Néron blowups and low-degree cohomological applications
aa r X i v : . [ m a t h . AG ] M a r N´ERON BLOWUPS AND LOW-DEGREE COHOMOLOGICAL APPLICATIONS
BY ARNAUD MAYEUX, TIMO RICHARZ AND MATTHIEU ROMAGNY
Abstract.
We define dilatations of general schemes and study their basic properties. Dilatationsof group schemes are –in favorable cases– again group schemes, called N´eron blowups. We givetwo applications to their cohomology in degree zero (integral points) and degree one (torsors):we prove a canonical Moy-Prasad isomorphism that identifies the graded pieces in the congruentfiltration of G with the graded pieces in its Lie algebra g , and we show that many level structureson moduli stacks of G -bundles are encoded in torsors under N´eron blowups of G . Contents
1. Introduction 12. Dilatations 33. N´eron blowups 124. Applications 16References 211.
Introduction
Motivation and goals.
N´eron blowups (or dilatations) provide a tool to modify group schemesover the fibers of a given Cartier divisor on the base. Classically, their integral points over discretevaluation rings appear as congruence subgroups of reductive groups over fields, see [Ana73, § § §§ § Results.
Let S be a scheme. Let S be an effective Cartier divisor on S , i.e., a closed subschemewhich is locally defined by a single non-zero divisor. We denote by Sch S -reg S the full subcategory ofschemes T → S such that T | S := T × S S is an effective Cartier divisor on T . This category containsall flat schemes over S . For a group scheme G → S together with a closed subgroup H ⊂ G | S over S , we define the contravariant functor G : Sch S -reg S → Groups given by all morphisms of S -schemes T → G such that the restriction T | S → G | S factors through H . Theorem. (1)
The functor G is representable by an open subscheme of the full scheme-theoreticblowup of G in H . The structure morphism G → S is an object in Sch S - reg S . (2.3, 2.4, 2.6)(2) The canonical map
G → G is affine. Its restriction over S \ S induces an isomorphism G| S \ S ∼ = G | S \ S . Its restriction over S factors as G| S → H ⊂ G | S . (2.4, 3.1)(3) If H → S has connected fibres and H ⊂ G | S is regularly immersed, then G| S → S has onnected fibres. (2.16, 3.2)(4) If G → S , H → S are flat and ( locally ) of finite presentation and H ⊂ G | S is regularlyimmersed, then G → S is flat and ( locally ) of finite presentation. If both G → S , H → S aresmooth, then G → S is smooth. (2.16, 3.2)(5) Assume that
G → S is flat. Then its formation commutes with base change S ′ → S in Sch S - reg S ,and it carries the structure of a group scheme such that the canonical map G → G is a morphismof S -group schemes. (2.7, 3.2)(6) Assume that G → S is flat, finitely presented and H → S is flat, regularly immersed in G | S .Locally over S , there is an exact sequence of S -group schemes → V → G| S → H → where V is the vector bundle given by restriction to the unit section of an explicit twist of the normal bundleof H in G | S . If H lifts to a flat S -subgroup scheme of G , this sequence is canonical; moreover itexists globally and is canonically split. (3.5)We call G → S the N´eron blowup (or dilatation ) of G in H along S . Note that G → S is a groupobject in Sch S -reg S by (1), but that G → S is a group scheme only if the self products G × S G and G × S G × S G are objects in Sch S -reg S which holds for example in (5), cf. § S is thespectrum of a discrete valuation ring and if S is defined by the vanishing of a uniformizer, then G → S is the group scheme constructed in [Ana73, § § §§ § Z in a scheme X along a divisor D . It is only later that we specialize to relativeschemes (over some base S , with the divisor coming from the base) and then further to groupschemes.The applications we give originate from a sheaf-theoretic viewpoint on N´eron blowups. Write j : S ֒ → S the closed immersion of the Cartier divisor, and assume that G → S and H → S areflat, locally finitely presented groups. In this context, the dilatation G → G sits in an exact sequenceof sheaves of pointed sets on the small syntomic site of S (Lemma 3.7):1 G G j ∗ ( G /H ) 1 , where G := G | S . If G → S and H → S are smooth, then the sequence is exact as a sequenceof sheaves on the small ´etale site of S . Considering the associated sequence on global sections, weobtain the following theorem which generalizes and unifies several results found in the literature(Remark 4.4) under the name of Moy-Prasad isomorphisms . Corollary 1.
Let r, s be integers such that r/ s r . Let ( O , π ) be a henselian pair where π ⊂ O is an invertible ideal. Let G be a smooth, separated O -group scheme. Let G r be the r -thiterated dilatation of the unit section and g r its Lie algebra. If O is local or G is affine, there is acanonical isomorphism G s ( O ) /G r ( O ) ∼ −→ g s ( O ) / g r ( O ) . (4.3)As another application, we are interested in comparing G -torsors with G -torsors. In light of theabove short exact sequence of sheaves, there is an equivalence between the category of G -torsors andthe category of G -torsors equipped with a section of their pushforward along G → j ∗ ( G /H ), see[Gi71, Chap. III, § § X is a smooth, projective, geometrically irreducible curve over afield k with a Cartier divisor N ⊂ X , that G → X is a smooth, affine group scheme and that H → N is a smooth closed subgroup scheme of G | N . In this case, the N´eron blowup G → X is a smooth, affine group scheme. Let Bun G (resp. Bun G ) denote the moduli stack of G -torsors(resp. G -torsors) on X . This is a quasi-separated, smooth algebraic stack locally of finite type over k (cf. e.g. [He10, Prop. 1] or [AH19, Thm. 2.5]). Pushforward of torsors along G → G induces amorphism Bun G → Bun G , E 7→ E × G G . We also consider the stack Bun ( G,H,N ) of G -torsors on X ith level-( H , N )-structures, cf. Definition 4.5. Its k -points parametrize pairs ( E , β ) consisting of a G -torsor E → X and a section β of the fppf quotient ( E| N /H ) → N , i.e., β is a reduction of E| N toan H -torsor. Corollary 2.
There is an equivalence of k -stacks Bun
G ∼ = −→ Bun ( G,H,N ) , E 7→ ( E × G G, β can ) , where β can denotes the canonical reduction induced from the factorization G| N → H ⊂ G | N givenby (2) in the Theorem. (4.7, 4.8)If H = { } is trivial, then Bun ( G,H,N ) is the moduli stack of G -torsors equipped with level- N -structures. If G → X is reductive, if N is reduced and if H is a parabolic subgroup in G | N , thenBun ( G,H,N ) is the moduli stack of G -torsors with quasi-parabolic structures as in [LS97]. In thiscase the restriction of G to the completed local rings of X are parahoric group schemes in the senseof [BT84] and the previous corollary was pointed out in [PR10, § k is a finite field. As a consequence of the corollary one naturally obtains integralmodels for moduli stacks of G -shtukas on X with level structures over N as in [Dri87] for G = GL n and in [Var04] for general split reductive G . We thank Alexis Bouthier for informing us that thiswas already pointed out in [NN08, § § G -shtukas with level structures as in [Dri87, Var04, Laf18]and G -shtukas as in [AH19, AHab19, Br]. We expect the point of view of G -shtukas, as opposed to G -shtukas with level structures, to be fruitful for investigations also outside the case of parahoriclevel structures. Acknowledgement.
We thank Anne-Marie Aubert, Patrick Bieker, Paul Breutmann, MichelBrion, Colin Bushnell, K¸estutis ˇCesnaviˇcius, Laurent Charles, Cyril Demarche, Antoine Ducros,Philippe Gille, Thomas Haines, Urs Hartl, Jochen Heinloth, Eugen Hellmann, Laurent Moret-Bailly,Gopal Prasad, Benoˆıt Stroh, Torsten Wedhorn and Jun Yu for useful discussions around the subjectof this note. Also we thank Alexis Bouthier for pointing us to the reference [NN08].2.
Dilatations
In this section, we define dilatations and give some properties. Dilatations (or affine blowups)are spectra of affine blowup algebras. We first introduce affine blowup algebras.2.1.
Definition.
Fix a scheme X . Let Z ⊂ D be closed subschemes in X , and assume that D is locally principal. Denote by J ⊂ I the associated quasi-coherent sheaves of ideals in O X sothat Z = V ( I ) ⊂ V ( J ) = D . Let Bl I O X = O X ⊕ I ⊕ I ⊕ . . . denotes the Rees algebra,it is a quasi-coherent Z > -graded O X -algebra. If J = ( b ) is principal with b ∈ Γ( X, J ), then (cid:0) Bl I O X (cid:1) [ J − ] := (cid:0) Bl I O X (cid:1) [ b − ] is well-defined independently of the choice of generators of J . If J is only locally principal, then we define the localization (cid:0) Bl I O X (cid:1) [ J − ] by glueing. This O X -algebra inherits a grading by giving local generators of J degree 1. In other words, the graduationof (cid:0) Bl I O X (cid:1) [ J − ] is given locally by deg( i/b k ) = n − k for i ∈ I n and b a local generator of J . Definition 2.1.
We use the following terminology:(1) The affine blowup algebra of O X in I along J is the quasi-coherent sheaf of O X -algebras O X [ IJ ] def = h(cid:0) Bl I O X (cid:1) [ J − ] i deg=0 , obtained as the subsheaf of degree 0 elements in (cid:0) Bl I O X (cid:1) [ J − ].
2) The dilatation (or affine blowup ) of X in Z along D is the X -affine schemeBl DZ X def = Spec (cid:0) O X [ IJ ] (cid:1) . The subscheme Z , or the pair ( Z, D ), is called the center of the dilatation.
Remark 2.2. If X is affine, affine blowup algebras are defined in [StaPro, 052P]. In this casewe denote B := Γ( X, O X ), I := Γ( X, I ), J := Γ( X, J ) and Bl I B := Γ( X, Bl I O X ) = ⊕ n > I n .Moreover, if J = ( b ) is principal, then B [ Ib ] := Γ( X, O X [ IJ ]) is the algebra whose elements areequivalence classes of fractions x/b n with x ∈ I n , where two representatives x/b n , y/b m with x ∈ I n , y ∈ I m define the same element in B [ Ib ] if and only if there exists an integer l > b l ( b m x − b n y ) = 0 inside B .By [StaPro, 07Z3], the image of b in B [ Ib ] is a non-zero divisor,(2.2) bB [ Ib ] = IB [ Ib ] , and(2.3) B [ Ib ][ b − ] = B [ b − ] . (2.4)In particular, the ring B [ Ib ] is the B -subalgebra of B [ b − ] generated by fractions x/b with x ∈ I .2.2. Basic properties.
We proceed with the notation from § § Lemma 2.3.
The affine blowup Bl DZ X is the open subscheme of the blowup Bl Z X = Proj(Bl I O X ) defined by the complement of V + ( J ) .Proof. Our claim is Zariski local on X . We reduce to the case where X = Spec( B ) is affine and J = ( b ) is principal. Then B [ Ib ] is the homogenous localization of B ⊕ I ⊕ I ⊕ . . . at b ∈ I viewed asan element in degree 1, cf. [StaPro, 052Q]. This shows that Spec( B [ Ib ]) is the complement of V + ( b )in Proj(Bl I B ). (cid:3) Lemma 2.4.
As closed subschemes of Bl DZ X , one has Bl DZ X × X Z = Bl DZ X × X D, which is an effective Cartier divisor on Bl DZ X .Proof. Our claim is Zariski local on X . We reduce to the case where X = Spec( B ) is affine and J = ( b ) is principal. We have to show that bB [ Ib ] = IB [ Ib ], and that b is a non-zero divisor in B [ Ib ].This is (2.2) and (2.3) above. (cid:3) When D is a Cartier divisor, we can also realize Bl DZ X as a closed suscheme of the affine projectingcone. Recall that classically, this cone is defined as the relative spectrum of the Rees algebra Bl I O X ,so the blowup and its affine cone are complementary to each other in the completed projective cone;see [EGA2, § J givesrise to different embeddings. Indeed, according to [EGA2, § Z X = Proj( ⊕ n > I n ⊗ J − n ). Here we define the affineprojecting cone of Bl Z X (with respect to the chosen presentation) asC Z X def = Spec (cid:0) ⊕ n > I n ⊗ J − n (cid:1) . Lemma 2.5. If D is a Cartier divisor, the affine blowup Bl DZ X is the closed subscheme of the affineprojecting cone C Z X defined by the equation ̺ − , where ̺ ∈ I ⊗ J − is the image of under theinclusion O X = J ⊗ J − ⊂ I ⊗ J − . roof. Let A = ⊕ n > I n ⊗ J − n . There is a surjective morphism of sheaves of algebras A → O X [ IJ ]defined by mapping a local section i ⊗ j − in degree 1 to i/j . To check that ̺ − U ⊂ X where the sheaf J is generated by a section b . Let t = b ∨ be the generator for J − , dual to b . Let B = Γ( X, O X )and I = Γ( X, I ). Then the map A ( U ) → O X [ IJ ]( U ) is given by( ⊕ n > I n t n ) −→ B [ Ib ] , P n > i n t n P n > i n /b n . This induces an isomorphism ( ⊕ n > I n t n ) / ( bt − ∼ −→ B [ Ib ]. (cid:3) Universal property.
In this text we will use regular immersions in a possibly non-noetheriansetting where the reference [EGA4.4, § §
19] is inadequate. In this case we refer to theStacks Project [StaPro]. There, four notions of regularity are studied: by decreasing order ofstrength, regular , Koszul-regular , H -regular , quasi-regular (see Sections 067M and 0638 in loc. cit. ).The useful ones for us are the first (which is regularity in its classical meaning) and the third: an H -regular sequence is a sequence whose Koszul complex has no homology in degree 1. In [StaPro]several results are stated under the weakest H -regular assumption. For simplicity we will state ourresults for regular immersions, although all of them hold also for H -regular immersions.Note that regularity and H -regularity coincide for sequences composed of one element x , becausefor them the Koszul complex has length one and the homology group in degree 1 is just the x -torsion. In particular, for locally principal subschemes the three notions regular, Koszul-regular and H -regular are equivalent.Let us denote by Sch D -reg X the full subcategory of schemes T → X such that T × X D ⊂ T isregularly immersed, or equivalently is an effective Cartier divisor (possibly the empty set) on T . If T ′ → T is flat and T → X is an object in this category, so is the composition T ′ → T → X . Inparticular, the category Sch D -reg X can be equipped with the fpqc/fppf/´etale/Zariski Grothendiecktopology so that the notion of sheaves is well-defined.As Bl DZ X → X defines an object in Sch D -reg X by Lemma 2.4, the contravariant functor(2.5) Sch D -reg X → Sets , ( T → X ) Hom X -Sch (cid:0) T, Bl DZ X (cid:1) together with id Bl DZ X determines Bl DZ X → X uniquely up to unique isomorphism. The next propo-sition gives the universal property of dilatations. Proposition 2.6.
The affine blowup Bl DZ X → X represents the contravariant functor Sch D - reg X → Sets given by (2.6) ( f : T → X ) ( {∗} , if f | T × X D factors through Z ⊂ X ; ∅ , else.Proof. Let F be the functor defined by (2.6). If T → Bl DZ X is a map of X -schemes, then thestructure map T → X restricted to T × X D factors through Z ⊂ X by Lemma 2.4. This defines amap(2.7) Hom X -Sch (cid:0) - , Bl DZ X (cid:1) −→ F of contravariant functors Sch D -reg X → Sets. We want to show that (2.7) is bijective when evaluatedat an object T → X in Sch D -reg X . As (2.7) is a morphism of Zariski sheaves, we reduce to the casewhere both X = Spec( B ), T = Spec( R ) are affine and J = ( b ) is principal.For injectivity, let g, g ′ : B [ Ib ] → R be two B -algebra maps. We need to show g = g ′ . Indeed, since B [ b − ] = B [ Ib ][ b − ] by (2.4), we get g [ b − ] = g ′ [ b − ]. As b is a non-zero divisor in R by assumption,this implies g = g ′ .For surjectivity, consider an element in F (Spec( R )) which corresponds to a ring morphism g : B → R such that I is contained in the kernel of B → R → R/bR . We need to show that g extends(necessarily unique) to an B -algebra morphism ˜ g : B [ Ib ] → R . Let [ x/b n ], x ∈ I n be a class in B [ Ib ]. Since g ( I n ) ⊂ ( b n ) in R , the b -torsion freeness of R implies that there is a unique element = r ( x, n ) ∈ R such that g ( x ) = b n · r . We define ˜ g ([ x/b n ]) := r ( x, n ). This is well-defined: If y/b m , y ∈ I m is another representative of [ x/b n ], then applying g to equation (2.1) yields b m g ( x ) = b n g ( y )in R . It follows that r ( x, n ) = r ( y, m ). Thus, ˜ g is well-defined. Similarly, one checks that ˜ g definesa morphism of B -algebras. (cid:3) Functoriality.
Let Z ′ ⊂ D ′ ⊂ X ′ be another triple as in § X ′ → X such thatits restriction to D ′ (resp. Z ′ ) factors through D (resp. Z ) induces a unique morphism Bl D ′ Z ′ X ′ → Bl DZ X such that the following diagram of schemesBl D ′ Z ′ X ′ Bl DZ XX ′ X. commutes. Indeed, the existence of Bl D ′ Z ′ X ′ → Bl DZ X follows directly from Definition 2.1. Theuniqueness can be tested Zariski locally on X and X ′ where it follows from (2.2) and (2.4).2.5. Base change.
Now let X ′ → X be a map of schemes, and denote by Z ′ ⊂ D ′ ⊂ X ′ thepreimage of Z ⊂ D ⊂ X . Then D ′ ⊂ X ′ is locally principal so that the affine blow Bl D ′ Z ′ X ′ → X ′ iswell-defined. By § X ′ -schemes(2.8) Bl D ′ Z ′ X ′ −→ Bl DZ X × X X ′ . Lemma 2.7. If Bl DZ X × X X ′ → X ′ is an object of Sch D ′ - reg X ′ , then (2.8) is an isomorphism.Proof. Our claim is Zariski local on X and X ′ . We reduce to the case where both X = Spec( B ), X ′ = Spec( B ′ ) are affine, and J = ( b ) is principal. We denote Z ′ = Spec( B ′ /I ′ ) and D ′ =Spec( B ′ /J ′ ). Then J ′ = ( b ′ ) is principal as well where b ′ is the image of b under B → B ′ . We needto show that the map of B ′ -algebras B ′ ⊗ B B (cid:2) Ib (cid:3) −→ B ′ (cid:2) I ′ b ′ (cid:3) is an isomorphism. However, this map is surjective with kernel the b ′ -torsion elements [StaPro,0BIP]. As b ′ is a non-zero divisor in B ′ ⊗ B B (cid:2) Ib (cid:3) by assumption, the lemma follows. (cid:3) Corollary 2.8.
If the morphism X ′ → X is flat and has some property P which is stable underbase change, then Bl D ′ Z ′ X ′ → Bl DZ X is flat and has P .Proof. Since flatness is stable under base change the projection p : Bl DZ X × X X ′ → Bl DZ X is flatand has property P . By Lemma 2.7, it is enough to check that the closed subscheme Bl DZ X × X D ′ defines an effective Cartier divisor on Bl DZ X × X X ′ . But this closed subscheme is the preimage ofthe effective Cartier divisor Bl DZ X × X D under the flat map p , and hence is an effective Cartierdivisor as well. (cid:3) Exceptional divisor.
For closed subschemes Z ⊂ D in X with D locally principal, we sawin Lemma 2.4 that the preimage of the center Bl DZ X × X Z = Bl DZ X × X D is an effective Cartierdivisor in Bl DZ X . It is called the exceptional divisor of the affine blowup.In order to describe it, as before we denote by I and J the sheaves of ideals of Z and D in O X .Also we let C Z/D = I / ( I + J ) and N Z/D = C ∨ Z/D be the conormal and normal sheaves of Z in D . Proposition 2.9.
Assume that D ⊂ X is an effective Cartier divisor, and Z ⊂ D is a regularimmersion. Write J Z := J | Z . (1) The exceptional divisor Bl DZ X × X Z → Z is an affine space fibration, Zariski locally over Z isomorphic to V ( C Z/D ⊗ J − Z ) → Z . (2) If H ( Z, N Z/D ⊗ J Z ) = 0 (for example if Z is affine), then Bl DZ X × X Z → Z is globally isomorphic to V ( C Z/D ⊗ J − Z ) → Z . If Z is a transversal intersection in the sense that there is a cartesian square of closedsubschemes whose vertical maps are regular immersions W XZ D (cid:3) then Bl DZ X × X Z → Z is globally and canonically isomorphic to V ( C Z/D ⊗ J − Z ) → Z .Proof. Using Lemma 2.5 we can view the affine blowup Bl DZ X as the closed subscheme with equation ̺ − Z X = Spec( A ) with A = ⊕ n > I n ⊗J − n . First we computethe preimage C Z X × X Z . We have: A ⊗ O X / I = ⊕ k > ( I k / I k +1 ) ⊗ J − k . Since the immersions Z ⊂ D ⊂ X are regular, the conormal sheaf C Z/X = I / I is finite locally free,and the canonical surjective morphism of O X -algebrasSym • ( C Z/X ) −→ Gr •I ( O X )is an isomorphism, that is, Sym k ( C Z/X ) → I k / I k +1 is an isomorphism for each k >
0. Taking intoaccount that J is an invertible sheaf, we obtainSym k ( C Z/X ⊗ J − Z ) ∼ −→ ( I k / I k +1 ) ⊗ J − k . It follows that Spec( A ⊗ O X / I ) = V ( C Z/X ⊗ J − Z ) and that Bl DZ X × X Z is the closed subschemecut out by ̺ − −→ C D/X | Z −→ C Z/X −→ C
Z/D −→ . By [EGA4.4, Prop. 16.9.13] or [StaPro, 063N] in the non-noetherian case, this sequence is exact andlocally split (beware that C Z/X is denoted N Z/X in [EGA4.4]). Using the fact that C D/X | Z ⊗ J − Z ≃O Z is freely generated by ̺ as a subsheaf of C Z/X ⊗ J − Z , we obtain an extension(2.9) 0 −→ ̺ O Z −→ C Z/X ⊗ J − Z −→ C Z/D ⊗ J − Z −→ . Now we consider the three cases listed in the proposition.(1) Since C Z/D ⊗ J − Z is locally free, locally over Z we can choose a splitting of the exact sequence(2.9) of conormal sheaves: C Z/X ⊗ J − Z = ̺ O Z ⊕ C Z/D ⊗ J − Z . Mapping ̺ O Z -modules C Z/X ⊗ J − Z −→ O Z ⊕ ( C Z/D ⊗ J − Z ) ⊂ Sym( C Z/D ⊗ J − Z )which extends to a surjection of algebras with kernel ( ̺ − C Z/X ⊗ J − Z ) −→ Sym( C Z/D ⊗ J − Z ) . This identifies Bl DZ X × X Z with the affine space bundle V ( C Z/D ⊗ J − Z ), locally over Z .(2) The exact sequence defines a class in Ext O X ( C Z/D ⊗ J − Z , O Z ). Because the conormal sheaf islocally free, we have:Ext O Z ( C Z/D ⊗ J − Z , O Z ) ≃ Ext O Z ( O Z , C ∨ Z/D ⊗ J Z ) ≃ H ( Z, N Z/D ⊗ J Z ) . By assumption this vanishes and we obtain a global splitting. From this one concludes as before.
3) If Z is the transversal intersection of W and D , then we have two exact, locally split sequences:0 0 C D/X | Z C Z/W C Z/X C W/X | Z C Z/D . We claim that the dashed arrow is an isomorphism. To see this, write I , J , K the defining ideals of Z, D, W . The composition C W/X | Z → C Z/D is the following map: K / ( K + IK ) −→ I / ( I + J ) . From the fact that I = J + K we deduce: • K + IK = K + J K , hence K / ( K + IK ) = K / ( K + J K ). • I = J + J K + K , hence I + J = K + J and I / ( I + J ) = ( J + K ) / ( K + J ) = K / ( K + J ∩ K ).Hence, the map above is an isomorphism if and only if
J K = J ∩ K , which holds because W cuts D transversally (this is another way of saying that a local equation for J remains a non-zero divisorin O W ). This provides a canonical splitting C Z/X = C D/X | Z ⊕ C Z/D . One concludes as before. (cid:3)
Remark 2.10.
In the course of the proof, we saw that the exceptional divisor has the followingexplicit description: as an affine space fibration over Z , its local sections over an open U ⊂ Z arethe O U -linear maps ϕ : C U/X ⊗ J − U → O U such that ϕ ( ̺ ) = 1.2.7. Iterated dilatations.
Here we study the behaviour of dilatations under iteration. Namely,we will prove that when the center Z of the affine blowup is a transversal intersection W ∩ D , it canbe dilated any finite number of times and the result of r dilatations can be seen as the dilatation ofthe single “thickened” center rZ (to be defined below) inside the multiple Cartier divisor rD . Tomake this precise, we first study the lifting of subschemes along an affine blowup. Lemma 2.11.
Let Z ⊂ D ⊂ X be closed subschemes with D ⊂ X a Cartier divisor. Let ι : W ֒ → X be an immersion such that ι − ( Z ) = ι − ( D ) is a Cartier divisor in W . Set X ′ := Bl DZ X and let ι ′ : W → X ′ be the lift of ι given by the universal property of the dilatation. (1) If ι is an open immersion, then ι ′ is an open immersion. (2) If ι is a closed immersion, then ι ′ is a closed immersion. (3) Write ι as the composition W U X where U ⊂ X is the largest open subschemesuch that W is a closed subscheme of U . Let J ( resp. K ) be the ideal sheaf of D ( resp. W ) in U . Then ι ′ is the composition W U ′ X ′ where U ′ = X ′ × X U is the preimageof U and W ֒ → U ′ is a closed immersion with sheaf of ideals KO U ′ ⊗ ( J O U ′ ) − .Proof. (1) In this case ι is flat, and the formation of the dilatation commutes with base change.That is, the canonical morphism of W -schemesBl D ∩ WZ ∩ W W −→ X ′ × X W is an isomorphism. But by the assumption Bl D ∩ WZ ∩ W W → W is an isomorphism, and this identifies W → X ′ with the preimage of W ֒ → X in X ′ .(2) Let K ⊂ O X be the ideal sheaf of W . We will prove that ι ′ : W → X ′ is a closed immersionwith ideal sheaf KO X ′ ⊗ ( J O X ′ ) − . First of all ι ′ is automatically a monomorphism of schemes,and a proper map because ι is proper and X ′ → X is separated. Therefore ι ′ is a closed immersionby [EGA4.4, Cor. 18.12.6]. The computation of the ideal sheaf is a local matter so we can supposethat X = Spec( A ) is affine and the ideal sheaf J is generated by a section b . We write I, J, K ⊂ A the ideals defining Z, D, W and t := b ∨ the generator of J − , dual to b . The assumptions of the emma mean that I + K = J + K and b is a non-zero divisor in A/K . From Lemma 2.5, we knowthat X ′ is the spectrum of the ring A ′ = ( ⊕ e > I e t e ) / ( bt − . In the present local situation, the map ι ′ : W → X ′ is given by a lifting of ι ♯ : A → A/K to amap ( ι ′ ) ♯ : A ′ → A/K . Since A ′ is generated by It as an A -algebra, this map is determined by theformula ( ι ′ ) ♯ ( it ) = j ♯ ( a ), for all i ∈ I written i = ab + k ∈ I ⊂ bA + K . In particular we see that( ι ′ ) ♯ ( Kt ) = 0. Now working modulo ( bt −
1) + Kt in the ring C = ⊕ e > I e t e , we have: It ⊂ btA + Kt ≡ A + Kt ≡ A which sits in the degree 0 part of C , whence a surjection A ֒ → C → A ′ /KtA ′ . Moreover bKt ≡ K implies that K in degree 0 belongs to the ideal generated by Kt , hence finally A ′ /KtA ′ ∼ −→ A/K as desired.(3) This is the conjunction of (1) and (2). (cid:3)
In view of the preceding lemma, if we fix a closed subscheme i : W ֒ → X such that i − ( Z ) = i − ( D ) is a Cartier divisor in W , we will be able to lift W to the dilatation and hence iterate theprocess. So we place ourselves in the following situation. Assumption 2.12.
The schemes Z ⊂ D ⊂ X sit in a cartesian diagram of closed subschemes ( D ) : W XZ D i (cid:3) such that the vertical maps are Cartier divisor inclusions. In this situation we can construct a sequence of dilatations . . . −→ X r −→ X r − −→ . . . . . . −→ X −→ X = X and closed immersions i r : W ֒ → X r , as follows. We let D = D , X = X , D = D , and i = i : W ֒ → X . Let u : X → X be the dilatation of Z in ( X , D ), and D the preimage of D in X . Since D is a cartesian diagram of closed subschemes, we have i − ( Z ) = i − ( D ) = X which isa Cartier divisor in W . So by the universal property and Lemma 2.11, there is a closed immersion i : W → X lifting i . Moreover,( i ) − ( D ) = ( i ) − ( u − ( D )) = i − ( D ) = Z. That is, we again have a cartesian diagram( D ) : W X Z D i (cid:3) where the vertical maps are Cartier divisor inclusions. Our sequence is obtained by iterating thisconstruction. Lemma 2.13.
Under Assumption 2.12, denote by I , J , K the ideal sheaves of Z, D, W in O X .Let rD be the r -th multiple of D as a Cartier divisor, and rZ := W ∩ rD . Then the dilatation v r : X ′ r → X of ( rZ, rD ) in X is characterized as being universal among all morphisms V → X withthe following two properties: (i) J O V is an invertible sheaf, (ii) KO V is divisible by J r O V , that is we have KO V = J r O V · K r for some sheaf of ideals K r ⊂ O V . roof. The defining properties of the dilatation v r say that it is universal among morphisms V → X such that rZ × X V = rD × X V is a Cartier divisor. Since the ideal sheaves of rD and rZ are J r and J r + K respectively, these properties mean that the ideal J r O V is invertible and J r O V =( J r + K ) O V . But the properties “ J is invertible” and “ J r is invertible” are equivalent, as followsfrom the isomorphism between the blow-up of J and the blow-up of J r , see [EGA2, Def. 8.1.3].This takes care of (i). Besides, J r O V = ( J r + K ) O V means that KO V ⊂ J r O V and in the situationwhere J O V is invertible, this is the same as saying that KO V = J r O V · K r with K r = ( KO V : J r O V ) as an ideal of O V . (Note that in this case ( KO V : J r O V ) ≃ ( K⊗J − r ) O V as an O V -module.) This takes care of (ii). (cid:3) Proposition 2.14.
In the situation of Assumption 2.12, let . . . −→ X r −→ X r − −→ . . . . . . −→ X −→ X = X be the sequence of dilatations constructed above. Let rD be the r -th multiple of D as a Cartierdivisor, and rZ := W ∩ rD . Then the composition X r → X is the dilatation of ( rZ, rD ) inside X .Proof. According to Lemma 2.13, dilating ( rZ, rD ) means making J invertible and K divisible by J r , all of this in a universal way. This can be done by steps: • make J invertible and make K = K divisible by J ; • keep J invertible and make K := ( K : J ) ≃ K ⊗ J − divisible by J ; • keep J invertible and make K := ( K : J ) ≃ K ⊗ J − ≃ K ⊗ J − divisible by J ; etc,and finally • keep J invertible and make K r − := ( K r − : J ) ≃ K ⊗ J − ( r − divisible by J .In view of Lemma 2.11, these steps amount to: • dilate Z in ( X, D ), • dilate Z in ( X , D ), • dilate Z in ( X , D ), etc. until • dilate Z in ( X r − , D r − ).In this way we see the equivalence between the dilatation of the thick pair ( rZ, rD ) and the sequenceof dilatations of Z constructed after 2.12. (cid:3) Flatness and smoothness.
Flatness and smoothness properties of blowups are discussed in[EGA4.4, § § S under X together with a locally principal closedsubscheme S ⊂ S fitting into a commutative diagram of schemes(2.10) Z D XS S, where the square is cartesian, that is D → X := X × S S is an isomorphism. Lemma 2.15.
Assume that S is an effective Cartier divisor in S . Let f : Y → S be a morphismof schemes such that Y := Y × S S is a Cartier divisor in Y . Assume that both restrictions of f above S \ S and S are flat. If one of the following holds: (i) S, Y are locally noetherian, (ii) Y → S is locally of finite presentation,then f is flat.Proof. Since by assumption u is flat at all points above the open subscheme S \ S , it is enough toprove that u is flat at all points y ∈ Y lying above a point s ∈ S . n case (i), the local criterion for flatness [EGA3.1, Chap. O, 10.2.2] (cf. also [StaPro, 00ML])shows that O S,s → O
Y,y is flat and we are done.In case (ii), we may localize around y and s and hence assume that Y and S are affine and smallenough so that the ideal sheaf of S in S is generated by an element f ∈ A = Γ( S, O S ). We write A = colim A i as the union of its subrings of finite type over Z . In each A i , the element f is a non-zerodivisor. Write S i := Spec( A i ) and S i, := Spec( A i /f ). Using the results of [EGA4.3, Chap. IV, § i and a morphism of finite presentation Y i → S i suchthat Y i, := Y i × S i S i, is a Cartier divisor in Y i and Y i, → S i, is flat, and such that the situation( S, S , Y ) is a pullback of ( S i , S i, , Y i ) by S → S i . More in detail, using the following results wefind indices which we increase at each step in order to have all the conditions simultaneously met:use [EGA4.3, Thm. 8.8.2] to find morphisms Y i → S i and Y i, → S i, , use [EGA4.3, Cor. 8.8.2.5]to make Y i × S i S i, and Y i, isomorphic over S i, , use [EGA4.3, Thm. 11.2.6] to ensure Y i, → S i, flat, and use [EGA4.3, Prop. 8.5.6] to ensure that f is a non-zero divisor in O Y i , i.e. Y i, ⊂ Y i is aCartier divisor. Since A i is noetherian, for ( S i , S i, , Y i ) we can apply case (i) and the result followsby base change. (cid:3) For our conventions on regular immersions, the reader is referred back to § Proposition 2.16.
Assume that S is an effective Cartier divisor on S . (1) If Z ⊂ D is regular, then Bl DZ X → X is of finite presentation. (2) If Z ⊂ D is regular, the fibers of Bl DZ X × S S → S are connected ( resp. irreducible,geometrically connected, geometrically irreducible ) if and only if the fibers of Z → S are. (3) If X → S is flat and if moreover one of the following holds: (i) Z ⊂ D is regular, Z → S is flat and S, X are locally noetherian, (ii) Z ⊂ D is regular, Z → S is flat and X → S is locally of finite presentation, (iii) the local rings of S are valuation rings,then Bl DZ X → S is flat. (4) If both X → S , Z → S are smooth, then Bl DZ X → S is smooth.Proof. For (1) recall that the blowup of a regularly immersed subscheme has an explicit structure,where generating relations between local generators of the blown up ideal are the obvious ones, infinite number; see [StaPro, 0BIQ]. This shows that Bl DZ X → X is locally of finite presentation.Being also affine, it is of finite presentation.For (2) about connectedness and irreducibility, recall from Proposition 2.9 (1) that the exceptionaldivisor is an affine space fibration over Z . In particular it is a submersion, so that the elementarytopological lemma [EGA4.2, Lem. 4.4.2] (cf. also [StaPro, 0377]) gives the assertion.For (3)(i)-(ii), we apply Lemma 2.15 to Y := Bl DZ X . The preimage of S under the affine blowup f : Y → S is equal to Bl DZ X × X D = Bl DZ X × X Z by Lemma 2.4. This implies that the restriction f | f − ( S \ S ) is equal to X \ D → S \ S which is flat by assumption. It remains to show flatness inpoints of Bl DZ X lying over S . For this note that the restriction f | f − ( S ) factors as(2.11) Bl DZ X × X D = Bl DZ X × X Z −→ Z −→ S , where the first map is smooth by Proposition 2.9 and the second map is flat by assumption. ThenLemma 2.15 applies and gives flatness of Y → S .For (3)(iii) we can work locally at a point of S and hence assume that S is the spectrum ofa valuation ring R . We use the fact that flat R -modules are the same as torsionfree R -modules.Locally over an open subscheme Spec( B ) ⊂ X , the Rees algebra Bl I B = B [ It ] is a subalgebra ofthe polynomial algebra B [ t ] and the affine blowup algebra is a localization of the latter. It followsthat if B is R -torsionfree then the affine blowup algebra also, hence it is flat.For (4) assume that X → S and Z → S are smooth. Then (4) follows from [EGA4.4, Thm. 17.5.1](cf. also [StaPro, 01V8]) once we know that Bl DZ X → S is locally of finite presentation, flat and hassmooth fibers. Applying [EGA4.4, Prop. 19.2.4] to the commutative triangle in (2.10) we see that Z ⊂ D is regularly immersed. Therefore, Bl DZ X → S is flat and locally of finite presentation by arts (1) and (3). The smoothness of the fiber over points in S \ S is clear, and follows from (2.11)over points in S . This proves (4). (cid:3) N´eron blowups
We extend the theory of N´eron blowups of affine group schemes over discrete valuation rings as in[Ana73, 2.1.2], [WW80, p. 551], [BLR90, § § §§ Definition.
Let S be a scheme, and let G → S be a group scheme. Let S ⊂ S be a locallyprincipal closed subscheme, and consider the base change G := G × S S . Let H ⊂ G be a closedsubgroup scheme over S . Let G := Bl G H G → G be the dilatation of G in H along the locallyprincipal, closed subscheme G ⊂ G in the sense of Definition 2.1. In this case, we also call G → S the N´eron blowup of G in H ( along S ). We denote by G := G × S S → S its exceptional divisor.Let Sch S -reg S be the full subcategory of schemes T → S such that T := T × S S defines aneffective Cartier divisor on T . By Lemma 2.4 the structure morphism G → S defines an object inSch S -reg S . Lemma 3.1.
Let
G → S be the N´eron blowup of G in H along S . (1) The scheme
G → S represents the contravariant functor Sch S - reg S → Sets given for T → S by the set of all S -morphisms T → G such that the induced morphism T → G factorsthrough H ⊂ G . (2) The map
G → G is affine. Its restriction over S \ S induces an isomorphism G| S \ S ∼ = G | S \ S . Its restriction over S factors as G → H ⊂ G .Proof. Part (1) is a reformulation of Proposition 2.6, and (2) is immediate from Lemmas 2.3 and2.4. (cid:3)
By virtue of Lemma 3.1 (1) the (forgetful) map
G → G defines a subgroup functor when restrictedto the category Sch S -reg S . As G → S is an object in Sch S -reg S , it is a group object in this category.Here we note that products in the category Sch S -reg S exist and are computed as Bl S ( X × S X ) bythe universal property of the blowup [StaPro, 085U]. This is the closed subscheme of X × S X whichis locally defined by the ideal of a -torsion elements for a local equation a of S in S . In particular,if G → S is flat, then it is equipped with the structure of a group scheme such that G → G is amorphism of S -group schemes.3.2. Properties.
We continue with the notation of § S is an effectiveCartier divisor in S . Again recall our conventions on regular immersions from § Theorem 3.2.
Let
G → G be the N´eron blowup of G in H along S . (1) If G → S is ( quasi- ) affine, then G → S is ( quasi- ) affine. (2) If G → S is ( locally ) of finite presentation and H ⊂ G is regular, then G → S is ( locally ) of finite presentation. (3) If H → S has connected fibres and H ⊂ G is regular, then G × S S → S has connectedfibres. (4) Assume that G → S is flat and one of the following holds: (i) H ⊂ G is regular, H → S is flat and S, G are locally noetherian, (ii) H ⊂ G is regular, H → S is flat and G → S is locally of finite presentation, (iii) the local rings of S are valuation rings,then G → S is flat. (5) If both G → S , H → S are smooth, then G → S is smooth. (6) Assume that
G → S is flat. If S ′ → S is a scheme such that S ′ := S ′ × S S is an effectiveCartier divisor on S ′ , then the base change G × S S ′ → S ′ is the N´eron blowup of G × S S ′ in H × S S ′ along S ′ . n cases (4) and (5) , the map G → S is a group scheme.Proof. The map
G → G is affine by Lemma 3.1 (2) which implies (1). Items (2) to (5) are adirect transcription of Proposition 2.16, noting for (3) that for schemes equipped with a sectionthe properties “with connected fibers” and “with geometrically connected fibers” are equivalent[EGA4.2, Cor. 4.5.14] (cf. also [StaPro, 04KV]). Part (6) follows from Lemma 2.7, noting that thepreimage of S ′ under the flat map G × S S ′ → S ′ defines an effective Cartier divisor. (cid:3) Example 3.3.
Let G → Spec( Z ) be a Chevalley group scheme, that is, a split reductive groupscheme with connected fibers [Co14, § G := G × Z A Z to the affine line. Let S = Spec( Z ) considered as the effective Cartier divisordefined by the zero section of S = A Z . Let P ⊂ G be a parabolic subgroup. By Theorem 3.2 (3)and (5), the N´eron blowup G → A Z of G in P is a smooth, affine group scheme with connectedfibers. In fact, it is an easy special case of the group schemes constructed in [PZ13, §
4] and [Lou, § ̟ denote a global coordinate on A Z . By Theorem 3.2 (6) the base changes have the followingproperties:(1) If k is any field, then G ( k [[ ̟ ]]) is the subgroup of those elements in G ( k [[ ̟ ]]) = G ( k [[ ̟ ]])whose reduction modulo ̟ lies in P ( k ).(2) If p is any prime number, then G ̟ p ( Z p ) is subgroup of those elements in G ̟ p ( Z p ) = G ( Z p ) whose reduction modulo p lies in P ( F p ).In other words the respective base changes G × A Z Spec( k [[ ̟ ]]) and G × A Z ,̟ p Spec( Z p ) are parahoricgroup schemes in the sense of [BT84], cf. also [PZ13, Cor. 4.2] and [Lou, § G → A Z can be viewed as a family of parahoric group schemes.3.3. Group structure on the exceptional divisor.
We continue with the notation of § X we write Γ( X ) = H ( X, O X ) its ring of global functions. Lemma 3.4.
Assume that S is affine. Let G be an S -group scheme and G := G × S S . Denote by i : G ֒ → G the closed immersion and K the corresponding ideal sheaf. Let m, pr , pr : G × S G → G be the multiplication and the projections, with corresponding morphisms: m ♯ , pr ♯ , pr ♯ : Γ( G ) −→ Γ( G × S G )( i × i ) ♯ : Γ( G × S G ) −→ Γ( G × S G ) . If δ := m ♯ − pr ♯ − pr ♯ , we have δ ( H ( G , K )) ⊂ ker (cid:0) ( i × i ) ♯ (cid:1) .Proof. Each of the maps f ∈ { m, pr , pr } fits in a commutative diagram: G × S G G × S GG G . i × if fi Since H ( G , K ) is the kernel of the map i ♯ : Γ( G ) → Γ( G ), by taking global sections we obtain(( i × i ) ♯ f ♯ )( H ( G , K )) = 0. (cid:3) We recall that for a group scheme G → S with unit section e : S → G , the Lie algebra Lie( G/S )is the S -group scheme V ( e ∗ Ω G/S ). Theorem 3.5.
With the notation of § G → S is flat, locally finitely presented and H → S is flat, regularly immersed in G . Let G → G be the dilatation of G in H with exceptionaldivisor G := G × S S . Let J be the ideal sheaf of G in G and J H := J | H . Let V be the restrictionof the normal bundle V ( C H/G ⊗ J − H ) → H along the unit section e : S → H . Locally over S , there is an exact sequence of S -group schemes → V → G → H → . (2) If H lifts to a flat S -subgroup scheme of G , there is globally an exact, canonically splitsequence → V → G → H → . (3) If G → S is smooth, separated and G → G is the dilatation of the unit section of G , thereis a canonical isomorphism of smooth S -group schemes G ∼ −→ Lie( G /S ) ⊗ N − S /S where N S /S is the normal bundle of S in S .Proof. (1) Let F = C H/G ⊗ J − H . According to Proposition 2.9(1), locally over S we have anisomorphism of S -schemes: ψ : G × G H ∼ −→ V ( F ) . Let K = ker( G → H ). To obtain the exact sequence of the statement, it is enough to prove thatthe restriction of ψ along the unit section e : S → H is an isomorphism of S -group schemes e ∗ ψ : K ∼ −→ V. For this we may localize further around a point of S , hence assume that S and S are affine andsmall enough so that J is trivial. Proving that e ∗ ψ is a morphism of groups is equivalent to checkingan equality between two morphisms K × S K → V . Since V = Spec(Sym( F )) where F := e ∗ F , thisis the same as checking an equality between two maps of Γ( S )-modules H ( S , F ) → Γ( K × S K ).More precisely, since K is affine we have Γ( K × S K ) = Γ( K ) ⊗ Γ( S ) Γ( K ) and what we have tocheck is that m ♯ ( x ) = x ⊗ ⊗ x for all x ∈ H ( S , F ), with m the multiplication of K . Thatis, we want to prove that δ ( H ( S , F )) = 0where δ : Γ( K ) → Γ( K × S K ) is defined by δ = m ♯ − pr ♯ − pr ♯ .In order to prove this, let I be the ideal sheaf of the closed immersion H ֒ → G , and let f ∗ I be itspreimage as a module under the dilatation morphism f : G → G . Consider the closed immersions K ֒ → G and i : G ֒ → G , and the diagram: H ( G , f ∗ I ) Γ( G ) Γ( G × S G )Γ( G × S G ) H ( S , F ) Γ( K ) Γ( K × S K ) . δ ( i × i ) ♯ δ We claim that the vertical map H ( G , f ∗ I ) → H ( S , F ) is surjective. To prove this, let e G : S → G be the unit section of G and j : S ֒ → S be the closed immersion, and decompose the said map asfollows: H ( G , f ∗ I ) e ∗G −→ H ( S, e ∗ I ) j ∗ −→ H ( S , j ∗ e ∗ I ) −→ H ( S , F ) . The first map is surjective because it has the section σ ∗ where σ : G → S is the structure map. Thesecond map is surjective because it is obtained by taking global sections on the affine scheme S ofthe surjective map of sheaves e ∗ I → j ∗ j ∗ e ∗ I . To show that the third map is surjective, start fromthe surjection of sheaves I| H → F . Since pullback is right exact, this gives rise to a surjection j ∗ e ∗ I = e ∗ I| H → e ∗ F = F . Taking global sections on the affine scheme S we obtain the desiredsurjection.Now let IO G = f − I · O G be the preimage of I as an ideal. Note that by property of thedilatation, we have IO G = J O G =: K . Therefore, according to Lemma 3.4, we have δ ( H ( G , IO G )) = δ ( H ( G , K )) ⊂ ker(( i × i ) ♯ ) . Precomposing with the surjection f ∗ I → IO G , we find that H ( G , f ∗ I ) is mapped into ker( i × i ) ♯ by δ . As a result H ( G , f ∗ I ) goes to zero in Γ( K × S K ). Since H ( G , f ∗ I ) → H ( S , F ) is surjective,the commutativity of the diagram implies that δ ( H ( S , F )) = 0 in Γ( K × S K ), as desired. Hence e ∗ ψ : K → V is an isomorphism of groups. This proves (1).
2) If ˜ H ⊂ G is a flat S -subgroup scheme lifting H , we have a transversal intersection H = ˜ H ∩ G .By Proposition 2.9(3), the preceding construction of the short exact sequence can be performedglobally over S . Moreover, by the universal property of the dilatation the map ˜ H → G lifts to amap ˜ H → G . In restriction to S this splits the short exact sequence previously obtained.(3) Finally if G → S is smooth, the unit section is a regular immersion with conormal sheaf ω G/S = e ∗ Ω G/S . In restriction to S the group V is the Lie algebra whence the canonical isomorphism G ∼ −→ Lie( G /S ) ⊗ N − S /S . (cid:3) Remark 3.6.
In the situation of Theorem 3.5 (2), the group H acts by conjugation on V = V ( e ∗ C H/G ⊗ J − S ). We checked on examples that this additive action is linear, and is in fact noneother than the “adjoint” representation of H on its normal bundle as in [SGA3.1, Exp. I, Prop. 6.8.6].To recall what this representation is, note that the normal sheaf J − S comes from the base and isendowed with the trivial action; for simplicity we describe the action Zariski locally on S and omitit from the notation. We start from the conormal sequence of the inclusions { } ⊂ H ⊂ G :0 e ∗ C H/G ω G ω H . The sequence is exact on the left if { } → H is a regular immersion. Taking the associated vectorbundles, we obtain an exact sequence of S -group schemes:0 Lie( H/S ) Lie( G /S ) V ( e ∗ C H/G ) 0 . The middle term Lie( G /S ) supports the adjoint action of G . The restricted action of H leavesstable the terms Lie( H/S ) and V ( e ∗ C H/G ), cf. [SGA3.1, Exp. III, Lem. 4.25]. The former is theadjoint action of H on its Lie algebra, and the latter is the action on the normal bundle along theunit section.3.4. N´eron blowups as syntomic sheaves.
We continue with the notation of § j : S ֒ → S is an effective Cartier divisor, that G → S is a flat, locally finitely presentedgroup scheme and that H ⊂ G := G × S S is a flat, locally finitely presented closed S -subgroupscheme. In this context, there is another viewpoint on the dilatation G of G in H , namely as thekernel of a certain map of syntomic sheaves.To explain this, let f : G → G /H be the morphism to the fppf quotient sheaf, which by Artin’stheorem ([Ar74, Cor. 6.3]) is representable by an algebraic space. By the structure theorem foralgebraic group schemes (see [SGA3.1, Exp. VII B , Cor. 5.5.1]) the morphisms G → S and H → S are syntomic. Since f : G → G /H makes G an H -torsor, it follows that f is syntomic also. Lemma 3.7.
Let S syn be the small syntomic site of S . Let η : G → j ∗ j ∗ G be the adjunction mapin the category of sheaves on S syn and consider the composition v = ( j ∗ f ) ◦ η : G j ∗ j ∗ G = j ∗ G j ∗ ( G /H ) . η j ∗ f Then the dilatation
G → G is the kernel of v . More precisely, we have an exact sequence of sheavesof pointed sets in S syn : G G j ∗ ( G /H ) 1 . v If G → S and H → S are smooth, then the sequence is exact as a sequence of sheaves on the small´etale site of S .Proof. That
G → G is the kernel of v follows directly from the universal property of the dilatation,restricted to syntomic S -schemes. It remains to prove that the map of sheaves v is surjective. Itis enough to prove that both maps η and j ∗ f are surjective. For η this is because if T → S is asyntomic morphism, any point t : T → j ∗ G lifts tautologically to G after the syntomic refinement T ′ = G × S T → T . For j ∗ f , we start from a syntomic morphism T → S and a point t : T → j ∗ ( G /H ), that is a morphism T → G /H . Using that G → G /H is syntomic, we can find asbefore a syntomic refinement T ′ → T and a lift T ′ → G . Using that syntomic coverings lift across losed immersions (see [StaPro, 04E4]), there is a syntomic covering T ′′ → T such that T ′′ refines T ′ . This provides a lift of t to j ∗ G .Finally if G → S and H → S are smooth, the existence of ´etale sections for smooth morphisms([EGA4.4, Cor. 17.16.3]) and the possibility to lift ´etale coverings across closed immersions (see[StaPro, 04E4] again) show that the sequence is exact also in the ´etale topology. (cid:3) Applications
Here we give two applications in cohomological degree 0 and 1 of the theory developed so far:integral points and torsors. In § G to the graded pieces of its Liealgebra g . In § § G -bundles on curves, and in § Integral points and the Moy-Prasad isomorphism.
In this subsection we prove an iso-morphism describing the graded pieces of the filtration by congruence subgroups on the integralpoints of reductive group schemes. For the benefit of the interested reader, we provide commentson the literature on this topic in Remark 4.4 below.We start with the following lemma.
Lemma 4.1.
Let O be a ring and π ⊂ O an invertible ideal such that ( O , π ) is a henselian pair.Let G be a smooth, separated O -group scheme and G → G the dilatation of the trivial subgroup over O /π . If either O is local or G is affine, then the exact sequence of Lemma 3.7 induces an exactsequence of groups: −→ G ( O ) −→ G ( O ) −→ G ( O /π ) −→ . Proof.
Write S = Spec( O ), S = Spec( O /π ) and set G = G × S S , G = G × S S . Consider theshort exact sequence of Lemma 3.7 on the ´etale site and take the global sections over S . It is thenenough to prove that the map G ( O ) → G ( O /π ) is surjective. For this start with an ( O /π )-pointof G , i.e. a section u : S → G to the map G → S . If either O is local or G is affine, u factors through an open affine subscheme U ⊂ G . In this situation, the classical existence result forlifting of sections for smooth schemes over a henselian local ring (as in for example [BLR90, 2.3/5])extends to henselian pairs, see [Gr72, Thm. 1.8]. In this way we see that u lifts to a section u of G → S . (cid:3) Remark 4.2.
The same proof gives a similar result with the dilatation of a split smooth unipotentclosed subgroup H ⊂ G , with the group G ( O /π ) replaced by the pointed set ( G /H )( O /π ).Indeed, such a group H is obtained by successive extensions of the additive group G a,S over S = Spec( O /π ). Since S is affine, the (´etale or syntomic) cohomology of G a,S vanishes, being thecoherent cohomology of O S . Using induction one concludes that H ( S , H ) is trivial. Now startingfrom a section u of G /H → S and pulling back the map G → G /H along u , we obtain an H -torsor G × G /H S → S . By the previous remarks this torsor has a section v . The latter liftsto a section of G by the same argument as in the proof of the lemma. Theorem 4.3.
Let r, s be integers such that r/ s r . Let ( O , π ) be a henselian pair where π ⊂ O is an invertible ideal. Let G be a smooth, separated O -group scheme. Let G r the r -th iterateddilatation of the unit section and g r its Lie algebra. If O is local or G is affine, there is a canonicalisomorphism: G s ( O ) /G r ( O ) ∼ −→ g s ( O ) / g r ( O ) . Proof.
The ( r − s )-th iterated dilatation of G s is naturally G r . But as we observed in Proposi-tion 2.14, the group scheme G r can also be seen as the dilatation of { } in ( O , π r ). For an integer n >
0, we write O n := O /π n . Putting these remarks together, the previous lemma applied to thedilatation of the group scheme G = G s with respect to ( O , π r − s ) yields an isomorphism:(4.1) G s ( O ) /G r ( O ) ∼ −→ G s ( O r − s ) . e now consider the statement of the theorem. If s = 0 we have r = 0, hence left-hand side andright-hand side are equal to { } and the result is clear. Therefore we may assume that s > { } in ( O , π s ) provides a canonical isomorphism G s | O s ∼ −→ Lie( G | O s ) ⊗ N − O s / O . Since the Lie algebra of a vector bundle V ( E ) is canonically isomorphic to V ( E ) itself ([SGA3.1,Exp. II, Ex. 4.4.2]), taking Lie algebras on both sides we deduce a canonical isomorphism G s | O s ∼ −→ g s | O s . Since r − s s , the ring O r − s is an O s -algebra and we can take O r − s -valued points in the previousisomorphism to obtain: G s ( O r − s ) ∼ −→ g s ( O r − s ) . Using (4.1) once for G s and once for g s , we end up with G s ( O ) /G r ( O ) ∼ −→ g s ( O ) / g r ( O ) , which is the desired canonical isomorphism. (cid:3) Remark 4.4.
In the literature on integral points of reductive groups over non-archimedean localfields, results similar to the isomorphism of Theorem 4.3 appeared with restrictions on the indices r, s , on the group schemes involved or on the ground ring. See for instance [Ser68, Prop. 6 (b)],[Ne99, Prop. 3.9 and 3.10] for the multiplicative group, [Ho77, p. 442 line 1], [Mo82], [BK93, p. 22]for general linear groups, [Sec04, p. 337] for general linear groups over division algebras, and [PR84, § § § §
1] for general reductive groups. In these examples,the isomorphisms are defined at the level of integral points using ad hoc explicit formulas. Theseisomorphisms are sometimes called Moy-Prasad isomorphisms as a tribute to [MP94], read [Yu15,0.4] and [De02, p. 242 lines 18-19] for informations. In the case of an affine, smooth group schemeover a discrete valuation ring, the isomorphism of Theorem 4.3 appears without proof in [Yu15,proof of Lemma 2.8].4.2.
Torsors and level structures.
In this subsection we adopt notations more specific to thestudy of torsors over curves. Let X be a scheme, and let N ⊂ X be an effective Cartier divisor.Let G → X be a smooth, finitely presented group scheme, and let H ⊂ G | N be an N -smooth closedsubgroup. We denote by G → G the N´eron blowup of G in H (over N ) which is a smooth, finitelypresented X -group scheme by Theorem 3.2.For a scheme T → X , let BG ( T ) (resp. B G ( T )) denote the groupoid of right G -torsors on T in the fppf topology. Here we note that every such torsor is representable by a smooth algebraicspace (of finite presentation), and hence admits sections ´etale locally. Whenever convenient we maytherefore work in the ´etale topology as opposed to the fppf topology.Pushforward of torsors along G → G induces a morphism of contravariant functors Sch X → Groupoids given by(4.2) B G →
BG,
E 7→ E × G G. Definition 4.5.
For a scheme T → X , let B ( G, H, N )( T ) be the groupoid whose objects are pairs( E , β ) where E → T is a right fppf G -torsor and β is a section of the fppf quotient (cid:0) E| T N (cid:14) H | T N (cid:1) → T N , where T N := T × X N , i.e., β is a reduction of E| T N to an H -torsor. Morphisms ( E , β ) → ( E ′ , β ′ ) aregiven by isomorphisms of torsors ϕ : E ∼ = E ′ such that ¯ ϕ ◦ β = β ′ where ¯ ϕ denotes the induced mapon the quotients. Note that if T N = ∅ , then there is no condition on the compatibility of β and β ′ .Each of the contravariant functors Sch X → Groupoids induced by B G , BG and B ( G, H, N )defines a stack over X in the fppf topology. We call B ( G, H, N ) the stack of G -torsors equipped withlevel- ( H, N ) -structures . emma 4.6. The map (4.2) factors as a map of X -stacks (4.3) B G → B ( G, H, N ) → BG, where the second arrow denotes the forgetful map.Proof.
By Lemma 3.1 (1) the map G| N → G | N factors as G| N → H ⊂ G | N . Thus, given a G -torsor E → T we get the H -equivariant map E × G| TN H | T N ⊂ E × G| TN G | T N . Passing to the fppf quotient for the right H -action defines the section β can . The association E 7→ ( E × G G, β can ) induces the desired map B G → B ( G, H, N ). (cid:3) Proposition 4.7.
The map (4.3) induces an equivalence of contravariant functors
Sch N - reg X → Groupoids given by B G ≃ −→ B ( G, H, N ) , E 7→ ( E × G G, β can ) . Proof.
For T → X in Sch N -reg X , we need to show that B G ( T ) → B ( G, H, N )( T ) is an equivalenceof groupoids. Since G → X is smooth, in particular flat, its formation commutes with base changealong T → X by Theorem 3.2. Hence, we may reduce to the case where T = X . Now recall fromLemma 3.7 the exact sequence of sheaves of pointed sets on the ´etale site of X ,1 G G j ∗ ( G | N /H ) 1 , where j : N ⊂ X denotes the inclusion. The desired equivalence is a consequence of [Gi71, Chap. III, § β : X → j ∗ ( G | N /H ) along the G -torsor G → j ∗ ( G | N /H ). Here we have used that by smoothness of the group schemes G → X , G → X , H → N and consequently of the quotient G | N /H we are allowed to work with the ´etaletopology as opposed to the fppf topology. (cid:3) Level structures on moduli stacks of bundles on curves.
We continue with the notation, andadditionally assume that X is a smooth, projective, geometrically irreducible curve over a field k ,and that G → X and hence G → X is affine.Let Bun G := Res X/k BG (resp. Bun G := Res X/k B G ) be the moduli stack of G -torsors (resp. G -torsors) on X ; here Res X/k stands for the Weil restriction along X → Spec( k ). This is a quasi-separated, smooth algebraic stack locally of finite type over k , cf. e.g. [He10, Prop. 1] or [AH19,Thm. 2.5]. Similarly, let Bun ( G,H,N ) := Res X/k B ( G, H, N ) be the stack parametrizing G -torsorsover X with level-( H, N )-structures as in Definition 4.5.
Theorem 4.8.
The map (4.3) induces equivalences of contravariant functors
Sch k → Groupoids given by
Bun
G ∼ = −→ Bun ( G,H,N ) , E 7→ ( E × G G, β can ) Proof.
For any k -scheme T , the projection X × k T → X is flat, and hence defines an object inSch N - reg X . The theorem follows from Proposition 4.7. (cid:3) Example 4.9. If H = { } is trivial, then Bun ( G,H,N ) is the moduli stack of G -torsors on X withlevel- N -structures. If G → X is split reductive, if N is reduced and if H a parabolic subgroup in G | N , then Bun ( G,H,N ) is the moduli stack of G -torsors with quasi-parabolic structures in the senseof Laszlo-Sorger [LS97], cf. [PR10, § §
1, Exam. (2)].We end this subsection by discussing Weil uniformizations. Let | X | ⊂ X be the set of closedpoints, and let η ∈ X be the generic point. We denote by F = κ ( η ) the function field of X .For each x ∈ | X | , we let O x be the completed local ring at x with fraction field F x and residuefield κ ( x ) = O x / m x . Let A := d ′ x ∈| X | F x be the ring of adeles with subring of integral elements O = d x ∈| X | O x . As in [N06, Lem. 1.1] or [Laf18, Rem. 8.21] one has the following result. roposition 4.10. Assume that k is either a finite field or a separably closed field, and that G → X has connected fibers. Then there is an equivalence of groupoids (4.4) Bun G ( k ) ≃ G γ G γ ( F ) (cid:15)(cid:0) G γ ( A ) /G ( O ) (cid:1) , where γ ranges over ker ( F, G ) := ker (cid:0) H ( F, G ) → d x ∈| X | H ( F x , G ) (cid:1) , and where G γ denotesthe associated pure inner form of G | F . The identification (4.4) is functorial in G among maps of X -group schemes which are isomorphisms in the generic fibre.Proof. Under our assumptions, Lang’s lemma implies that H ( O x , G ) is trivial for all x ∈ | X | :use that H ( κ ( x ) , G ) is trivial because G | κ ( x ) smooth, affine, connected and κ ( x ) is either finite orseparably closed; then an approximation argument as in e.g. [RS20, Lem. A.4.3]. In particular, forevery G -torsor E → X the class of its generic fibre [ E| F ] lies in ker ( F, G ). For each γ ∈ ker ( F, G ),we fix a G -torsor E γ → Spec( F ) of class γ . We denote by G γ its group of automorphisms which isan inner form of G . We also fix an identification G γ ( F x ) = G ( F x ) for all x ∈ | X | , γ ∈ ker ( F, G ).In particular, G γ ( A ) = G ( A ) so that the right hand quotient in (4.4) is well-defined. Now considerthe groupoid Σ γ := (cid:8) ( E , δ, ( ǫ x ) x ∈| X | ) (cid:12)(cid:12) δ : E| F ≃ E γ , ǫ x : E ≃ E| O x (cid:9) . For each x ∈ | X | , we have g x := δ | F x ◦ ǫ x | F x ∈ Aut( E γ | F x ) = G γ ( F x ) = G ( F x ) , and further g x ∈ G ( O x ) for almost all x ∈ | X | . Thus, the collection ( g x ) x ∈| X | defines a point in G ( A ) = G γ ( A ). In this way, we obtain an G γ ( F ) × G ( O )-equivariant map π γ : Σ γ → G γ ( A ), andthus a commutative diagram of groupoids F γ Σ γ F γ G γ ( A )Bun G F γ G γ ( F ) (cid:15)(cid:0) G γ ( A ) /G ( O ) (cid:1) . ⊔ γ π γ As the vertical maps are disjoint unions of G γ ( F ) × G ( O )-torsors, the dashed arrow is fully faithful.Hence, it suffices to show that it is a bijection on isomorphism classes, i.e., a bijection of sets. Weconstruct an inverse of the dashed arrow as follows: Given a representative ( g x ) x ∈| X | ∈ G γ ( A ) = G ( A ) of some class, there is a non-empty open subset U ⊂ X such that g x ∈ G ( O x ) for all x ∈ | U | ,and such that E γ is defined over U . Let X \ U = { x , . . . , x n } for some n >
0. We define theassociated G -torsor by gluing the torsor E γ on U with the trivial G -torsor onSpec( O x ) ⊔ . . . ⊔ Spec( O x n )using the elements g x , . . . , g x n and the identification G γ ( F x ) = G ( F x ). The gluing is justified bythe Beauville-Laszlo lemma [BL95], or alternatively [He10, Lem. 5]. This shows (4.4). From theconstruction of the map ⊔ γ π γ , one sees that (4.4) is functorial in G among generic isomorphisms. (cid:3) Note that N defines an effective Cartier divisor on Spec( O ) so that the map of groups G ( O ) → G ( O ) is injective. As subgroups of G ( O ) we have(4.5) G ( O ) = ker (cid:0) G ( O ) → G ( O N ) → G ( O N ) /H ( O N ) (cid:1) , where O N denotes the ring of functions on N viewed as a quotient ring O X → O N . Corollary 4.11.
Under the assumptions of Proposition 4.10, the N´eron blowup
G → X is smooth,affine with connected fibers by Theorem 3.2, and there is a commutative diagram of groupoids Bun G ( k ) F γ G γ ( F ) (cid:15)(cid:0) G γ ( A ) / G ( O ) (cid:1) Bun G ( k ) F γ G γ ( F ) (cid:15)(cid:0) G γ ( A ) /G ( O ) (cid:1) , ≃≃ dentifying the vertical maps as the level maps. Remark 4.12.
Let G | F be reductive.(1) If k is algebraically closed, then F is C by Tsen’s theorem and in particular of cohomologicaldimension
1, cf. [Ser65, II.3]. In this case, H ( F, G ), and hence ker ( F, G ), is trivial by[BS68, 8.6].(2) If k is a finite field, then ker ( F, G ) is dual to ker ( F, Z ( ˆ G )( ¯ Q )) where Z ( ˆ G ) denotes thecenter of the Langlands dual group ˆ G , formed, say, over ¯ Q , cf. [Ko84, Ko86] and [NQT11]for global fields of positive characteristic. In particular, if G | F is either simply connected orsplit reductive, then ker ( F, G ) is trivial, cf. also [Laf18, Rem. 8.21, 12.2].4.2.2.
Integral models of moduli stacks of shtukas.
Here we point out that Theorem 4.8 immediatelyapplies to construct certain integral models of moduli stacks of shtukas. We proceed with thenotation of § k is a finite field. Our presentation follows [Laf18, §§ I = I ⊔ . . . ⊔ I r , r ∈ Z > of a finite index set, the moduli stack of iterated G -shtukas is the contravariant functor of groupoids Sch k → Groupoids given by(4.6) Sht
G,I • def = n E r α r I r E r − α r I r − . . . α I E α I E = τ E r o , where τ E := (id X × Frob
T/k ) ∗ E denotes the pullback under the relative Frobenius Frob T/k . Herethe dashed arrows in (4.6) indicate that the maps α j between G -bundles are rationally defined.More precisely, Sht G,I • ( T ) classifies data (cid:0) ( E j ) j =1 ,...,r , { x i } i ∈ I , ( α j ) j =1 ,...,r (cid:1) where E j ∈ Bun G ( T )are torsors, { x i } i ∈ I ∈ X I ( T ) are points, and α j : E j | X T \ ( ∪ i ∈ Ij Γ xi ) → E j − | X T \ ( ∪ i ∈ Ij Γ xi ) are isomorphisms of torsors. Here Γ x i ⊂ X T denotes the graph of x i viewed as a relative effectiveCartier divisor on X T → T . We have a forgetful map Sht G,I • → X I . Similarly, we have the modulistack Sht G ,I • → X I defined by replacing G with G . By [Var04] for split reductive groups and by[AH19, Thm. 3.15] for general smooth, affine group schemes both stacks are ind-(Deligne-Mumford)stacks which are ind-(separated and of locally finite type) over k . Furthermore, pushforward oftorsors along G → G induces a map of X I -stacks(4.7) Sht G ,I • → Sht
G,I • , cf. [Br]. We also consider the moduli stack of iterated G -shtukas with level- ( H, N ) -structures ,Sht ( G,H,N ) ,I • → X I , i.e., Sht ( G,H,N ) ,I • ( T ) classifies data (cid:0) ( E j , β j ) j =1 ,...,r , { x i } i ∈ I , ( α j ) j =1 ,...,r (cid:1) , where ( E j , β j ) ∈ Bun ( G,H,N ) ( T ) are G -torsors with a level-( H, N )-structure, { x i } i ∈ I ∈ X I ( T ) arepoints, and(4.8) α j : ( E j , β j ) | X T \ ( ∪ i ∈ Ij Γ xi ) → ( E j − , β j − ) | X T \ ( ∪ i ∈ Ij Γ xi ) are maps of G -torsors with a level-( H, N )-structure where ( E , β ) := ( τ E r , τ β r ). We have a forgetfulmap of X I -stacks(4.9) Sht ( G,H,N ) ,I • → Sht
G,I • . Corollary 4.13.
Let
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