to congradients. (v) If G (cid:48) → G is a map of reductive groups inducing an isomorphism of adjoint groups, π is anirreducible smooth representation of G ( E ) and π (cid:48) is an irreducible constitutent of π | G (cid:48) ( E ) , then ϕ π (cid:48) is the image of ϕ π under the induced map (cid:98) G → (cid:99) G (cid:48) . (vi) If G = G × G is a product of two groups and π is an irreducible smooth representation of G ( E ) , then π = π (cid:2) π for irreducible smooth representations π i of G i ( E ) , and ϕ π = ϕ π × ϕ π under (cid:98) G = (cid:98) G × (cid:98) G . (vii) If G = Res E (cid:48) | E G (cid:48) is the Weil restriction of scalars of a reductive group G (cid:48) over some ﬁnite sep-arable extension E (cid:48) | E , so that G ( E ) = G (cid:48) ( E (cid:48) ) , then L -parameters for G | E agree with L -parametersfor G (cid:48) | E (cid:48) . (viii) The correspondence π (cid:55)→ ϕ π is compatible with parabolic induction. (ix) For G = GL n and supercuspidal π , the correspondence π (cid:55)→ ϕ π agrees with the usual localLanglands correspondence [ LRS93 ] , [ HT01 ] , [ Hen00 ] .

10 IX. THE HECKE ACTION

IX.1. Condensed ∞ -categories In order to meaningfully talk about W IE -equivariant objects in D lis (Bun G , Λ), we need to give D lis (Bun G , Λ) the structure of a condensed ∞ -category. This is in fact easy to do: We can associateto any extremally disconnected proﬁnite set S the ∞ -category D lis (Bun G × S, Λ). This is a fullcondensed ∞ -subcategory of the condensed ∞ -category D (cid:4) (Bun G , Λ), taking any proﬁnite S to D (cid:4) (Bun G × S, Λ). The latter deﬁnes a hypersheaf in S , by v-hyperdescent of D (cid:4) ( X, Λ) (as followsfrom the case of D ( X v , Λ)). With this deﬁnition, it becomes a direct consequence of descent that D (cid:4) (Bun G × [ ∗ /W IE ] , Λ) ∼ = D (cid:4) (Bun G , Λ) BW IE , where the latter is the evaluation of the condensed ∞ -category D (cid:4) (Bun G , Λ) on the condensed an-ima BW IE . More concretely, this is the ∞ -category of objects A ∈ D (cid:4) (Bun G , Λ) together with a mapof condensed animated groups W IE → Aut( A ). We see in particular that to deﬁne D (cid:4) (Bun G , Λ) BW IE ,we do not need to know the full structure as a condensed ∞ -category. Rather, we only need thestructure as an ∞ -category enriched in condensed anima. This structure on D (cid:4) (Bun G , Λ) inducesa similar structure on D lis (Bun G , Λ).For the discussion of Hecke operators, we observe in particular the following result, that followsdirectly from the discussion above.

Proposition

IX.1.1 . Pullback under

Bun G × (Div ) I → Bun G × [ ∗ /W IE ] induces a fully faithfulfunctor D lis (Bun G , Λ) BW IE (cid:44) → D (cid:4) (Bun G , Λ) BW IE ∼ = D (cid:4) (Bun G × [ ∗ /W IE ] , Λ) (cid:44) → D (cid:4) (Bun G × (Div ) I , Λ) . The essential image consists of all objects A ∈ D (cid:4) (Bun G × [ ∗ /W IE ] , Λ) whose pullback to Bun G liesin D lis (Bun G , Λ) . (cid:3) In fact, this structure of D lis (Bun G , Λ) as an ∞ -category enriched in condensed anima, in factcondensed animated Λ-modules, can be obtained in the following way from its structure as a Λ-linear stable ∞ -category. Proposition

IX.1.2 . For A ∈ D lis (Bun G , Λ) ω and B ∈ D lis (Bun G , Λ) , the condensed animated Λ -module Hom D lis (Bun G , Λ) ( A, B ) is relatively discrete over Z (cid:96) . In other words, the condensed structure on D lis (Bun G , Λ) can also be deﬁned as the relativelydiscrete condensed structure when restricted to compact objects, and in general induced from this.In particular, when restricting attention to the compact objects D lis (Bun G , Λ) ω , it is simply therelatively discrete condensed structure. Proof.

Take some b ∈ B ( G ) and K ⊂ G b ( E ) an open pro- p -subgroup, and let f K : (cid:102) M b /K → Bun G be the local chart. We can assume A = f K(cid:92) Z (cid:96) , as these form a family of generators. By ad-junction, it is enough to show that for any B (cid:48) ∈ D lis ( (cid:102) M b /K, Λ), the global sections R Γ( (cid:102) M b /K, B (cid:48) )have the relatively discrete condensed Z (cid:96) -module structure. We claim that the restriction map R Γ( (cid:102) M b /K, B (cid:48)(cid:48) ) → R Γ([ ∗ /K ] , B (cid:48) ) is an isomorphism, where [ ∗ /K ] ⊂ (cid:102) M b /K is the base point.Without the condensed structure, this was proved in the proof of Proposition VII.7.2, but actu-ally the proof applies with condensed structure (as Theorem VII.2.10 remembers the condensed X.2. HECKE OPERATORS 311 structure). But R Γ([ ∗ /K ] , B (cid:48) ) is a direct summand of the stalk of B (cid:48) at ∗ , which has the relativelydiscrete condensed Z (cid:96) -module structure (as this is true for all objects of D lis ( ∗ , Λ)). (cid:3)

IX.2. Hecke operators

The geometric Satake equivalence gives exact Rep Z (cid:96) [ √ q ] ( Q I )-linear monoidal functorsRep Z (cid:96) [ √ q ] ( (cid:98) G (cid:111) Q ) I → Perv

ULA ( H ck IG , Z (cid:96) [ √ q ]) : V (cid:55)→ S V . Moreover, this association is functorial in I . We can compose with the functor A (cid:55)→ D ( A ) ∨ (wherethe Verdier duality is relative to the projection H ck IG → [(Div ) I /L + G ]) to get exact Rep Z (cid:96) [ √ q ] ( Q I )-linear monoidal functors Rep Z (cid:96) [ √ q ] ( (cid:98) G (cid:111) Q ) I → D (cid:4) ( H ck IG , Z (cid:96) [ √ q ]) , functorially in I . Here, the functor A (cid:55)→ D ( A ) ∨ is monoidal with respect to the usual convolutionon perverse sheaves, and the convolution of Section VII.5 on the right. We note that as theconvolution on D (cid:4) makes use only of pullback, tensor product, and π (cid:92) -functors, all of which aredeﬁned naturally on ∞ -categories, this monoidal structure is actually a monoidal structure onthe ∞ -category D (cid:4) ( H ck IG , Z (cid:96) [ √ q ]). (We would have to work harder to obtain this structure whenemploying lower-!-functors, as we have not deﬁned them in a suﬃciently structured way.) Also,the functor from Rep Z (cid:96) [ √ q ] ( (cid:98) G (cid:111) Q ) I is monoidal in this setting, as on perverse sheaves there are nohigher coherences to take care of.This extends by linearity uniquely to an exact Rep Λ ( Q I )-linear monoidal functorRep Λ ( (cid:98) G (cid:111) Q ) I → D (cid:4) ( H ck IG , Λ) : V (cid:55)→ S (cid:48) V ;here, we implicitly use highest weight theory to showPerf( B ( (cid:98) G (cid:111) Q ) I Z (cid:96) [ √ q ] ) ⊗ Perf( BQ I Z (cid:96) [ √ q ] ) Perf( BQ I Λ ) ∼ = Perf( B ( (cid:98) G (cid:111) Q ) I Λ ) , and that the free stable ∞ -category with an exact functor from Rep Λ ( (cid:98) G (cid:111) Q ) I is Perf( B ( (cid:98) G (cid:111) Q ) I Λ ).Pulling back to the global Hecke stack, we get exact Rep Λ ( Q I )-linear monoidal functorsRep Λ ( (cid:98) G (cid:111) Q ) I → D (cid:4) (Hck IG , Λ) . On the other hand, there is a natural exact Rep Λ ( Q I )-linear monoidal functor D (cid:4) (Hck IG , Λ) → End D (cid:4) ((Div ) I , Λ) ( D (cid:4) (Bun G × (Div ) I , Λ)) , where the right-hand side denotes the D (cid:4) ((Div ) I , Λ)-linear endofunctors. In particular, any V ∈ Rep Λ ( (cid:98) G (cid:111) Q ) I gives rise to a functor T V : D lis (Bun G , Λ) → D (cid:4) (Bun G × (Div ) I , Λ)via T V ( A ) = p (cid:92) ( p ∗ A (cid:4) ⊗ L Λ S (cid:48) V )

12 IX. THE HECKE ACTION where we consider the usual diagram Hck

IGp (cid:123) (cid:123) p (cid:39) (cid:39) Bun G Bun G × (Div ) I . Note that we have thus essentially used the translation of Proposition VII.5.2 to extend the Heckeoperators from the case of torsion rings Λ to all Λ.We note that if we pull back to the diagonal geometric point Spd C → (Div ) I , where C = (cid:98) E ,then this functor depends only on the compositeRep Λ ( (cid:98) G (cid:111) Q ) I → D (cid:4) (Hck IG , Λ) → D (cid:4) (Hck IG × (Div ) I Spd C, Λ) , and this composite factors naturally over Rep Λ ( (cid:98) G I ). Proposition

IX.2.1 . For any V ∈ Rep Λ ( (cid:98) G I ) , the functor T V : D (cid:4) (Bun G × Spd C, Λ) → D (cid:4) (Bun G × Spd C, Λ) restricts to a functor T V : D lis (Bun G , Λ) → D lis (Bun G , Λ) . Proof.

By highest weight theory, one can reduce to the case that V is an exterior tensorproduct of representations of (cid:98) G , and then by using that V (cid:55)→ T V is monoidal, we can reduce to thetensor factors, which reduces us to the case I = {∗} . Consider the Hecke diagramBun G,C h ←− Hck

G,C h −→ Bun

G,C where Hck

G,C parametrizes over S ∈ Perf C pairs of G -torsors E , E on X S together with an isomor-phism over X S \ S (cid:93) meromorphic along S (cid:93) . It suﬃces to see that for all B ∈ D ULA ( H ck G, Spd C/ Div X , Z (cid:96) ),the object h (cid:92) ( h ∗ A (cid:4) ⊗ L q ∗ B ∨ ) ∈ D lis (Bun G,C , Λ) . Now the category of such B is generated (under colimits) by the objects Rf ˙ w ∗ Z (cid:96) for f ˙ w : L + I\ Dem ˙ w → H ck G, Spd C/ Div X a Demazure resolution (modulo action of Iwahori) of some Schubert variety in the aﬃne ﬂag variety.Using Proposition VII.4.3, it thus suﬃces to see that for the corresponding push-pull correspondenceon Bun G,C with kernel given by the Demazure resolution, one has preservation of D lis (Bun G,C , Λ).But this is a proper and cohomologically smooth correspondence. (cid:3)

Theorem

IX.2.2 . For any V ∈ Rep Λ ( (cid:98) G I ) , the action of T V on D lis (Bun G , Λ) preserves alllimits and colimits, and the full subcategories of compact objects, and of universally locally acyclicobjects. Moreover, for the automorphism sw ∗ of Rep Λ ( (cid:98) G I ) given by Proposition VI.12.1, there arenatural isomorphisms D BZ ( T V ( A )) ∼ = T sw ∗ V ∨ ( D BZ ( A )) , R H om lis ( T V ( A ) , Λ) ∼ = T sw ∗ V ∨ R H om lis ( A, Λ) . X.2. HECKE OPERATORS 313

Proof.

The functor V (cid:55)→ T V is monoidal. As V is dualizable in the Satake category, withdual V ∨ , it follows that T V has a left and a right adjoint, given by T V ∨ , and hence it fol-lows formally that it preserves all limits and colimits, and compact objects. Now recall that A ∈ D lis (Bun G , Λ) is universally locally acyclic if and only if for all compact B ∈ D lis (Bun G , Λ),the object R Hom Λ ( B, A ) ∈ D (Λ) is perfect, by Proposition VII.7.9. Thus, the preservation ofuniversally locally acyclic objects follows by adjointness from the preservation of compact objects.For the duality statements, we note that, for π : Bun G → ∗ the projection, there are naturalisomorphisms π (cid:92) ( T V ( A ) (cid:4) ⊗ L Λ B ) ∼ = π (cid:92) ( A (cid:4) ⊗ L Λ T sw ∗ V ( B )) , as follows from the deﬁnition of the Hecke operator, and Proposition VI.12.1: Both sides identifywith the homology of Hck IG × (Div ) I Spd C with coeﬃcients in h ∗ A (cid:4) ⊗ L Λ h ∗ B (cid:4) ⊗ L Λ S (cid:48) V . The displayedequation implies the statement for Bernstein–Zelevinsky duals by also using that T sw ∗ V ∨ is rightadjoint to T sw ∗ V , and the statement for naive duals by using that T sw ∗ V ∨ is left adjoint to T sw ∗ V . (cid:3) Composing Hecke operators, we get the following corollary.

Corollary

IX.2.3 . For any V ∈ Rep Λ ( (cid:98) G (cid:111) Q ) I , the functor T V : D lis (Bun G , Λ) → D (cid:4) (Bun G × (Div X ) I , Λ) takes image in the full subcategory D (cid:4) (Bun G × [ ∗ /W IE ] , Λ) ; moreover, all objects in the image havethe property that their pullback to D (cid:4) (Bun G , Λ) lies in D lis (Bun G , Λ) , so by Proposition IX.1.1 thefunctor T V induces a functor D lis (Bun G , Λ) → D lis (Bun G , Λ) BW IE . Proof.

We only need to see that the image lands in D (cid:4) (Bun G × [ ∗ /W IE ] , Λ); the rest followsfrom Proposition IX.2.1. One can reduce to the case that V is an exterior tensor product of | I | representations V i ∈ Rep Λ ( (cid:98) G (cid:111) Q ) — one can always ﬁnd a, possibly inﬁnite, resolution by suchexterior tensor products that involves only ﬁnitely many weights of (cid:98) G I , and thus induces a resolutionin D (cid:4) ( H ck IG , Λ) — and thus reduce to I = {∗} . By Corollary VII.2.7, it suﬃces to see that thepullback to D (cid:4) (Bun G × Spd C, Λ) lies in D (cid:4) (Bun G , Λ). But by Proposition IX.2.1, we know thatit lies in D lis (Bun G × Spd C, Λ), and D lis (Bun G , Λ) → D lis (Bun G × Spd C, Λ) is an equivalence byProposition VII.7.3. (cid:3)

Finally, we get the following Hecke action.

Corollary

IX.2.4 . Endowing the stable Z (cid:96) -linear ∞ -category D lis (Bun G , Λ) ω with the rela-tively discrete condensed structure, the Hecke action deﬁnes exact Rep Λ ( Q I ) -linear monoidal func-tors Rep Λ ( (cid:98) G (cid:111) Q ) I → End Λ ( D lis (Bun G , Λ) ω ) BW IE , functorially in I .

14 IX. THE HECKE ACTION

IX.3. Cohomology of local Shimura varieties

Theorem IX.2.2 encodes strong ﬁniteness properties for the cohomology of local Shimura vari-eties, giving unconditional proofs, and reﬁnements, of the results of [

RV14 , Section 6]. For this, weﬁrst specialize to E = Q p as this is the standard setting of local Shimura varieties. Consider anylocal Shimura datum, consisting of a reductive group G over Q p , a conjugacy class of minusculecocharacters µ : G m → G Q p with ﬁeld of deﬁnition E | Q p and some element b ∈ B ( G, µ ) ⊂ B ( G ).(Beware that we are making a small sin here in changing the meaning of the letter E , using it nowin its usual meaning as a reﬂex ﬁeld.) In [ SW20 , Lecture 24], we construct a tower of partiallyproper smooth rigid-analytic spaces ( M ( G,b,µ ) ,K ) K ⊂ G ( Q p ) over ˘ E , equipped with a Weil descent datum. Each object in the tower carries an action of G b ( Q p ),and the tower carries an action of G ( Q p ). Following Huber [ Hub98 ], one deﬁnes R Γ c ( M ( G,b,µ ) ,K,C , Z (cid:96) ) = lim −→ U R Γ c ( U, Z (cid:96) )where U ⊂ M ( G,b,µ ) ,K,C runs through quasicompact open subsets, and one deﬁnes R Γ c ( U, Z (cid:96) ) =lim ←− m R Γ c ( U, Z /(cid:96) m Z ). This carries an action of G b ( Q p ) as well as an action of the Weil group W E . Theorem

IX.3.1 . The complex R Γ c ( M ( G,b,µ ) ,K,C , Z (cid:96) ) is naturally a complex of smooth G b ( Q p ) -representations, and, if K is pro- p , a compact object in D ( G b ( Q p ) , Z (cid:96) ) . Moreover, the action of W E is continuous. In particular, each H ic ( M ( G,b,µ ) ,K,C , Z (cid:96) ) is a ﬁnitely generated smooth G b ( Q p )-representation.By descent, this is true even for all K (not necessarily pro- p ). Proof.

Let f K : M ( G,b,µ ) ,K,C → Spa C be the projection. Up to shift, we can replace Z (cid:96) bythe dualizing complex Rf ! K Z (cid:96) . Now by Proposition VII.5.2, one has Rf K ! Rf ! K Z (cid:96) | U ∼ = f K(cid:92) Z (cid:96) | U for any quasicompact U ⊂ M ( G,b,µ ) ,K,C . As the left-hand side is perfect, it is given by its limitover reductions modulo (cid:96) m . We see that H ic ( M ( G,b,µ ) ,K,C , Z (cid:96) ) can be identiﬁed with H i ( f K(cid:92) Z (cid:96) ) upto shift.Now µ gives rise to a Hecke operator T µ = T V µ where V µ is the highest weight representationof weight µ . It corresponds to the Hecke correspondence on Bun G,C parametrizing modiﬁcationsof type µ ; this Hecke correspondence is proper and smooth over both factors. We apply T µ to thecompact object A = j ! c -Ind G ( Q p ) K Z (cid:96) ∈ D lis (Bun G , Z (cid:96) )where j : Bun G ∼ = [ ∗ /G ( Q p )] (cid:44) → Bun G is the open immersion. By Theorem IX.2.2, also T µ ( A ) iscompact. By Proposition VII.7.4, it follows that also i b ∗ T µ ( A ) ∈ D lis (Bun bG , Z (cid:96) ) ∼ = D ( G b ( Q p ) , Z (cid:96) )is compact. But this is, up to shift again, precisely f K(cid:92) Z (cid:96) , by the identiﬁcation of M ( G,b,µ ) ,K,C withthe space of modiﬁcations of G -torsors of type µ from the G -bundle E b to the G -bundle E , up tothe action of K (cf. [ SW20 , Lecture 23, 24]).

X.3. COHOMOLOGY OF LOCAL SHIMURA VARIETIES 315

Descending to E , note that T µ can be deﬁned with values in D (cid:4) (Bun G × Spd

E/ϕ Z , Z (cid:96) ), andtakes values in those sheaves whose pullback to Bun G,C lies in D lis (Bun G , Z (cid:96) ). Thus W E , as acondensed group, acts on i b ∗ T µ ( A ) ∈ D lis (Bun bG , Z (cid:96) ) ∼ = D ( G b ( Q p ) , Z (cid:96) ) considered as representationson condensed Z (cid:96) -modules. In classical language, this means that the action is continuous. (cid:3) In particular, for each admissible representation ρ of G b ( Q p ) on a Z (cid:96) -algebra Λ, the complex R Hom G b ( Q p ) ( R Γ c ( M ( G,b,µ ) ,K,C , Z (cid:96) ) , ρ )is a perfect complex of Λ-modules. Passing to the colimit over K , one obtains at least on eachcohomology group an admissible G ( Q p )-representation. In fact, as T µ is left adjoint to T µ ∨ , we seethat this is (up to shift) given by i ∗ T µ ∨ ( Ri b ∗ [ ρ ]) ∈ D lis (Bun G , Λ) ∼ = D ( G ( Q p ) , Λ) . Here i b : Bun bG (cid:44) → Bun G is the inclusion, and [ ρ ] ∈ D ( G b ( Q p ) , Λ) ∼ = D lis (Bun bG , Λ) can be a complexof smooth G b ( Q p )-representations. This shows in particular that there is in fact a natural complexof admissible G ( Q p )-representations underlying R Hom G b ( Q p ) ( R Γ c ( M ( G,b,µ ) ,K,C , Z (cid:96) ) , ρ ) . Assuming again that ρ is admissible, one can pull through Verdier duality, i ∗ T µ ∨ ( Ri b ∗ [ ρ ]) ∼ = i ∗ T µ ∨ ( Ri b ∗ D ([ ρ ∨ ])) ∼ = i ∗ T µ ∨ ( D ( i b ! [ ρ ∨ ])) ∼ = i ∗ D ( T sw ∗ µ ( i b ! [ ρ ∨ ])) ∼ = D ( i ∗ T sw ∗ µ ( i b ! [ ρ ∨ ])) . As T sw ∗ µ also preserves compact objects, it follows that [ RV14 , Remark 6.2 (iii)] has a positiveanswer: If Λ = Q (cid:96) and ρ has ﬁnite length, then also each cohomology group of R Hom G b ( Q p ) ( R Γ c ( M ( G,b,µ ) ,K,C , Z (cid:96) ) , ρ )has ﬁnite length. Indeed, with Q (cid:96) -coeﬃcients, the category of smooth representations has ﬁniteglobal dimension, and hence being compact is equivalent to each cohomology group being ﬁnitelygenerated. Compact objects are preserved under the Hecke operators, and so we see that eachcohomology group is ﬁnitely generated. Being also admissible, it is then of ﬁnite length by Howe’stheorem [ Ren10 , VI.6.3].The same arguments apply to prove Corollary I.7.3. Let us recall the setup. We start witha general E now. As in [ SW20 , Lecture XXIII], for any collection { µ i } i of conjugacy classes ofcocharacters with ﬁelds of deﬁnition E i /E and b ∈ B ( G ), there is a tower of moduli spaces of localshtukas f K : (Sht ( G,b,µ • ) ,K ) K ⊂ G ( E ) → (cid:89) i ∈ I Spd ˘ E i as K ranges over compact open subgroups of G ( E ), equipped with compatible ´etale period maps π K : Sht ( G,b,µ • ) ,K → Gr tw G, (cid:81) i ∈ I Spd ˘ E i , ≤ µ • .

16 IX. THE HECKE ACTION

Here, Gr tw G, (cid:81) i ∈ I Spd ˘ E i → (cid:81) i ∈ I Spd ˘ E is a certain twisted form of the convolution aﬃne Grassman-nian, cf. [ SW20 , Section 23.5]. Let W be the exterior tensor product (cid:2) i ∈ I V µ i of highest weightrepresentations, and S W the corresponding sheaf on Gr tw G, (cid:81) i ∈ I Spd ˘ E i . More precisely, away fromFrobenius-twisted partial diagonals, Gr tw G, (cid:81) i ∈ I Spd ˘ E i is isomorphic to the Beilinson–Drinfeld Grass-mannian Gr G, (cid:81) i ∈ I Spd ˘ E i , and we have deﬁned S W on this locus. One can uniquely extend over theseFrobenius-twisted partial diagonals to universally locally acyclic, necessarily perverse, sheaves, as inthe discussion of the fusion product. We continue to write S W for its pullback to Sht ( G,b,µ • ) ,K . Let S (cid:48) W = D ( S W ) ∨ be the corresponding solid sheaf. By Proposition VII.5.2, with torsion coeﬃcients f K(cid:92) S (cid:48) W agrees with Rf K ! S W , but f K(cid:92) S (cid:48) W is well-deﬁned in general. Proposition

IX.3.2 . The sheaf f K(cid:92) S (cid:48) W ∈ D (cid:4) ([ ∗ /G b ( E )] × (cid:89) i ∈ I Spd ˘ E i , Λ) is equipped with partial Frobenii, thus descends to an object of D (cid:4) ([ ∗ /G b ( E )] × (cid:89) i ∈ I Spd ˘ E i /ϕ Z i , Λ) . This object lives in the full ∞ -subcategory D ( G b ( E ) , Λ) B (cid:81) i ∈ I W Ei ⊂ D (cid:4) ([ ∗ /G b ( E ))] × (cid:89) i ∈ I Spd ˘ E i /ϕ Z i , Λ) , and its restriction to D ( G b ( E ) , Λ) is compact. In particular, for any admissible representation ρ of G b ( E ) , the object R Hom G b ( E ) ( f K(cid:92) S (cid:48) W , ρ ) ∈ D (Λ) B (cid:81) i ∈ I W Ei is a representation of (cid:81) i ∈ I W E i on a perfect complex of Λ -modules. Taking the colimit over K , thisgives rise to a complex of admissible G ( E ) -representations lim −→ K R Hom G b ( E ) ( f K(cid:92) S (cid:48) W , ρ ) equipped with a (cid:81) i ∈ I W E i -action.If ρ is compact, then so is lim −→ K R Hom G b ( E ) ( f K(cid:92) S (cid:48) W , ρ ) as a complex of G ( E ) -representations. Proof.

The key observation is that f K(cid:92) S (cid:48) W can be identiﬁed with T W ( j ! [ c -Ind G ( E ) K Λ]) | Bun bG . Apriori, for the latter, we have to look at the moduli space M of modiﬁcations of type bounded by µ • from E b to the trivial vector bundle, up to the action of K , and take the homology of M withcoeﬃcients in S (cid:48) W ; more precisely, the relative homology of M → (cid:81) i ∈ I Spd ˘ E i /ϕ Z i . After pull backto (cid:81) i ∈ I Spd ˘ E i , there is a natural map from M to Sht ( G,b,µ • ,K ) that is an isomorphism away fromFrobenius-twisted partial diagonals. Indeed, Sht ( G,b,µ • ,K ) parametrizes G -torsors over Y S togetherwith an isomorphism with their Frobenius pullback away from the given points, together with alevel- K -trivialization of the G -bundle near { π = 0 } . This induces two vector bundles on X S , givenby the bundles near { π = 0 } and near { [ (cid:36) ] = 0 } , and these are identiﬁed away from the images of X.5. THE BERNSTEIN CENTER 317 the punctures in X S . As long as their images in X S are disjoint, one can reverse this procedure.Now the fusion compatibility of S W (and thus S (cid:48) W ) implies the desired result.In particular, this shows that f K(cid:92) S (cid:48) W admits natural partial Frobenius operators. The rest ofthe proof is now as before. (cid:3) IX.4. L -parameter We can now deﬁne L -parameter. Definition/Proposition

IX.4.1 . Let L be an algebraically closed ﬁeld over Z (cid:96) [ √ q ] , and let A ∈ D lis (Bun G , L ) be a Schur-irreducible object, i.e. End( A ) = L as condensed algebras. Thenthere is a unique semisimple L -parameter ϕ A : W E → (cid:98) G ( L ) (cid:111) Q such that for all excursion data ( I, V, α, β, ( γ i ) i ∈ I ) consisting of a ﬁnite set I , V ∈ Rep(( (cid:98) G (cid:111) Q ) I ) , α : 1 → V | (cid:98) G , β : V | (cid:98) G → and γ i ∈ W E for i ∈ I , the endomorphism A = T ( A ) α −→ T V ( A ) ( γ i ) i ∈ I −−−−→ T V ( A ) β −→ T ( A ) = A is given by the scalar L α −→ V ( ϕ A ( γ i )) i ∈ I −−−−−−−→ V β −→ L. Proof.

By the arguments of Section VIII.4, we can build excursion data as required for Propo-sition VIII.3.8. (cid:3)

IX.5. The Bernstein center

As before, there is the problem that the stack Z ( W E , (cid:98) G ) of L -parameters is not quasicompact,but an inﬁnite disjoint union. We can now actually decompose D lis (Bun G , Λ) into a direct productaccording to the connected components of Z ( W E , (cid:98) G ). We start with the following observation. Proposition

IX.5.1 . Let A ∈ D lis (Bun G , Λ) ω be any compact object. Then there is an opensubgroup P ⊂ W E of the wild inertia subgroup such that for all ﬁnite sets I and all V ∈ Rep(( (cid:98) G (cid:111) Q ) I ) , the object T V ( A ) ∈ D lis (Bun G , Λ) BW IE lies in the full ∞ -subcategory D lis (Bun G , Λ) B ( W E /P ) I ⊂ D lis (Bun G , Λ) BW IE . Proof.

First, note that indeed the functor D lis (Bun G , Λ) B ( W E /P ) I → D lis (Bun G , Λ) BW IE . is fully faithful; this follows from fully faithfulness of the pullback functor f ∗ : D (cid:4) (Bun G × [ ∗ / ( W E /P ) I ] , Λ) → D (cid:4) (Bun G × [ ∗ /W IE ] , Λ) , which in turn follows from f (cid:92) Λ ∼ = Λ (and the projection formula for f (cid:92) ), which can be deducedvia base change from the case of [ ∗ /W IE ] → [ ∗ / ( W E /P ) I ], or after pullback to a v-cover Spa C →

18 IX. THE HECKE ACTION [ ∗ / ( W E /P ) I ], for [Spa C/P I ] → Spa C , where it amounts to the vanishing of the Λ-homology of P I . Now note that if P I acts trivially on T V ( A ) and on T W ( A ) for two V, W ∈ Rep Z (cid:96) (( (cid:98) G (cid:111) Q ) I ),then it also acts trivially on T V ⊗ W ( A ) = T V ( T W ( A )) = T W ( T V ( A )): Indeed, the W I (cid:116) IE -action on T V ( T W ( A )) ∼ = T V (cid:2) W ( A ) ∼ = T W ( T V ( A )) is trivial on P I (cid:116)∅ and P ∅(cid:116) I , thus on P I (cid:116) I , and hence thediagonal W IE -action is trivial on P I . Using reductions to exterior tensor products, we can alsoreduce to I = {∗} . Then if V ∈ Rep Z (cid:96) ( (cid:98) G (cid:111) Q ) is a ⊗ -generator, it follows that it suﬃces that P acts trivially on T V ( A ). But D lis (Bun G , Λ) BW E = (cid:91) P D lis (Bun G , Λ) B ( W E /P ) as for any relatively discrete condensed animated Z (cid:96) -module M with a map W E → Aut( M ), themap factors over W E /P for some P . Indeed, one can reduce to M concentrated in degree 0, andthen to M ﬁnitely generated, so that Aut( M ) is proﬁnite, and locally pro- (cid:96) . (cid:3) Fix some open subgroup P of the wild inertia subgroup of W E , and let D P lis (Bun G , Λ) ω ⊂ D lis (Bun G , Λ) ω be the full ∞ -subcategory of all A such that P I acts trivially on T V ( A ) for all V ∈ Rep(( (cid:98) G (cid:111) Q ) I ).Pick W ⊂ W E /P a discrete dense subgroup, by discretizing the tame inertia, as before. ThenTheorem VIII.4.1 gives a canonical map of algebrasExc( W, (cid:98) G ) → Z ( D P lis (Bun G , Λ) ω ) = π End(id D P lis (Bun G , Λ) ω ) . As Exc( W, (cid:98) G ) ⊗ Λ → O ( Z ( W E /P, (cid:98) G ) Λ ) (cid:98) G is a universal homeomorphism, there are in partic-ular idempotents corresponding to the connected components of Z ( W E /P, (cid:98) G ) Λ . Their action on D P lis (Bun G , Λ) ω then induces a direct sum decomposition D P lis (Bun G , Λ) ω = (cid:77) c ∈ π Z ( W E /P, (cid:98) G ) Λ D c lis (Bun G , Λ) ω . Taking now a union over all P , we get a direct sum decomposition D lis (Bun G , Λ) ω = (cid:77) c ∈ π Z ( W E , (cid:98) G ) Λ D c lis (Bun G , Λ) ω . On the level of Ind-categories, this gives a direct product D lis (Bun G , Λ) = (cid:89) c ∈ π Z ( W E , (cid:98) G ) Λ D c lis (Bun G , Λ) . Note in particular that any Schur-irreducible object A ∈ D lis (Bun G , Λ) necessarily lies in oneof these factors, given by some connected component c of Z ( W E , (cid:98) G ) Λ ; and then the L -parameter ϕ A of A necessarily lies in this connected component. X.5. THE BERNSTEIN CENTER 319

Using excursion operators, we get the following result on the “Bernstein center”. Note that for G = GL n , all (cid:96) are allowed, so this is a generalization of results of Helm–Moss, [ HM18 ], notingthat by the fully faithful functor D ( G ( E ) , Λ) (cid:44) → D lis (Bun G , Λ), there is a map of algebras Z geom ( G, Λ) → Z ( G ( E ) , Λ)to the usual Bernstein center of smooth G ( E )-representations on Λ-modules. Theorem

IX.5.2 . Assume that (cid:96) is invertible in Λ , or that (cid:96) is a very good prime for (cid:98) G . Thenthere is a natural map Z spec ( G, Λ) → Z geom ( G, Λ) compatible with the above decomposition into connected components. Moreover, for all ﬁnite sets I ,all V ∈ Rep Λ ( (cid:98) G I ) , and all A ∈ D lis (Bun G , Λ) , the diagram Z spec ( G, Λ) (cid:47) (cid:47) (cid:39) (cid:39) End( A ) (cid:15) (cid:15) End( T V ( A )) commutes, so the map factors over Z geomHecke ( G, Λ) ⊂ Z geom ( G, Λ) . Of course, as noted above, in general we still have mapsExc( W, (cid:98) G ) → Z ( D P lis (Bun G , Λ) ω )and Exc( W, (cid:98) G ) → O ( Z ( W E /P, (cid:98) G )) (cid:98) G is a universal homeomorphism. The commutation with theHecke action is true here as well. Proof.

This follows from the decomposition into connected components, the map Exc( W, (cid:98) G ) Λ →Z ( D P lis (Bun G , Λ) ω ) above, and Theorem VIII.3.6. The statement about commutation with Heckeoperators follows from the construction of excursion operators and the commutation of Hecke op-erators. (cid:3) Before going on, we make the following observation regarding duality. The Bernstein–Zelevinskyduality functor D BZ on D lis (Bun G , Λ) induces an involution D geom of Z geom ( G, Λ). On the otherhand, on Z ( W E , (cid:98) G ), the Cartan involution of (cid:98) G induces an involution; after passing to the quotientby the conjugation action of (cid:98) G , we can also forget about the inner automorphism appearing inProposition VI.12.1. Let D spec be the induced involution of Z spec ( G, Λ).

Proposition

IX.5.3 . Assume that (cid:96) is very good for (cid:98) G , or (cid:96) is invertible in Λ . Then thediagram Z spec ( G, Λ) D spec (cid:15) (cid:15) (cid:47) (cid:47) Z geom ( G, Λ) D geom (cid:15) (cid:15) Z spec ( G, Λ) (cid:47) (cid:47) Z geom ( G, Λ) commutes.The formation of L -parameters for irreducible smooth representations of G ( E ) is compatiblewith passage to Bernstein–Zelevinsky duals, and to smooth duals.

20 IX. THE HECKE ACTION

Proof.

The commutation follows easily from the construction of excursion operators andProposition VI.12.1. For the ﬁnal part, it now follows that the formation of L -parameters iscompatible with passage to Bernstein–Zelevinsky duals. For supercuspidal representations, thisagrees with the smooth dual. In general, the claim for smooth duals follows from the compatibilitywith parabolic induction proved below. (cid:3) IX.6. Properties of the correspondence

As usual, we ﬁx an open subgroup P of the wild inertia of W E (mapping trivially to Q ), and adiscretization W ⊂ W E /P . IX.6.1. Isogenies.

Theorem

IX.6.1 . Let G (cid:48) → G be a map of reductive groups inducing an isomorphism of adjointgroups, inducing a dual map (cid:98) G → (cid:99) G (cid:48) , and π : Bun G (cid:48) → Bun G . Then for any A ∈ D P lis (Bun G , Λ) the diagram Exc( W, (cid:99) G (cid:48) ) Λ (cid:47) (cid:47) (cid:15) (cid:15) End( π ∗ A )Exc( W, (cid:98) G ) Λ (cid:47) (cid:47) End( A ) (cid:79) (cid:79) commutes. In particular, if L is an algebraically closed ﬁeld, A is Schur-irreducible and A (cid:48) is aSchur-irreducible constituent of π ∗ A , then ϕ A (cid:48) is the composite of ϕ A with (cid:98) G → (cid:99) G (cid:48) . Moreover, ifeither (cid:96) is invertible in Λ or (cid:96) is very good for (cid:98) G and (cid:99) G (cid:48) , the diagram Z spec ( (cid:99) G (cid:48) , Λ) (cid:47) (cid:47) (cid:15) (cid:15) End( π ∗ A ) Z spec ( (cid:98) G, Λ) (cid:47) (cid:47) End( A ) (cid:79) (cid:79) commutes. Proof.

Consider any excursion operator for G (cid:48) , given by some ﬁnite set I , a representation V (cid:48) ∈ Rep Λ (( (cid:99) G (cid:48) (cid:111) Q ) I ), maps α : 1 → V (cid:48) | (cid:99) G (cid:48) , β : V (cid:48) | (cid:99) G (cid:48) → γ i ∈ Γ as usual. Considerthe diagram Bun G (cid:48) π (cid:15) (cid:15) Hck IG (cid:48) h (cid:48) (cid:111) (cid:111) π H (cid:15) (cid:15) h (cid:48) (cid:47) (cid:47) Bun G (cid:48) × (Div ) Iπ (cid:15) (cid:15) Bun G Hck

IGh (cid:111) (cid:111) h (cid:47) (cid:47) Bun G × (Div ) I . Then T V (cid:48) ( π ∗ A ) = h (cid:48) (cid:92) ( h (cid:48)∗ π ∗ A (cid:4) ⊗ L Λ S (cid:48) V (cid:48) ) . X.6. PROPERTIES OF THE CORRESPONDENCE 321

We are interested in computing an endomorphism of π ∗ A ; in particular, it is enough to compute π (cid:92) T V (cid:48) ( π ∗ A ). But π (cid:92) T V (cid:48) ( π ∗ A ) = π (cid:92) h (cid:48) (cid:92) ( h (cid:48)∗ π ∗ A (cid:4) ⊗ L Λ S (cid:48) V (cid:48) ) ∼ = h (cid:92) π H (cid:92) ( π ∗ H h ∗ A (cid:4) ⊗ L Λ S V (cid:48) ) ∼ = h (cid:92) ( h ∗ A (cid:4) ⊗ L Λ π H(cid:92) S (cid:48) V (cid:48) ) ∼ = h (cid:92) ( h ∗ A (cid:4) ⊗ L Λ S (cid:48) V ) = T V ( A ) . This identiﬁcation is functorial in V (cid:48) and I , and is over Bun G × (Div ) I , hence implies the desiredequality of excursion operators. Here, to identify π H(cid:92) S V (cid:48) , we use that the diagramHck IG (cid:48) (cid:47) (cid:47) π H (cid:15) (cid:15) H ck IG (cid:48) (cid:15) (cid:15) Hck IG (cid:47) (cid:47) H ck IG is cartesian, and the compatibility of the geometric Satake equivalence with the map G → G (cid:48) inducing isomorphisms of adjoint groups, as in the proof of Theorem VI.11.1. (cid:3) IX.6.2. Products.

Proposition

IX.6.2 . If G = G × G is a product of two groups, then the diagram Exc( W, (cid:99) G ) Λ ⊗ Λ Exc( W, (cid:99) G ) Λ ∼ = (cid:15) (cid:15) (cid:47) (cid:47) Z ( D P lis (Bun G , Λ)) ⊗ Λ Z ( D P lis (Bun G , Λ)) (cid:15) (cid:15)

Exc( W, (cid:98) G ) (cid:47) (cid:47) Z ( D P lis (Bun G , Λ)) commutes.In particular, if

Λ = L is an algebraically closed ﬁeld and A , A ∈ D lis (Bun G , L ) are Schur-irreducible, and A is a Schur-irreducible constituent of A (cid:2) A , then ϕ A = ( ϕ A , ϕ A ) : W E → (cid:98) G ( L ) ∼ = (cid:99) G ( L ) × (cid:99) G ( L ) . If (cid:96) is invertible in Λ or is very good for (cid:99) G and (cid:99) G , then the diagram Z spec ( G , Λ) ⊗ Λ Z spec ( G , Λ) ∼ = (cid:15) (cid:15) (cid:47) (cid:47) Z geom ( G , Λ) ⊗ Λ Z geom ( G , Λ) (cid:15) (cid:15) Z spec ( G, Λ) (cid:47) (cid:47) Z geom ( G, Λ) commutes. Proof.

The statement can be checked using excursion operators, and the proof is a straight-forward diagram chase, noting that everything decomposes into products. (cid:3)

22 IX. THE HECKE ACTION

IX.6.3. Weil restriction.

Proposition

IX.6.3 . If G = Res E (cid:48) | E G (cid:48) is a Weil restriction of scalars of some reductive group G (cid:48) over some ﬁnite separable extension E (cid:48) of E . Choose P to be an open subgroup of the wild inertiaof W E (cid:48) ⊂ W E , and let W (cid:48) ⊂ W E (cid:48) /P be the preimage of W ⊂ W E /P . Then there are canonicalidentiﬁcations Bun G (cid:48) ∼ = Bun G , Z ( W E , (cid:98) G ) ∼ = Z ( W E (cid:48) , (cid:99) G (cid:48) ) and Exc( W, (cid:98) G ) ∼ = Exc( W (cid:48) , (cid:99) G (cid:48) ) , and thediagram Exc( W (cid:48) , (cid:99) G (cid:48) ) ∼ = (cid:15) (cid:15) (cid:47) (cid:47) Z ( D P lis (Bun G (cid:48) , Λ)) ∼ = (cid:15) (cid:15) Exc( W, (cid:98) G ) (cid:47) (cid:47) Z ( D P lis (Bun G , Λ)) commutes. In particular, L -parameters are compatible with Weil restriction. If (cid:96) is invertible in Λ or is very good for (cid:98) G and (cid:99) G (cid:48) , then the diagram Z spec ( G (cid:48) , Λ) ∼ = (cid:15) (cid:15) (cid:47) (cid:47) Z geom ( G (cid:48) , Λ) ∼ = (cid:15) (cid:15) Z spec ( G, Λ) (cid:47) (cid:47) Z geom ( G, Λ) commutes. Proof.

The most nontrivial of these identiﬁcations is the identiﬁcationExc( W, (cid:98) G ) ∼ = Exc( W (cid:48) , (cid:99) G (cid:48) ) . One way to understand this is to use the presentationExc( W, (cid:98) G ) = colim ( n,F n → W ) O ( Z ( F n , (cid:98) G )) (cid:98) G (and the similar presentation for Exc( W (cid:48) , (cid:99) G (cid:48) )) and the natural isomorphism Z ( F n , (cid:98) G ) (cid:12) (cid:98) G ∼ = Z ( F n × W W (cid:48) , (cid:99) G (cid:48) ) (cid:12) (cid:99) G (cid:48) of aﬃne schemes (and then passing to global sections), noting that F n × W W (cid:48) ⊂ F n is a subgroup of ﬁnite index, and thus itself a ﬁnitely generated free group. This shows infact that restricting to those maps F n → W factoring over W (cid:48) produces the same colimit, and soExc( W, (cid:98) G ) = colim ( n,F n → W ) O ( Z ( F n , (cid:98) G )) (cid:98) G ∼ ←− colim ( n,F n → W (cid:48) ) O ( Z ( F n , (cid:98) G )) (cid:98) G ∼ = colim ( n,F n → W (cid:48) ) O ( Z ( F n , (cid:99) G (cid:48) )) (cid:99) G (cid:48) = Exc( W (cid:48) , (cid:99) G (cid:48) ) . Now consider an excursion operator for G (cid:48) , including a representation V (cid:48) of ( (cid:99) G (cid:48) (cid:111) W E (cid:48) ) I . Notethat (cid:98) G (cid:111) W E contains (cid:98) G (cid:111) W E (cid:48) as a subgroup, and this admits a surjection onto (cid:99) G (cid:48) (cid:111) W E (cid:48) (notingthat (cid:98) G = (cid:81) E (cid:48) (cid:44) → E (cid:99) G (cid:48) , where we picked out an embedding E (cid:48) (cid:44) → E and hence a projection (cid:98) G → (cid:99) G (cid:48) when we regarded W E (cid:48) ⊂ W E as a subgroup). In this way, one can inﬂate V (cid:48) to a representation of X.6. PROPERTIES OF THE CORRESPONDENCE 323 ( (cid:98) G (cid:111) W E (cid:48) ) I and then induce to ( (cid:98) G (cid:111) W E ) I to get a representation V of ( (cid:98) G (cid:111) W E ) I . Geometrically,this procedure amounts to the commutative diagramBun G (cid:48) ∼ = (cid:15) (cid:15) Hck IG (cid:48) h (cid:48) (cid:111) (cid:111) h (cid:48) (cid:47) (cid:47) ψ (cid:15) (cid:15) Bun G (cid:48) × (Div (cid:48) ) I (cid:15) (cid:15) Bun G Hck

IGh (cid:111) (cid:111) h (cid:47) (cid:47) Bun G × (Div ) I and taking ψ ∗ on sheaves. More precisely, we note thatHck IG (cid:48) → Hck IG × (Div ) I (Div (cid:48) ) I is a closed immersion (and arises via pullback from a similar closed immersion of local Hecke stacks).Now the claim follows from a diagram chase. (cid:3) IX.6.4. Tori. If G = T is a torus, then D lis (Bun T , Λ) ∼ = (cid:89) b ∈ B ( T )= π ( T ) Γ D ( T ( E ) , Λ)and in particular Z geom ( T, Λ) = (cid:89) b ∈ B ( T ) Z ( T ( E ) , Λ)where Z ( T ( E ) , Λ) is the Bernstein center of T ( E ); explicitly, this is Z ( T ( E ) , Λ) = lim ←− K ⊂ T ( E ) Λ[ T ( E ) /K ]where K runs over open subgroups of T ( E ).For tori, the representation theory of (cid:98) T is semisimple even integrally, so we get a map Z spec ( T, Λ) → Z geom ( T, Λ) . Proposition

IX.6.4 . There is a natural isomorphism Z spec ( T, Λ) ∼ = lim ←− K ⊂ T ( E ) Λ[ T ( E ) /K ] . Proof.

One can resolve T by products of induced tori and then reduce to the case that T is induced, and then by Weil restrictions of scalars to T = G m . In that case Z ( W E , G m ) =Hom( E ∗ , G m ) by local class ﬁeld theory, giving the result. (cid:3) Proposition

IX.6.5 . Under the above identiﬁcations Z spec ( T, Λ) = lim ←− K ⊂ T ( E ) Λ[ T ( E ) /K ] and Z geom ( T, Λ) = (cid:89) b ∈ B ( T ) lim ←− K ⊂ T ( E ) Λ[ T ( E ) /K ] , the map Z spec ( T, Λ) → Z geom ( T, Λ)

24 IX. THE HECKE ACTION is the diagonal embedding.

Proof.

We may resolve T by induced tori and use Theorem IX.6.1, Proposition IX.6.2 andProposition IX.6.3 to reduce to the case of T = G m . It is enough to compute the excursion operatorscorresponding to I = { , } , V = std (cid:2) std ∨ and the tautological maps α : 1 → std ⊗ std ∨ and β : std ⊗ std ∨ →

1. It is then an easy consequence of Section II.2.1. (cid:3)

Proposition IX.6.5 in particular shows that the L -parameters we construct for tori are theusual L -parameters, and together with Theorem IX.6.1 and Proposition IX.6.2 implies that L -parameters are compatible with central characters and twisting, by applying Theorem IX.6.1 tothe maps Z × G → G and G → G × D where Z ⊂ G is the center and G → D is the quotient bythe derived group. IX.7. Applications to representations of G ( E )Finally, we apply the preceding results to representations of G ( E ). To simplify the statements,we assume from now on that either (cid:96) is invertible in Λ, or that (cid:96) is very good for (cid:98) G ; in general,similar statements hold true up to replacing O ( Z ( W E /P, (cid:98) G )) (cid:98) G by the algebra Exc( W, (cid:98) G ) as above.We get the following map to the Bernstein center. Definition

IX.7.1 . The map Ψ G : Z spec ( G, Λ) → Z ( G ( E ) , Λ) is the composite Z spec ( G, Λ) → Z geom ( G, Λ) → Z ( G ( E ) , Λ) induced by the fully faithful functor j ! : D ( G ( E ) , Λ) ∼ = D lis (Bun G , Λ) → D lis (Bun G , Λ) . More generally, for any b ∈ B ( G ), we can deﬁne a mapΨ bG : Z spec ( G, Λ) → Z ( G b ( E ) , Λ)to the Bernstein center for G b ( E ) by using the fully faithful embedding D ( G b ( E ) , Λ) ∼ = D lis (Bun bG , Λ) → D lis (Bun G , Λ)determined for example by the left adjoint to i b ∗ , where i b : Bun bG (cid:44) → Bun G is the locally closedembedding (see Proposition VII.7.2). (Recall that in the D lis -setting, we do not have a general i b ! -functor, although it can be deﬁned in the present situation. All these maps will induce the samemap to the Bernstein center.) X.7. APPLICATIONS TO REPRESENTATIONS OF G ( E ) 325 IX.7.1. Compatibility with G b . One can describe the maps Ψ bG for b (cid:54) = 1 in terms of themaps Ψ G b . Note that (cid:99) G b is naturally a Levi subgroup of (cid:98) G , as G b, ˘ E ⊂ G ˘ E is the centralizer of theslope morphism ν b : D → G ˘ E . This extends naturally to a morphism of L -groups (cid:99) G b (cid:111) Q → (cid:98) G (cid:111) Q where as usual Q is a ﬁnite quotient of W E over which the action on (cid:98) G factors. However, fromgeometric Satake we rather get the natural inclusion G b (cid:86) (cid:111) W E → G (cid:86) (cid:111) W E where the W E -actions include the cyclotomic twist. The latter induces a map Z ( W E , (cid:99) G b ) → Z ( W E , (cid:98) G )that in terms of the usual W E -action is given by sending a 1-cocycle ϕ : W E → (cid:99) G b ( A ) to the1-cocycle W E → (cid:98) G ( A ) : w (cid:55)→ (2 ρ (cid:98) G − ρ (cid:99) G b )( √ q ) | w | ϕ ( w )where | · | : W E → W E /I E ∼ = Z is normalized as usual by sending a geometric Frobenius to 1. Theorem

IX.7.2 . For all G and b ∈ B ( G ) , the diagram Z spec ( G, Λ) Ψ bG (cid:47) (cid:47) (cid:15) (cid:15) Z ( D ( G b ( E ) , Λ)) Z spec ( G b , Λ) Ψ Gb (cid:54) (cid:54) commutes. Proof.

We note that to prove the theorem, we can assume that Λ is killed by power of (cid:96) (if (cid:96) is not very good for (cid:98) G , replacing the left-hand side with an algebra of excursion operators), as theresult for Λ = Z (cid:96) [ √ q ] implies it in general, and the right-hand side Z ( D ( G b ( E ) , Λ)) = lim ←− K ⊂ G b ( E ) Z (Λ[ G b ( E ) (cid:12) K ])is (cid:96) -adically separated in that case. This means we can avoid the subtleties of D lis in place of D ´et .If b is basic, the theorem follows from the identiﬁcation Bun G ∼ = Bun G b of Corollary III.4.3,which is equivariant for the Hecke action.In general, we ﬁrst reduce to the case that G is quasisplit. Take a z-embedding G (cid:44) → G (cid:48) as in [ Kal18 , Section 5], with quotient a torus D , so that the center Z ( G (cid:48) ) is connected. ThenBun G = Bun G (cid:48) × Bun D {∗} and the map B ( G ) → B ( G (cid:48) ) is injective. To see the latter, by thedescription of the stacks, it suﬃces to see that for all b ∈ B ( G ) with image b (cid:48) ∈ B ( G (cid:48) ), the map G (cid:48) b (cid:48) ( E ) → D ( E ) is surjective. But for any b ∈ B ( G ), the map G b → G (cid:48) b (cid:48) is a z-embedding withquotient D , and Z (cid:48) ( E ) → D ( E ) is surjective by [ Kal18 , Fact 5.5], where also Z (cid:48) ⊂ G (cid:48) b (cid:48) , so inparticular G (cid:48) b (cid:48) ( E ) → D ( E ) is surjective. An element of Z ( D ( G b ( E ) , Λ)) = lim ←− K ⊂ G b ( E ) Z (Λ[ G b ( E ) (cid:12) K ])

26 IX. THE HECKE ACTION of the Bernstein center of G b ( E ) is determined by its action on π (cid:48) | G b ( E ) for representations π (cid:48) of G (cid:48) b (cid:48) ( E ). By Theorem IX.6.1, we can thus reduce to G (cid:48) in place of G , i.e. that the center of G isconnected. When Z ( G ) is connected, there is some basic b ∈ B ( G ) such that G b is quasisplit.Using the Hecke-equivariant isomorphism Bun G ∼ = Bun G b we can thus assume that G is quasisplit.Now if G is quasisplit, ﬁx a Borel B ⊂ G . Any b ∈ B ( G ) then admits a reduction to a canonicalparabolic P = P b ⊂ G containing B . Pick a cocharacter µ : G m → G with dynamical parabolic P .For any N ≥

0, let b N = bµ ( π N ): This is a sequence of elements of B ( G ) associated to the sameparabolic P but increasingly instable. Moreover, G b = G b N . We note that the diagram Z spec ( G, Λ) Ψ bG (cid:47) (cid:47) = (cid:15) (cid:15) Z ( G b ( E ) , Λ) = (cid:15) (cid:15) Z spec ( G, Λ) Ψ bNG (cid:47) (cid:47) Z ( G b N ( E ) , Λ)commutes. For this, take any representation σ of G b ( E ) and consider the sheaf A N ∈ D ´et (Bun G , Λ)concentrated on Bun b N G , corresponding to the representation σ . Let V ∈ Rep (cid:98) G be the highestweight representation with weight µ N . We claim that T V ( A N ) | Bun bG is given by the representation σ . As Hecke operators commute with excursion operators, this implies the desired result. Tocompute T V ( A N ) | Bun bG , we have to analyze the moduli space of modiﬁcations of E b of type boundedby µ N that are isomorphic to E b N . There is in fact precisely one such modiﬁcation, given by pushoutof the standard modiﬁcation of line bundles from O to O (1) via µ N : G m → G ; its type is exactly µ N . This gives the claim.Now to prove the theorem, we have to prove the commutativity of the diagram for any excursionoperator, given by excursion data ( I, V, α, β, ( γ i ) i ∈ I ). For any such excursion data, we can pick N large enough so that any modiﬁcation of E b N to itself, of type bounded by V , is automaticallycompatible with the Harder–Narasimhan reduction to P . In that case, for σ and A N as above, toanalyze the excursion operators A N = T ( A N ) α −→ T V ( A N ) ( γ i ) i ∈ I −−−−→ T V ( A N ) β −→ T ( A N ) = A N , we have to analyze the moduli space of modiﬁcations of E b N , at I varying points, of type boundedby V , and that are isomorphic to E b N . By assumption on N , this is the same as the modulispace of such modiﬁcations as P -bundles. This maps to the similar moduli space parametrizingmodiﬁcations as M -bundles, where M is the Levi of P (and G b = M b M for a basic b M ∈ B ( M )).We want to compute T V ( A N ). Note that A N comes from A (cid:48) N ∈ D ´et (Bun P , Λ) as it is concentratedon Bun b N G ∼ = Bun b N P ⊂ Bun P , by the Harder–Narasimhan reduction. X.7. APPLICATIONS TO REPRESENTATIONS OF G ( E ) 327 Consider the diagram Bun Mψ (cid:15) (cid:15) Hck

IM,Pψ H (cid:15) (cid:15) h (cid:48)(cid:48) (cid:111) (cid:111) h (cid:48)(cid:48) (cid:47) (cid:47) Bun P × (Div ) I = (cid:15) (cid:15) Bun Pπ (cid:15) (cid:15) Hck

IPh (cid:48) (cid:111) (cid:111) π H (cid:15) (cid:15) h (cid:48) (cid:47) (cid:47) Bun P × (Div ) Iπ (cid:15) (cid:15) Bun M Hck

IMh (cid:111) (cid:111) h (cid:47) (cid:47) Bun M × (Div ) I where Hck IM,P is deﬁned as the ﬁbre product Hck IP × Bun P Bun M , and thus parametrizes modiﬁca-tions from an M -bundle to a P -bundle.We need to compute T V ( A N ) | Bun bNG , which by the above argument that any modiﬁcation of E b N to itself of type bounded by V is a modiﬁcation as P -bundles, can be computed in terms of themiddle diagram, as Rh (cid:48) ( h (cid:48)∗ A (cid:48) N ⊗ L Λ S V ) | Bun bNP where S V ∈ H ck IG is the perverse sheaf determined by V under the geometric Satake equivalence,and we continue to denote by S V any of its pullbacks.There is some B N ∈ D ´et (Bun M , Λ) such that A (cid:48) N = Rψ ! B N . In fact, one can take B N = Rπ ! A (cid:48) N ,noting that on the support of A (cid:48) N , the map π : Bun P → Bun M is (cohomologically) smooth, so Rπ ! is deﬁned on A (cid:48) N (although π is a stacky map). (Indeed, everything is concentrated on onestratum, and the relevant categories are all equivalent to D ( G b ( E ) , Λ).) Moreover, to compute therestriction to Bun b N P it is enough to do the computation after applying Rπ ! . We compute: Rπ ! Rh (cid:48) ( h (cid:48)∗ A (cid:48) N ⊗ L Λ S V ) = Rπ ! Rh (cid:48) Rψ H ! ( h (cid:48)(cid:48)∗ B N ⊗ L Λ S V )= Rh ( h ∗ B N ⊗ L Λ Rg ! S V )where g : Hck IM,P → Hck IM is the projection. But this is the pullback of the map L + M \ Gr IP → L + M \ Gr IM = H ck IM under Hck IM → H ck IM . This means that Rg ! S V arises via pullback fromCT P ( S V ) ∈ D ´et ( H ck IM , Λ). Up to the shift [deg P ], this agrees with S V | ( (cid:99) M (cid:111) Q ) I , where the restrictioninvolves a cyclotomic twist, as above. (It is the canonical restriction along M (cid:86) → G (cid:86) for the canonical W E -actions arising geometrically.) Now the excursion operators, which involve maps from and tothe sheaf corresponding to V = 1, require only the connected component where deg P = 0, so wecan ignore the shift.With these translations, we see that the excursion operators on Bun b N G and on Bun b N M agree,giving the desired result. (cid:3) IX.7.2. Parabolic induction.

A corollary of this result is compatibility with parabolic in-duction.

28 IX. THE HECKE ACTION

Corollary

IX.7.3 . Let G be a reductive group with a parabolic P ⊂ G and Levi P → M . Thenfor all representations σ of M ( E ) with (unnormalized) parabolic induction Ind G ( E ) P ( E ) σ , the diagram Z spec ( G, Λ) (cid:47) (cid:47) (cid:15) (cid:15) End(Ind G ( E ) P ( E ) σ ) Z spec ( M, Λ) (cid:47) (cid:47) End( σ ) (cid:79) (cid:79) commutes. In particular, the formation of L -parameters is compatible with parabolic induction:If Λ = L is an algebraically closed ﬁeld, σ is irreducible and (cid:101) σ is an irreducible subquotient of Ind G ( E ) P ( E ) σ , then ϕ (cid:101) σ is conjugate to the composite W E ϕ σ −−→ (cid:99) M ( L ) (cid:111) W E → (cid:98) G ( L ) (cid:111) W E where the map (cid:99) M (cid:111) W E → (cid:98) G (cid:111) W E is deﬁned as above, involving the cyclotomic twist. Proof.

It suﬃces to prove the result for σ = c -Ind M ( E ) K Λ for K ⊂ M ( E ) an open pro- p -subgroup, and then one can assume Λ = Z (cid:96) [ √ q ], where one can further by (cid:96) -adic separatednessreduce to torsion coeﬃcients.Let µ : G m → G be a cocharacter with dynamical parabolic P and let b = µ ( π ) ∈ B ( G ).Then G b = M , and we can build a sheaf A ∈ D ´et (Bun G , Λ) concentrated on Bun bG , given by therepresentation σ . Let V ∈ Rep( (cid:98) G ) be a highest weight representation of weight µ . Then T V ( A ) | Bun G is given by Ind G ( E ) P ( E ) σ : To see this, we have to understand the moduli space of modiﬁcation of thetrivial G -torsor of type bounded by µ that are isomorphic to E b . This is in fact given by G ( E ) /P ( E ),the G ( E )-orbit of the pushout of the modiﬁcation from O to O (1) via µ . All of these modiﬁcationsare of type exactly µ . This easily gives the claim on T V ( A ) | Bun G . Now as Hecke operators commutewith excursion operators, the excursion operators on Ind G ( E ) P ( E ) σ agree with those on A , and theseare determined by Theorem IX.7.2, giving the result. (cid:3) IX.7.3. The case G = GL n . For the group G = GL n , we can identify the L -parameterswith the usual L -parameters of [ LRS93 ], [

HT01 ], [

Hen00 ]. This is the only place of this paperwhere we rely on previous work on the local Langlands correspondence, or (implicitly) rely onglobal arguments. More precisely, we use the identiﬁcation of the cohomology of the Lubin–Tateand Drinfeld tower, see [

Boy99 ], [

Har97 ], [

HT01 ], [

Hau05 ], [

Dat07 ]. In the proof, we use thetranslation between Hecke operators and local Shimura varieties as Section IX.3, together with thedescription of these as the Lubin–Tate tower and Drinfeld tower in special cases, see [

SW13 ]. Theorem

IX.7.4 . Let π be any irreducible smooth representation of GL n ( E ) over some alge-braically closed ﬁeld L over Q (cid:96) ( √ q ) . Then the L -parameter ϕ π agrees with the usual (semisimpliﬁed) L -parameter. Proof.

By Corollary IX.7.3, we can assume that π is supercuspidal. We only need to evaluatethe excursion operators for the excursion data given by I = { , } , the representation V = std (cid:2) std ∨ X.7. APPLICATIONS TO REPRESENTATIONS OF G ( E ) 329 of (cid:100) GL n , and the tautological maps α : 1 → std ⊗ std ∨ and β : std ⊗ std ∨ →

1, as these excursionoperators determine the trace of the representation (and thus the semisimpliﬁed representation).First, we analyze these excursion operators on the sheaf B which is the sheaf on Bun b GL n for b corresponding to the bundle O ( − n ), given by the representation σ = JL( π ) of D × ; here D isthe division algebra of invariant n . The Hecke operator T V is the composite of two operators.The ﬁrst Hecke operator, corresponding to std, takes minuscule modiﬁcations O ( − n ) ⊂ E withcokernel a skyscraper sheaf of rank 1. Such an E is necessarily isomorphic to O n , and the Heckeoperator will then produce the σ -isotypic part of the cohomology of the Lubin–Tate tower, whichis π ⊗ ρ π , where ρ π is the irreducible n -dimensional W E -representation associated to π by thelocal Langlands correspondence. (Note that the shift [ n − n − )that usually appears, is hidden inside the normalization of the perverse sheaf corresponding to thestandard representation.) Now the second Hecke operator, when restricted to Bun b GL n , producesthe π -isotypic component of the cohomology of the Drinfeld tower, which is σ ⊗ ρ ∗ π . In total, we seethat T V ( B ) | Bun b GL n is given by π ⊗ ρ π ⊗ ρ ∗ π as representation of D × × W E × W E . By irreducibilityof ρ π , the W E -equivariant map σ → σ ⊗ ρ π ⊗ ρ ∗ π induced by α (and the similar backwards inducedby β ) must agree up to scalar with the obvious map. The scalar of the total composite can beidentiﬁed by taking both elements of W E to be equal to 1. This shows that B has the correct L -parameter. Now use that the sheaf corresponding to π appears as a summand of T std ( B ) (afterforgetting the W E -action) to conclude the same for π . (cid:3) In particular, it follows that the map Z spec (GL n , Q (cid:96) ) → Z (GL n ( E ) , Q (cid:96) )to the Bernstein center agrees with the usual map. We recall that we have already proved that thismap reﬁnes to a map Z spec (GL n , Λ) → Z (GL n ( E ) , Λ)to the integral Bernstein center (where Λ = Z (cid:96) [ √ q ]), recovering a result of Helm–Moss [ HM18 ].HAPTER X

The spectral action

As a ﬁnal topic, we construct the spectral action. We will ﬁrst construct it with characteristic0 coeﬃcients, and then explain reﬁnements with integral coeﬃcients.Let Λ be the ring of integers in a ﬁnite extension of Q (cid:96) ( √ q ). We have the stable ∞ -category C = D lis (Bun G , Λ) ω of compact objects, which is linear over Λ, and functorially in the ﬁnite set I an exact monoidal functor Rep Λ ( (cid:98) G (cid:111) Q ) I → End Λ ( C ) BW IE that is linear over Rep Λ ( Q I ). A ﬁrstversion of the following theorem is due to Nadler–Yun [ NY19 ] in the context of Betti geometricLanglands, and a more general version appeared in the work of Gaitsgory–Kazhdan–Rozenblyum–Varshavsky [

GKRV19 ]. Both references, however, eﬀectively assume that G is split, work onlywith characteristic 0 coeﬃcients, and work with a discrete group in place of W E . At least theextension to Z (cid:96) -coeﬃcients is a nontrivial matter.Note that Z ( W E , (cid:98) G ) is not quasicompact, as it has inﬁnitely many connected components; itcan be written as the increasing union of open and closed quasicompact subschemes Z ( W E /P, (cid:98) G ).We say that an action of Perf( Z ( W E , (cid:98) G ) / (cid:98) G ) on a stable ∞ -category C is compactly supported iffor all X ∈ C the functor Perf( Z ( W E , (cid:98) G ) / (cid:98) G ) → C (induced by acting on X ) factors over somePerf( Z ( W E /P, (cid:98) G ) / (cid:98) G ).The goal of this chapter is to prove the following theorem. Recall that for G = GL n , all (cid:96) arevery good, and for classical groups, all (cid:96) (cid:54) = 2 are very good. Theorem

X.0.1 . Assume that (cid:96) is a very good prime for (cid:98) G . Let C be a small idempotent-complete Λ -linear stable ∞ -category. Then giving, functorially in the ﬁnite set I , an exact Rep Λ ( Q I ) -linear monoidal functor Rep Λ ( (cid:98) G (cid:111) Q ) I → End Λ ( C ) BW IE is equivalent to giving a compactly supported Λ -linear action of Perf( Z ( W E , (cid:98) G ) Λ / (cid:98) G ) . Here, given a Λ -linear action of Perf( Z ( W E , (cid:98) G ) Λ / (cid:98) G ) , one can produce such an exact Rep Λ ( Q I ) -linear monoidal functor Rep Λ ( (cid:98) G (cid:111) Q ) I → End Λ ( C ) BW IE functorially in I by composing the exact Rep Λ ( Q I ) -linear symmetric monoidal functor Rep Λ ( (cid:98) G (cid:111) Q ) I → Perf( Z ( W E , (cid:98) G ) Λ / (cid:98) G ) BW IE with the action of Perf( Z ( W E , (cid:98) G ) Λ / (cid:98) G ) .The same result holds true if Λ is a ﬁeld over Q (cid:96) ( √ q ) , for any prime (cid:96) . Here, the exact Rep Λ ( Q I )-linear symmetric monoidal functorRep Λ ( (cid:98) G (cid:111) Q ) I → Perf( Z ( W E , (cid:98) G ) Λ / (cid:98) G ) BW IE is induced by tensor products and the exact Rep Λ ( Q )-linear symmetric monoidal functorRep Λ ( (cid:98) G (cid:111) Q ) → Perf( Z ( W E , (cid:98) G ) Λ / (cid:98) G ) BW E corresponding to the universal (cid:98) G (cid:111) Q -torsor, with the universal W E -equivariance as parametrizedby Z ( W E , (cid:98) G ) / (cid:98) G .Before starting the proof, we note that the proof of Proposition IX.5.1 shows that we mayreplace W E by W E /P in the statement of Theorem X.0.1. Choosing moreover a discretization W ⊂ W E /P , we reduce to the following variant. Theorem

X.0.2 . Assume that (cid:96) is a very good prime for (cid:98) G . Let C be a small idempotent-complete Λ -linear stable ∞ -category. Then giving, functorially in the ﬁnite set I , an exact Rep Λ ( Q I ) -linear monoidal functor Rep Λ ( (cid:98) G (cid:111) Q ) I → End Λ ( C ) BW I is equivalent to giving a Λ -linear action of Perf( Z ( W, (cid:98) G ) Λ / (cid:98) G ) , with the same compatibility as above. The same result holds true if Λ is a ﬁeld over Q (cid:96) , for anyprime (cid:96) . X.1. Rational coeﬃcients

With rational coeﬃcients, we can prove a much more general result, following [

GKRV19 ].Consider a reductive group H over a ﬁeld L of characteristic 0 (like (cid:98) G over Q (cid:96) ) with an action ofa ﬁnite group Q . Let S be any anima over BQ (like BW , where W ⊂ W E /P is a discretization of W E /P for an open subgroup of the wild inertia, as usual). We can then consider the (derived) stackMap BQ ( S, B ( H (cid:111) Q )) over L , whose values in an animated L -algebra A are the maps of anima S → B ( H (cid:111) Q )( A ) over BQ . This recovers the stack [ Z ( W, (cid:98) G ) Q (cid:96) / (cid:98) G ] in the above example, usingProposition VIII.3.5.In general, Map BQ ( S, B ( H (cid:111) Q )) is the fpqc quotient of an aﬃne derived scheme by a powerof H . Indeed, pick a surjection S (cid:48) → S × BQ ∗ from a set S (cid:48) . Then Map BQ ( S, B ( H (cid:111) Q )) mapsto BH S (cid:48) ; we claim that the ﬁbre is an aﬃne derived scheme, i.e. representable by an animated L -algebra. For this, note thatMap BQ ( S, B ( H (cid:111) Q )) → Map( S × BQ ∗ , BH )is relatively representable, as it is given by the Q -ﬁxed points. To show that the right-hand sideis relatively representable over BH S (cid:48) , we can replace S × BQ ∗ by a connected anima T , and S (cid:48) by a point. Then Map( T, BH ) × BH ∗ parametrizes pointed maps T → BH , which are equivalentto maps of E -groups Ω( T ) → H . Writing Ω( T ) as a sifted colimit of ﬁnite free groups F n , onereduces to representability of maps of groups F n → H , which is representable by H n . .1. RATIONAL COEFFICIENTS 333 Theorem

X.1.1 . Let C be an idempotent-complete small stable L -linear ∞ -category. Giving,functorially in ﬁnite sets I , an exact Rep L ( Q I ) -linear monoidal functor Rep L (( H (cid:111) Q ) I ) → End L ( C ) S I is equivalent to giving an L -linear action of Perf(Map BQ ( S, B ( H (cid:111) Q ))) on C . Here, given such anaction of Perf(Map BQ ( S, B ( H (cid:111) Q ))) , one gets exact Rep L ( Q I ) -linear monoidal functors Rep L (( H (cid:111) Q ) I ) → End L ( C ) S I by precomposing the exact monoidal functor Perf(Map BQ ( S, B ( H (cid:111) Q ))) → End L ( C ) with the nat-ural exact Rep L ( Q I ) -linear symmetric monoidal functor Rep L (( H (cid:111) Q ) I ) → Perf(Map BQ ( S, B ( H (cid:111) Q ))) S I given by I -fold tensor product of the exact Rep L ( Q ) -linear symmetric monoidal functor Rep L ( H (cid:111) Q ) → Perf(Map BQ ( S, B ( H (cid:111) Q ))) S assigning to each s ∈ S pullback along evaluation at s , Map BQ ( S, B ( H (cid:111) Q )) → B ( H (cid:111) Q ) . Proof.

Note ﬁrst that, for any L -linear idempotent-complete small stable ∞ -category C , givingan exact L -linear functor Rep L (( H (cid:111) Q ) I ) → C is equivalent to giving an exact L -linear functor ofstable ∞ -categories Perf(( H (cid:111) Q ) I ) → C , as the ∞ -category of perfect complexes is freely generatedby the exact category of representations. Indeed, such functors extend to the ∞ -category obtainedby inverting quasi-isomorphisms in Ch b (Rep L ( H (cid:111) Q ) I ), and this is Perf( B ( H (cid:111) Q ) I ).For any S , we have the anima F ( S ) of L -linear actions of Perf(Map BQ ( S, B ( H (cid:111) Q ))) on C ,and the anima F ( S ) of functorial exact monoidal functorsRep L (( H (cid:111) Q ) I ) → End L ( C ) S I linear over Rep L ( Q I ), and a natural map F ( S ) → F ( S ) functorial in S (where both F and F are contravariant functors of S ). Both functors take sifted colimits in S to limits. This is clearfor F (as S (cid:55)→ S I commutes with sifted colimits). For F , it is enough to see that taking S toPerf(Map BQ ( S, B ( H (cid:111) Q ))) commutes with sifted colimits (taken in idempotent-complete stable ∞ -categories), which is Lemma X.1.2 below.Therefore it suﬃces to handle the case that S is a ﬁnite set, for which the map S → BQ canbe factored over ∗ . Then Map BQ ( S, B ( H (cid:111) Q )) ∼ = BH S . Similarly, exact monoidal functorsRep L (( H (cid:111) Q ) I ) → End L ( C ) S I linear over Rep L ( Q I ) are equivalent to exact monoidal functorsRep L ( H I ) → End L ( C ) S I linear over L . Here, we use Perf( B ( H (cid:111) Q ) I ) ⊗ Perf( BQ I ) Perf( L ) ∼ = Perf( BH I ), which follows easilyfrom highest weight theory.The latter data is equivalent to mapsHom( I, S ) = S I → Fun monex ,L (Rep L ( H I ) , End L ( C ))

34 X. THE SPECTRAL ACTION functorially in I , where Fun monex ,L denotes the exact L -linear monoidal functors. Both sides here arefunctors in I , and on the left-hand side we have a representable functor. By the Yoneda lemma, itfollows that this data is equivalent to L -linear exact monoidal functorsRep L ( H S ) → End L ( C ) . Such actions extend uniquely to Perf( BH S ), giving the desired result. (cid:3) Lemma

X.1.2 . The functor taking an anima S over BQ to Perf(Map BQ ( S, B ( H (cid:111) Q ))) , regardedas an idempotent-complete stable ∞ -category, commutes with sifted colimits. More precisely, as afunctor into L -linear symmetric monoidal idempotent-complete stable ∞ -categories, it commuteswith all colimits. Proof.

We ﬁrst check that it commutes with ﬁltered colimits. For this, let S i , i ∈ I , be aﬁltered diagram of anima over BQ , and choose compatible surjections S (cid:48) i → S i × BQ ∗ from sets S (cid:48) i .Let S = colim i S i and S (cid:48) = colim i S (cid:48) i , which is a set surjecting onto S × BQ ∗ . Letting G i = H S (cid:48) i and G = H S (cid:48) , we get presentations Map BQ ( S i , B ( H (cid:111) Q )) = X i /G i as quotients of aﬃne derived L -schemes X i by the pro-reductive group G i , and similarly Map BQ ( S, B ( H (cid:111) Q )) = X/G , with X = lim ←− i X i and G = lim ←− i G i . We claim that in this generalitylim −→ i Perf( X i /G i ) → Perf(

X/G )is an isomorphism of idempotent-complete stable ∞ -categories.Assume ﬁrst that all X = Spec L are a point. Then note that Perf( BG ) is generated by Rep( G ),which is easily seen to be the ﬁltered colimit lim −→ i Rep( G i ), and (by writing it as limit of reductivegroups) is seen to be semisimple. The claim is easily checked in this case.In general, Perf( X/G ) is generated by Rep( G ) as an idempotent complete stable ∞ -category.Indeed, given any perfect complex A ∈ Perf(

X/G ), we can look at the largest n for which thecohomology sheaf H n ( A ) is nonzero; after shift, n = 0. Pick V ∈ Rep( G ) with a map V → H ( A )such that V ⊗ L O X/G → H ( A ) is surjective. By semisimplicity of Rep( G ), we can lift V → H ( A )to V → A , and then pass to the cone of V ⊗ L O X/G → A to reduce the projective amplitude until A is a vector bundle. In that case the homotopy ﬁbre B of V ⊗ L O X/G → A is again a vectorbundle, and the map V ⊗ L O X/G → A splits, as the obstruction is H ( X/G, A ∨ ⊗ O X/G B ), whichvanishes by semisimplicity of Rep( G ).This already proves essential surjectivity. For fully faithfulness, it suﬃces by passage tointernal Hom’s to show that for all A i ∈ Perf( X i /G i ) (for some chosen i ) with pullbacks A i ∈ Perf( X i /G i ) for i → i and A ∈ Perf(

X/G ), the map lim −→ i R Γ( X i /G i , A i ) → R Γ( X/G, A ) isan isomorphism. By semisimplicity of Rep( G i ) and Rep( G ), it suﬃces to see that lim −→ i R Γ( X i , A i ) → R Γ( X, A ) is an isomorphism, which is clear by aﬃneness.This handles the case of ﬁltered colimits. For the more precise claim, it is also easy to see thatit commutes with disjoint unions. It is now enough to handle pushouts, so consider a diagram S ← S → S of anima over BQ , with pushout S . We can assume that the maps S → S and S → S are surjective, as otherwise we can use compatibility with disjoint unions (replacing S bythe disjoint union of the image of S and its complement). Then choose a surjection S (cid:48) → S × BQ ∗ , .1. RATIONAL COEFFICIENTS 335 which induces similar surjections in the other cases. Thus, we get aﬃne derived L -schemes X → X ← X with actions by G = H S (cid:48) , and X = X × X X , and we want to see that the functorPerf( X /G ) ⊗ Perf( X /G ) Perf( X /G ) → Perf( X × X X /G )is an equivalence. On the level of Ind-categories, Ind Perf( X i /G ) is the ∞ -category of O ( X i )-modules in Ind Perf( BG ): This is a consequence of Barr–Beck and the fact observed above thatPerf( BG ) generates Perf( X i /G ), so that the forgetful functor Ind Perf( X i /G ) → Ind Perf( BG )is conservative. It follows that the tensor product is the ∞ -category of O ( X ) ⊗ O ( X ) O ( X )-modules in Ind Perf( BG ), the tensor product taken in the symmetric monoidal stable ∞ -categoryInd Perf( BG ). The map O ( X ) ⊗ O ( X ) O ( X ) → O ( X ) is an isomorphism in Ind Perf( BG ): Thiscan be checked after the forgetful functor Ind Perf( BG ) → D ( L ) as it is conservative (using that G is pro-reductive, hence Rep( G ) is semisimple), and then it amounts to X = X × X X . (cid:3) In particular, we get the following corollary.

Corollary

X.1.3 . Let L be a ﬁeld over Q (cid:96) ( √ q ) . There is a natural compactly supported L -linear action of Perf( Z ( W E , (cid:98) G ) L / (cid:98) G ) on D lis (Bun G , L ) ω , uniquely characterized by the requirementthat by restricting along the Rep L ( Q I ) -linear maps Rep L (( (cid:98) G (cid:111) Q ) I ) → Perf( Z ( W E , (cid:98) G ) L / (cid:98) G ) BW IE it induces the Hecke action, which gives functorially in the ﬁnite set I exact Rep L ( Q I ) -linear func-tors Rep L (( (cid:98) G (cid:111) Q ) I ) → End L ( D lis (Bun G , L ) ω ) BW IE . Proof.

We can reduce to the subcategories D P lis (Bun G , L ) ω ⊂ D lis (Bun G , L ) for open sub-groups P of the wild inertia of W E , acting trivially on (cid:98) G . Then we can replace W E by W E /P throughout. In that case, restricting the given Hecke action to W ⊂ W E /P , Theorem X.1.1 givesan action of Perf( Z ( W, (cid:98) G ) L / (cid:98) G ), and Z ( W, (cid:98) G ) = Z ( W E /P, (cid:98) G ), so we get the desired action ofPerf( Z ( W E /P, (cid:98) G ) L / (cid:98) G ). (cid:3) With this action, we can formulate the main conjecture, “the categorical form of the geometricLanglands conjecture on the Fargues–Fontaine curve”. Recall that for a quasisplit reductive group G over E , Whittaker data consist of a choice of a Borel B ⊂ G with unipotent radical U ⊂ B ,together with a generic character ψ : U ( E ) → Q × (cid:96) . As usual, we also ﬁx √ q ∈ Q (cid:96) . Conjecture

X.1.4 . There is an equivalence of Q (cid:96) -linear small stable ∞ -categories D lis (Bun G , L ) ω ∼ = D b, qccoh ( Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G ) linear over Perf( Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G ) , under which the structure sheaf on the right corresponds tothe Whittaker sheaf W ψ , which is the sheaf concentrated on Bun G ⊂ Bun G corresponding to therepresentation c - Ind G ( E ) U ( E ) ψ of G ( E ) . To be precise, D b, qccoh ( Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G ) refers here to the ∞ -category of those bounded complexeswith coherent cohomology that also have quasicompact support, i.e. only live on ﬁnitely manyconnected components, and the ﬁnal compatibility statement is really on the level of Ind-objects

36 X. THE SPECTRAL ACTION (or after restriction to connected components). The linearity condition over Perf( Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G )means that this equivalence is compatible with the spectral action. Remark

X.1.5 . There is an orthogonal decomposition D lis (Bun G , Q (cid:96) ) ω = (cid:77) α ∈ π ( G ) Γ D lis (Bun c = αG , Q (cid:96) ) ω given by the connected components of Bun G . There is a morphism Z ( (cid:98) G ) Γ → Id Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G , as Z ( (cid:98) G ) Γ ⊂ (cid:98) G acts trivially on Z ( W E , (cid:98) G ). There is an associated “eigenspace” decomposition D b, qccoh ( Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G ) = (cid:77) χ ∈ X ∗ ( Z ( (cid:98) G ) Γ ) D b, qccoh ( Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G ) χ . Compatibility with the spectral action implies that via the identiﬁcation π ( G ) Γ = X ∗ ( Z ( (cid:98) G ) Γ )those two decompositions should match.Another way to phrase the preceding conjecture is to say that, noting ∗ the spectral action, the“non-abelian Fourier transform”Perf qc ( Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G ) −→ D lis (Bun G , Q (cid:96) ) M (cid:55)−→ M ∗ W ψ is fully faithful and extends to an equivalence of Q (cid:96) -linear small stable ∞ -categories D b, qccoh ( Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G ) ∼ −→ D lis (Bun G , Q (cid:96) ) ω . Example

X.1.6 . Fully faithfulness in the categorical conjecture, applied to the structure sheaf,implies that Z spec ( G, Q (cid:96) ) ∼ −→ End( c -Ind G ( E ) U ( E ) ψ ) . Example

X.1.7 (Kernel of functoriality) . Conjecture X.1.4 implies the existence of a kernel offunctoriality for the local Langlands correspondence in the following way. Let f : L H → L G be an L -morphism between the L -groups of two quasi-split reductive groups H and G over E . Thisdeﬁnes a morphism of stacks Z ( W E , (cid:98) H ) Q (cid:96) / (cid:98) H −→ Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G, and pushforward along this map induces a functorInd D b, qccoh ( Z ( W E , (cid:98) H ) Q (cid:96) / (cid:98) H ) → Ind D b, qccoh ( Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G ) . (There may be slightly diﬀerent ways of handling the singularities here. One way to argue is toobserve that pushforward is naturally a functor D ≥ ( Z ( W E , (cid:98) H ) Q (cid:96) / (cid:98) H ) → D ≥ ( Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G ) , and D ≥ = Ind D b, qc , ≥ , and then extend by shifts.) The categorical equivalence then leads to acanonical functor D lis (Bun H , Q (cid:96) ) → D lis (Bun G , Q (cid:96) ) . .2. ELLIPTIC PARAMETERS 337 By the self-duality of D lis coming from Bernstein–Zelevinsky duality, and Proposition VII.7.10, anysuch functor is given by a kernel A f ∈ D lis (Bun H × Bun G , Q (cid:96) ) . One could, in fact, identify the image of A f under the categorical equivalence for H × G ; upto minor twists, it should be given by the structure sheaf of the graph of Z ( W E , (cid:98) H ) Q (cid:96) / (cid:98) H −→ Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G . It would be very interesting if some examples of such kernels A f can be con-structed explicitly.Since D ( H ( E ) , Q (cid:96) ), resp. D ( G ( E ) , Q (cid:96) ), are direct factors of D lis (Bun H , Q (cid:96) ), resp. D lis (Bun G , Q (cid:96) ),this should give rise to the “classical” Langlands functoriality D ( H ( E ) , Q (cid:96) ) → D ( G ( E ) , Q (cid:96) ). Remark

X.1.8 . A kernel of functoriality should only exist in the quasi-split case. In fact, as iswell-known, the local Jacquet–Langlands correspondence can not be functorial since any discreteseries representation of GL n ( E ) is deﬁned over its ﬁeld of moduli but this is not the case for smoothirreducible representations of D × if D is a division algebra over E .Let us now explain how Fargues’s original conjecture ﬁts into this context. Let ϕ : W E → (cid:98) G ( Q (cid:96) )be a Langlands parameter. Consider the map i : Spec Q (cid:96) → Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G corresponding to ϕ ,and let E ϕ = i ∗ Q (cid:96) ∈ D qcoh ( Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G ) = Ind Perf qc ( Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G ) . Factoring the map i via [Spec Q (cid:96) /S ϕ ], one actually sees that E ϕ carries naturally an action of S ϕ .Moreover, if one acts via tensoring with a representation V of (cid:98) G (cid:111) Q , then by the projection formulathe sheaf E ϕ gets taken to itself, tensored with the W E -representation V ◦ ϕ . Using the spectralaction, we ﬁnd an S ϕ -equivariant “automorphic complex”Aut ϕ = E ϕ ∗ W ψ ∈ D lis (Bun G , Q (cid:96) ) . It already follows that Aut ϕ ∈ D lis (Bun G , Q (cid:96) ) is a Hecke eigensheaf, with eigenvalue ϕ , so thespectral action produces Hecke eigensheaves. Except, it is not clear whether Aut ϕ (cid:54) = 0. Under thefully faithfulness part of the categorical conjecture, one sees that it must be nonzero, and moreoverhave some of the properties stated in [ Far16 ], in particular regarding the relation to L -packets.The particular case of elliptic parameters is further spelled out in the next section. X.2. Elliptic parameters

Let us make explicit what the spectral action, and Conjecture X.1.4, entails in the case ofelliptic parameters. As coeﬃcients, we take L = Q (cid:96) for simplicity. Definition

X.2.1 . An L -parameter ϕ : W E → (cid:98) G ( Q (cid:96) ) is elliptic if it is semisimple and thecentralizer S ϕ ⊂ (cid:98) G Q (cid:96) has the property that S ϕ /Z ( (cid:98) G ) Γ Q (cid:96) is ﬁnite. By deformation theory, it follows that the unramiﬁed twists of ϕ deﬁne a connected component C ϕ (cid:44) → [ Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G ] .

38 X. THE SPECTRAL ACTION

Thus, the spectral action (in fact, the excursion operators are enough for this, see the discussionaround Theorem IX.5.2) implies that there is a corresponding direct summand D C ϕ lis (Bun G , Q (cid:96) ) ω ⊂ D lis (Bun G , Q (cid:96) ) ω . For any Schur-irreducible A ∈ D C ϕ lis (Bun G , Q (cid:96) ) ω , the excursion operators act via scalars on A ,as determined by an unramiﬁed twist of ϕ . In particular, they act in this way on i b ∗ A for any b ∈ B ( G ). By compatibility with parabolic induction, it follows that for any A ∈ D C ϕ lis (Bun G , Q (cid:96) ) ω ,the restriction i b ∗ A is equal to 0 if b is not basic (if it was not zero, one could ﬁnd an irreduciblesubquotient to which this argument applies). Thus, D C ϕ lis (Bun G , Q (cid:96) ) ω ∼ = (cid:77) b ∈ B ( G ) basic D C ϕ ( G b ( E ) , Q (cid:96) ) ω . Moreover, all A ∈ D C ϕ ( G b ( E ) , Q (cid:96) ) ω must lie in only supercuspidal components of the Bernsteincenter, again by compatibility with parabolic induction. If Z ( (cid:98) G ) Γ is ﬁnite (equivalently, if theconnected split center of G is trivial), then C ϕ = [ ∗ /S ϕ ] is a point and it follows that all A areﬁnite direct sums of shifts supercuspidal representations of G b ( E ), and so D C ϕ lis (Bun G , Q (cid:96) ) ω ∼ = (cid:77) b ∈ B ( G ) basic (cid:77) π Perf( Q (cid:96) ) ⊗ π, where π runs over supercuspidal Q (cid:96) -representations of G b ( E ) with L -parameter ϕ π = ϕ .In general, acting on D C ϕ lis (Bun G , Q (cid:96) ) ω , we have the direct summandPerf( C ϕ )of Perf([ Z ( W E , (cid:98) G ) Q (cid:96) / (cid:98) G ]) . If Z ( (cid:98) G ) Γ is ﬁnite, one has C ϕ = [ ∗ /S ϕ ], and hence we get an action of Rep( S ϕ ) on D C ϕ lis (Bun G , Q (cid:96) ) ω .In general, one can get a similar picture by ﬁxing central characters; let us for simplicity only spellout the case when Z ( (cid:98) G ) Γ is ﬁnite, i.e. the connected split center of G is trivial.If π b is a supercuspidal representation of some G b ( E ) with ϕ π b = ϕ , and W ∈ Rep( S ϕ ) thenacting via W on π b we get some objectAct W ( π b ) ∈ (cid:77) b (cid:48) ∈ B ( G ) basic (cid:77) π Perf( Q (cid:96) ) ⊗ π. Assume that W | Z ( (cid:98) G ) Γ is isotypic, given by some character χ : Z ( (cid:98) G ) Γ → Q × (cid:96) . As Z ( (cid:98) G ) Γ is thediagonalizable group with characters π ( G ) Γ , it follows that we get an element b χ ∈ π ( G ) Γ = B ( G ) basic . Then Act W ( π b ) is concentrated on b (cid:48) = b + b χ , and soAct W ( π b ) ∼ = (cid:77) π b (cid:48) V π b (cid:48) ⊗ π b (cid:48) for a certain multiplicity space V π b (cid:48) ∈ Perf( Q (cid:96) ), where π b (cid:48) runs over supercuspidal representationsof G b (cid:48) ( E ), b (cid:48) = b + b χ , with L -parameter ϕ π b (cid:48) = ϕ . .3. INTEGRAL COEFFICIENTS 339 The conjectural description of L -packets [ Kal14 ] then suggests the following conjecture, whichis (up to the added t -exactness) the specialization of Conjecture X.1.4 to the case of elliptic L -parameters. Conjecture

X.2.2 . Assume that G is quasisplit, with a ﬁxed Whittaker datum, and that theconnected split center of G is trivial. Then there is a unique generic supercuspidal representation π of G ( E ) with L -parameter ϕ π = ϕ , and the functor Perf([Spec Q (cid:96) /S ϕ ]) → D C ϕ lis (Bun G , Q (cid:96) ) ω : W (cid:55)→ Act W ( π ) is an equivalence. In particular, the set of irreducible supercuspidal representations of some G b ( E ) with L -parameter ϕ is in bijection with the set of irreducible representations of S ϕ .Moreover, the equivalence is t -exact for the standard t -structures on source and target. Thus, the conjecture gives an explicit parametrization of L -packets.Let us explain what the compatibility of the spectral action with Hecke operators entails in thiscase. Given V ∈ Rep( (cid:98) G (cid:111) Q ), the restriction of V to S ϕ admits a commuting W E -action given by ϕ . This deﬁnes a functor Rep( (cid:98) G (cid:111) Q ) → Rep( S ϕ ) BW E . Now the diagram of monoidal functorsRep( (cid:98) G (cid:111) Q ) (cid:47) (cid:47) (cid:15) (cid:15) End Q (cid:96) ( D C ϕ lis (Bun G , Q (cid:96) )) BW E Rep( S ϕ ) BW E (cid:53) (cid:53) commutes; this follows from the compatibility of the spectral action with the Hecke action.Concretely, given π as above and V ∈ Rep( (cid:98) G (cid:111) Q ), decompose the image of V in Rep( S ϕ ) BW E as a direct sum (cid:76) i ∈ I W i ⊗ σ i where W i ∈ Rep( S ϕ ) is irreducible and σ i is some continuous repre-sentation of W E on a ﬁnite-dimensional Q (cid:96) -space. Then T V ( π ) ∼ = (cid:77) i ∈ I Act W i ( π ) ⊗ σ i . Recall that T V ( π ) can be calculated concretely through the cohomology of local Shimura va-rieties, or in general moduli spaces of local shtukas. Noting that the functor Act W i is realizing aform of the Jacquet–Langlands correspondence relating diﬀerent inner forms, the formula above isessentially the conjecture of Kottwitz [ RV14 , Conjecture 7.3]. In fact, assuming Conjecture X.2.2,it is an easy exercise to deduce [

RV14 , Conjecture 7.3], assuming that the parametrization of theConjecture X.2.2 agrees with the parametrization implicit in [

RV14 , Conjecture 7.3].

X.3. Integral coeﬃcients

We want to construct the spectral action with integral coeﬃcients. Unfortunately, the naiveanalogue of Theorem X.1.1 is not true, the problem being that the analogue of Lemma X.1.2 fails.However, the rest of the argument still works, and gives the following result.

40 X. THE SPECTRAL ACTION

Consider a split reductive group H over a discrete valuation ring R with an action of a ﬁnitegroup Q . Let S be any anima over BQ . As before, we can deﬁne a derived stack Map BQ ( S, B ( H (cid:111) Q )) over R , whose values in an animated R -algebra A are the maps of anima S → B ( H (cid:111) Q )( A )over BQ . In general, the functor S (cid:55)→ Perf(Map BQ ( S, B ( H (cid:111) Q ))) does not commute with siftedcolimits in S .However, we can consider the best approximation to it that does commute with sifted colimits.Note that the ∞ -category of anima over BQ is the animation of the category of sets equippedwith a Q -torsor; it is freely generated under sifted colimits by the category of ﬁnite sets equippedwith a Q -torsor. Thus, the sifted-colimit approximation to S (cid:55)→ Perf(Map BQ ( S, B ( H (cid:111) Q ))) is theanimation of its restriction to ﬁnite sets with Q -torsors; we denote it by S (cid:55)→ Perf(Map Σ BQ ( S, B ( H (cid:111) Q ))) , with the idea in mind that it is like the ∞ -category of perfect complexes on some (nonexistent)derived stack Map Σ BQ ( S, B ( H (cid:111) Q )), gotten as a (co-)sifted limit approximation to Map Σ BQ ( S, B ( H (cid:111) Q )). The symbol Σ here is in reference to the notation used in [ Lur09 , Section 5.5.8] in relation tosifted colimits. Thus Perf(Map Σ BQ ( S, B ( H (cid:111) Q ))) is an R -linear idempotent-complete small stable ∞ -category, mapping to Perf(Map BQ ( S, B ( H (cid:111) Q ))). Proposition

X.3.1 . Let C be an R -linear idempotent-complete small stable ∞ -category. Giving,functorially in ﬁnite sets I , an exact Rep R ( Q I ) -linear monoidal functor Rep R (( H (cid:111) Q ) I ) → End R ( C ) S I is equivalent to giving an R -linear action of Perf(Map Σ BQ ( S, B ( H (cid:111) Q ))) on C . Here, given such anaction of Perf(Map Σ BQ ( S, B ( H (cid:111) Q ))) , one gets exact Rep R ( Q I ) -linear monoidal functors Rep R (( H (cid:111) Q ) I ) → End R ( C ) S I by composing the exact monoidal functor Perf(Map Σ BQ ( S, B ( H (cid:111) Q ))) → End R ( C ) with the naturalexact Rep R ( Q I ) -linear symmetric monoidal functor Rep R (( H (cid:111) Q ) I ) → Perf(Map Σ BQ ( S, B ( H (cid:111) Q ))) S I given by I -fold tensor product of the exact Rep R ( Q ) -linear symmetric monoidal functor Rep R ( H (cid:111) Q ) → Perf(Map Σ BQ ( S, B ( H (cid:111) Q ))) S assigning to each s ∈ S pullback along evaluation at s , Map BQ ( S, B ( H (cid:111) Q )) → B ( H (cid:111) Q ) ; moreprecisely, it is deﬁned in this way if S is a ﬁnite set, and in general by animation. Proof.

This follows from the proof of Theorem X.1.1. (cid:3)

To make use of Proposition X.3.1, we need to ﬁnd suﬃciently many situations in which thefunctor Perf(Map Σ BQ ( S, B ( H (cid:111) Q ))) → Perf(Map BQ ( S, B ( H (cid:111) Q )))is an equivalence, and speciﬁcally we need to prove this for Map BQ ( BW, B ( (cid:98) G (cid:111) Q )) = Z ( W, (cid:98) G ) / (cid:98) G .First, we have the following result. .3. INTEGRAL COEFFICIENTS 341 Proposition

X.3.2 . The functor S (cid:55)→ Perf(Map Σ BQ ( S, B ( H (cid:111) Q ))) from anima over BQ tosymmetric monoidal idempotent-complete stable R -linear ∞ -categories commutes with all colimits. Proof.

As the functor commutes with sifted colimits by deﬁnition, it suﬃces to show thatwhen restricted to ﬁnite sets S equipped with Q -torsors, it commutes with disjoint unions. But forsuch S , the map S → BQ can be factored over a point, and then Map BQ ( S, B ( H (cid:111) Q )) = BH S .Thus, one has to see that for two ﬁnite sets S , S , the functorPerf( BH S ) ⊗ Perf( R ) Perf( BH S ) → Perf( BH S (cid:116) S )is an equivalence. But this follows easily from highest weight theory, which for any split reductivegroup H ﬁlters Perf( BH ) in terms of copies of Perf( R ) enumerated by highest weights. (cid:3) Pro