Compact generation of the category of D-modules on the stack of G-bundles on a curve
aa r X i v : . [ m a t h . AG ] M a y COMPACT GENERATION OF THE CATEGORY OF D-MODULES ONTHE STACK OF G -BUNDLES ON A CURVE V. DRINFELD AND D. GAITSGORY
Abstract.
Let G be a reductive group. Let Bun G denote the stack of G -bundles on asmooth complete curve over a field of characteristic 0, and let D-mod(Bun G ) denote the DGcategory of D-modules on Bun G . The main goal of the paper is to show that D-mod(Bun G ) iscompactly generated (this is not automatic because Bun G is not quasi-compact). The proofis based on the following observation: Bun G can be written as a union of quasi-compactopen substacks j : U ֒ → Bun G , which are ”co-truncative”, i.e., the functor j ! is defined onthe entire category D-mod( U ). Contents
Introduction 30.1. The main result 30.2. Truncativeness, co-truncativeness and truncatability 40.3. Duality 50.4. Generalizations and open questions 60.5. Organization of the paper 60.6. Conventions and notation 80.7. Acknowledgements 91. DG categories 91.1. The setting 91.2. Compactness and compact generation 101.3. Ind-completions 111.4. Karoubi-completions 121.5. Symmetric monoidal structure and duality 131.6. Limits of DG categories 141.7. Colimits in DGCat cont
Date : May 4, 2015. SL ( ≤ n ) G G G P -admissible sets 528.1. Some elementary geometry 528.2. P -admissible subsets of Λ + , Q G P -admissible subsets 539. Proof of the main theorem 569.1. The main result of this section 569.2. A key proposition 579.3. Proof of Theorem 9.1.2 579.4. A variant of the proof 5910. The estimates 5910.1. The vanishing of H and H c i : proof of Proposition 9.2.2(a) 5910.3. The notion of strangeness 6010.4. Proof of Proposition 10.1.3 6110.5. Remarks on the positive characteristic case 6211. Constructing the contraction 6211.1. Morphisms between Bun P , Bun P − , Bun M , and Bun G A on Bun P − HE CATEGORY OF D-MODULES ON Bun G A.2. Continuous maps 68Appendix B. The Langlands retraction and coarsenings of the Harder-Narsimhan-Shatzstratification 68B.1. Recollections on the Langlands retraction 68B.2. The η -stratification 69B.3. The case where η is “deep inside” Λ + , Q G Introduction
The main result.
Let k be an algebraically closed field of characteristic 0. Let X be asmooth complete connected curve over k and let Bun G denote the moduli stack of principal G -bundles on X , where G is a connected reductive group.0.1.1. The object of study of this paper is the DG category D-mod(Bun G ) of D-modules onBun G . Our main goal is to prove the following theorem: Theorem 0.1.2.
The DG category
D-mod(Bun G ) is compactly generated. For the reader’s convenience we will review the theory of DG categories, and the notion ofcompact generation in Sect. 1.Essentially, the property of compact generation is what makes a DG category manageable.0.1.3. The above theorem is somewhat surprising for the following reason.It is known that if an algebraic stack Y is quasi-compact and the automorphism group of everyfield-valued point of Y is affine, then the DG category D-mod( Y ) is compactly generated. Thisresult is established in [DrGa1, Theorem 0.2.2]. In fact, the compact generation of D-mod( Y ) formost stacks Y that one encounters in practice is much easier than the above-mentioned theoremof [DrGa1]: it is nearly obvious for stacks of the form Z/H , where Z is a quasi-compact schemeand H an algebraic group acting on it.However, if Y is not quasi-compact then D-mod( Y ) does not have to be compactly generated.We will exhibit two such examples in Sect. 12.1; in both of them Y will actually be a smoothnon quasi-compact scheme (non-separated in the first example, and separated in the secondone). V. DRINFELD AND D. GAITSGORY Y ) encodes a certain geometric property of thestack Y . We do not know how to formulate a necessary and sufficient condition for D-mod( Y )to be compactly generated.But we do formulate a sufficient condition, which we call “truncatibility” (see Sect. 0.2.3or Definition 4.1.1). The idea is that Y is truncatable if it can be represented as a union ofquasi-compact open substacks U so that for each of them direct image functor j ∗ : D-mod( U ) → D-mod( Y )has a particularly nice property explained below.0.2. Truncativeness, co-truncativeness and truncatability. Y be a quasi-compact algebraic stack with affine automorphism groups of points,and let Z i ֒ → Y be a closed embedding. By [DrGa1, Theorem 0.2.2], both categories D-mod( Z )and D-mod( Y ) are compactly generated.We have a pair of adjoint functors i dR , ∗ : D-mod( Z ) ⇄ D-mod( Y ) : i ! . Being a left adjoint, the functor i dR , ∗ preserves compactness. But there is no reason for i ! to have this property. We will say that Z is truncative in Y if i ! does preserve compactness.Truncativeness is a purely “stacky” phenomenon. In Sect. 3.2.1 we will show that it neveroccurs for schemes, unless Z is a union of connected components of Y .Let U j ֒ → Y be the embedding of the complementary open substack. We say that U is co-truncative in Y if Z is truncative. This property can be reformulated as saying that thefunctor j ∗ : D-mod( U ) → D-mod( Y )preserves compactness. We show that the property of co-truncativeness can be also reformulatedas the existence of the functor j ! : D-mod( U ) → D-mod( Y ), left adjoint to the restriction functor j ∗ . (A priori, j ! is only defined on the holonomic subcategory.) Remark . The property of being compact for an object in D-mod( Y ) is somewhat subtle(e.g., it is not local in the smooth topology). In Sect. 3.5 we reformulate the notion of trunca-tiveness and co-truncativeness in terms of the more accessible property of coherence instead ofcompactness.0.2.3. Let us now drop the assumption that Y be quasi-compact. We say that a closed substack Z (resp., open substack U ) is truncative (resp., co-truncative), if for every quasi-compact open ◦ Y ⊂ Y , the intersection Z ∩ ◦ Y (resp., U ∩ ◦ Y ) is truncative (resp., co-truncative) in ◦ Y .We say that Y is truncatable if it equals the union of its quasi-compact co-truncative opensubstacks. We will show that a union of two co-truncative open substacks is co-truncative. So Y is truncatable if and only if every open quasi-compact substack of Y is contained in one whichis co-truncative.We will show (see Proposition 4.1.6) that if Y is truncatable, then D-mod( Y ) is compactlygenerated . (This is an easy consequence of [DrGa1, Theorem 0.2.2].) HE CATEGORY OF D-MODULES ON Bun G Theorem 0.2.5.
The stack
Bun G is truncatable. Let us explain how to cover Bun G by quasi-compact open co-truncative substacks. For everydominant rational coweight θ let Bun ( ≤ θ ) G ⊂ Bun G denote the open substack parameterizing G -bundles whose Harder-Narasimhan coweight is ≤ G θ (the partial ordering ≤ G on coweights isdefined as usual: λ ≤ G λ if λ − λ is a linear combination of simple coroots with non-negativecoefficients). Equivalently, Bun ( ≤ θ ) G parameterizes those G -bundles P G that have the followingproperty: for every reduction P B to the Borel, the degree of P B (which is a coweight of G ) is ≤ G θ .The substacks Bun ( ≤ θ ) G are quasi-compact and cover Bun G . So Theorem 0.2.5 is a conse-quence of the following fact proved in Sect. 9: The substack
Bun ( ≤ θ ) G is co-truncative if for every simple root ˇ α i one has (0.1) h θ , ˇ α i i ≥ g − , where g is the genus of X . E.g., if G = GL (2) this means that the open substackBun ( ≤ m ) GL ∩ Bun nGL ⊂ Bun GL that parameterizes rank 2 vector bundles of degree n all of whose line sub-bundles have degree ≤ m , is co-truncative provided that 2 m − n ≥ g − θ is “deep enough” inside the dominant chamber (of course, if g ≤ θ ).0.2.6. Establishing truncativeness.
To prove Theorem 0.2.5, we will have to show that certainexplicitly defined locally closed substacks of Bun G are truncative.We will do this by using a “contraction principle”, see Proposition 5.1.2. In its simplestform, it says that the substack { } / G m ֒ → A n / G m is truncative (here G m acts on A n byhomotheties).0.3. Duality. Y ) when Y is a truncatable algebraic stack.0.3.2. However, more is true. As we recall in Sect. 2.2.14, if Y is quasi-compact, not only isthe category D-mod( Y ) dualizable, but Verdier duality defines an equivalenceD-mod( Y ) ∨ ≃ D-mod( Y ) . It is natural to ask for a description of the dual category D-mod( Y ) ∨ when Y is no longerquasi-compact, but just truncatable. This rational coweight was defined by Harder-Narasimhan [HN] in the case G = GL ( n ) and by A. Ra-manathan [R1] for any G . V. DRINFELD AND D. GAITSGORY Y ) ∨ can be described explicitly, butit is a priori different from D-mod( Y ).There exists a naturally defined functor D Verdier Y , naive : D-mod( Y ) ∨ → D-mod( Y ) , but we show (see Proposition 4.4.5) that this functor is not an equivalence unless the closureof every quasi-compact open in Y is again quasi-compact.0.3.4. However, in Sect. 4.4.8 we define a less obvious functor D Verdier Y , ! : D-mod( Y ) ∨ → D-mod( Y ) , which may differ from D Verdier Y , naive even for Y quasi-compact.In general, D Verdier Y , ! is not an equivalence, but there are important and nontrivial examplesof quasi-compact and non-quasi-compact stacks Y for which D Verdier Y , ! is an equivalence.In particular, in a subsequent publication it will be shown that the functor D Verdier Y , ! is anequivalence if Y = Bun G , where G is any reductive group. Thus, for any reductive G , the DG category D-mod(Bun G ) identifies with its dual (in anon-trivial way and for non-trivial reasons).0.4. Generalizations and open questions.
Let us return to the main result of this paper,namely, Theorem 0.1.2.0.4.1. In the situation of Quantum Geometric Langlands, one needs to consider the categoriesof twisted D-modules on Bun G . The corresponding analog of Theorem 0.1.2, with the sameproof, holds in this more general context.0.4.2. Let x , . . . , x n ∈ X . Instead of Bun G , consider the stack of G -bundles on X with areduction to a parabolic P i at x i , 1 ≤ i ≤ n . Most probably, in this situation an analog ofTheorem 0.1.2 holds and can be proved in a similar way.0.4.3. Suppose now that instead of reductions to parabolics (as in Sect. 0.4.2), one considersdeeper level structures at x , . . . , x n (the simplest case being reduction to the unipotent radicalof the Borel).We do not know whether an analog of Theorem 0.1.2 holds in this case, and we do not knowwhat to expect. In any case, our strategy of the proof of Theorem 0.1.2 fails in this context.0.4.4. Here are some more questions: Question 0.4.5.
Does the assertion of Theorem 0.1.2 (and its strengthening, Theorem 0.2.5)hold for Y being one of the stacks Bun B , Bun P , Bun P and g Bun P , where B is the Borel, and P a general parabolic? We are quite confident that the answer is “yes” for Bun B , but are less sure in other cases. Question 0.4.6.
Does the assertion of Theorem 0.1.2 hold for an arbitrary connected affinealgebraic group G (i.e., without the assumption that G be reductive)? Organization of the paper. For a draft see [Ga2].
HE CATEGORY OF D-MODULES ON Bun G Y . We first consider the case when Y is quasi-compact and make a summary of the relevantresults from [DrGa1]. We then consider the case when Y is not quasi-compact and characterizethe subcategory of D-mod( Y ) formed by compact objects.0.5.3. In Sect. 3, we introduce some of the main definitions for this paper: the notions oftruncativeness (for a locally closed substack) and co-truncativeness (for an open substack). Westudy the behavior of these notions under morphisms, base change, refinement of stratification,etc. We also discuss the “non-standard” functors associated to a truncative closed (or locallyclosed) substack (see Sects. 3.3 and Remark 3.4.5), in particular, the very unusual functors i ? and j ? .0.5.4. Sect. 4 is, philosophically, the heart of this article.In Sect. 4.1 we introduce the notion of truncatable stack. We show that if Y is truncatablethen the category D-mod( Y ) is compactly generated. In particular, we obtain that Theo-rem 0.2.5 implies Theorem 0.1.2.In Sects. 4.2–4.5 we discuss the behavior of Verdier duality on truncatable stacks and therelation beween the category D-mod( Y ) and its dual.0.5.5. In Sect. 5 we formulate a contraction principle , see Proposition 5.1.2. It shows that aclosed substack with the property that we call contractiveness is truncative.In Sect. 5.3 we explicitly describe the non-standard functors i ∗ and i ? in the setting ofProposition 5.1.2.0.5.6. In Sect. 6 we prove Theorem 0.2.5 in the particular case of G = SL . The proof in thegeneral case follows the same idea, but is more involved combinatorially.0.5.7. In Sect. 7 we recall the stratification of Bun G according to the Harder-Narasihmancoweight of the G -bundle. We briefly indicate a way to establish the existence of such a strati-fication using the relative compactification of the map Bun P → Bun G .0.5.8. In Sect. 8 we introduce a book-keeping device that allows to produce locally closedsubstacks of Bun G from locally closed substacks of Bun M , where M is a Levi subgroup of G .Certain locally closed substacks of Bun G obtained in this way, will turn out to be contractive ,and hence truncative , and as such will play a crucial role in the proof of Theorem 0.2.5.0.5.9. In Sect. 9–11 we finally prove Theorem 0.2.5. The proof amounts to combining theHarder-Narasimhan-Shatz strata of Bun G (i.e., the strata corresponding to a fixed value ofthe Harder-Narasihman coweight) into certain larger locally closed substacks and applying thecontraction principle. A more detailed explanation of the idea of the proof can be found inSect. 9.1.In Sect. 9 we prove Theorem 0.2.5 modulo a key Proposition 9.2.2. The latter is proved inSect. 10–11.0.5.10. In Sect. 12 we prove the existence of non quasi-compact stacks Y such that the categoryD-mod( Y ) is not compactly generated.Namely, we show that if Y = Y is a smooth scheme containing a non quasi-compact divisor,then the category D-mod( Y ) is not generated by compact objects. More precisely, we show that(locally) coherent D-modules on Y that belong to the full subcategory generated by compactobjects cannot have all of T ∗ ( Y ) as their singular support. In particular, the D-module D Y does not belong to the subcategory. V. DRINFELD AND D. GAITSGORY G , see Sect. 7.4.10 and Corollary 7.4.11.0.5.12. In Appendix B we give a variant of the proof of Theorem 9.1.2 that has some ad-vantages compared with the one from Sect. 9.3. The method is to define a coarsening of theHarder-Narasimhan-Shatz stratification such that each stratum is contractive (and thereforetruncative). This is done using the Langlands retraction of the space of rational coweights ontothe dominant cone.0.5.13. In Appendix C we prove a “stacky” generalization of the contraction principle fromSect. 5.1 and of the adjunction from Proposition 5.3.2.0.6.
Conventions and notation. ∞ -categories follow those of [DrGa1, Sect. 0.6.1]. Whenever we say“category”, by default we mean an ( ∞ , ∞ -Cat the ( ∞ , ∞ -categories.We denote by ∞ -Grpd ⊂ ∞ -Cat in the ( ∞ , ∞ -groupoids, a.k.a.,spaces. We denote by C C grpd the functor ∞ -Cat → ∞ -Grpd right adjoint to the aboveembedding. Explicitly, C grpd is obtained from C by discarding non-invertible 1-morphisms.For C ∈ ∞ -Cat and objects c , c ∈ C we denote by Maps C ( c , c ) ∈ ∞ -Grpd the corre-sponding space of maps. We let Hom C ( c , c ) denote the set π (Maps C ( c , c )).0.6.2. Schemes and stacks.
This paper deals with categorical aspects of the category of D-modules, i.e., we do not need derived algebraic geometry for this paper. Therefore, by a scheme we shall understand a classical scheme . We let Sch (resp., Sch aff ) denote the category of schemes(resp., affine schemes) over k , and Sch lft (resp., Sch affft ) its full subcategory consisting of affineschemes locally of finite type (resp., affine schemes of finite type).By a prestack we shall mean an arbitrary functor (Sch aff ) op → ∞ -Grpd.By a stack we shall mean a prestack that satisfies the fppf descent condition. For the generalnotion of Artin stack we refer the reader to [GL:Stacks, Sect. 4.2]. However, neither generalstacks nor Artin stacks are necessary for this paper. What we need is the more restricted (andstandard) notion of algebraic stack . We adopt the following conventions: a stack Y is said tobe an algebraic stack if: • The diagonal morphism Y → Y × Y is schematic, quasi-compact and quasi-separated; • There exists a scheme Z equipped with a morphism f : Z → Y (this morphism is auto-matically schematic, by the previous condition) such that f is smooth and surjective.The pair ( Z, f ) is called a presentation or atlas of Y .We note that this definition is slightly more restrictive than the one in [GL:Stacks, Sect.4.2.8]. HE CATEGORY OF D-MODULES ON Bun G Finite type(ness).
All schemes, algebraic stacks and prestacks considered in this paperwill be locally of finite type over k .We recall that a classical prestack, i.e., a functor (Sch aff ) op → ∞ -Grpd, is said to be locallyof finite type if it takes limits in Sch aff to colimits in ∞ -Grpd. Equivalently, a classical prestackis locally of finite type if it is the left Kan extension from the full subcategory Sch affft ⊂ Sch aff .The upshot is that when considering prestacks locally of finite type, one can forget about allaffine schemes altogether and restrict one’s attention to Sch affft .An algebraic stack is said to be locally of finite type if it is such when considered as a prestack.This is equivalent to requiring that it admit an atlas ( Z, f ) with Z being locally of finite type.Or, still equivalently, that for any Z ∈ Sch equipped with a smooth map to Y , the scheme Z is offinite type. The equivalence of these conditions is established, e.g., in [GL:Stacks, Proposition4.9.2].0.6.4. D-modules.
We refer the reader to the paper [GR] for the theory of D-modules (a.k.a.crystals) on prestacks locally of finite type.For a morphism f : Y → Y of prestacks we have a tautologically defined functor f ! : D-mod( Y ) → D-mod( Y ) . This functor may or may not have a left adjoint, which we denote by f ! .If f is schematic and quasi-compact, we also have a functor of direct image f dR , ∗ : D-mod( Y ) → D-mod( Y ) . However, when f is an open embedding, we will use the notation j ∗ instead of j dR , ∗ , and j ∗ instead of j ! , for reasons of tradition. This is not supposed to cause confusion, as the abovefunctors go to the same-named functors for the underlying O -modules.0.7. Acknowledgements.
The research of V. D. is partially supported by NSF grant DMS-1001660. The research of D. G. is partially supported by NSF grant DMS-1063470. We thankR. Bezrukavnikov for drawing our attention to Langlands’ article [La]. We are grateful toS. Schieder for comments on the previous version of the paper.1.
DG categories
Sects. 1.1-1.6 are devoted to recollections and conventions regarding DG categories. InSects. 1.7-1.9 we provide a categorical framework for Sects. 4.2-4.3; this material can definitelybe skipped until it is used.1.1.
The setting. k . Werefer the reader to [GL:DG] for a survey. We let Vect denote the DG category of chain complexes of k -vector spaces.We let DGCat denote the ∞ -category of all DG categories. Recall that f is said to be schematic if Y × Y S is a scheme for any scheme S equipped with a morphism S → Y . Whenever we talk about a DG category C , we will always assume that it is pre-triangulated , which bydefinition means that Ho( C ) is triangulated . We will ignore set-theoretic issues; however, the reader can assume that all DG categories and functors are accessible in the sense of [Lu1, Sect. 5.4.2].
Cocomplete DG categories.
Our basic object of study is the ( ∞ , cont whose objects are cocomplete DG categories (i.e., ones that contain arbitrary direct sums, orequivalently, colimits), and where 1-morphisms are continuous functors (i.e., exact functors thatcommute with arbitrary direct sums, or equivalently all colimits).The construction of DGCat cont as an ( ∞ , cont by the equivalent ( ∞ , ∞ -categoriestensored over k , whose construction is a consequence of [Lu2, Sects. 4.2 and 6.3].We have a forgetful functor DGCat cont → DGCat that induces an isomorphism on 2-morphisms and higher.1.1.3.
Terminological deviation (i). We will sometimes encounter non-cocomplete DG categories(e.g., the subcategory of compact objects in a given DG category). Every time that this happens,we will say so explicitly.1.1.4. The category DGCat cont has a natural symmetric monoidal structure given by Lurie’stensor product, denoted by ⊗ (see [Lu2, Sect. 6.3] or [GL:DG, Sect. 1.4] for a brief review).Its unit object is the category Vect of chain complexes of k -vector spaces.1.1.5. Functors.
For C , C ∈ DGCat cont we will denote by Funct cont ( C , C ) their internalHom in DGCat cont , which is therefore another DG category.1.1.6. Terminological deviation (ii). For two DG categories C and C we will sometimes en-counter functors C → C that are not continuous (but still exact). For example, for a non-compact object c ∈ C , such is the functor M aps C ( c , − ) : C → Vect (see below for the notation).Every time when we encounter a non-continuous functor, we will say so explicitly.All exact functors C → C also form a DG category, which we denote by Funct( C , C ).1.1.7. Mapping spaces.
Any DG category C can be thought of as an ∞ -category enriched overVect with the same set of objects. For two objects c , c , we will denote by M aps C ( c , c ) ∈ Vect the corresponding Hom object.We let Maps C ( c , c ) ∈ ∞ -Grpd denote the Hom-space, when we consider C as a plain ∞ -category. The object Maps C ( c , c ) equals the image of τ ≤ ( M aps C ( c , c )) under theDold-Kan functor Vect ≤ → ∞ -Grpd . We denote by Hom C ( c , c ) the object H ( M aps C ( c , c )) ∈ Vect ♥ . Its underlying setidentifies with π (Maps C ( c , c )).1.1.8. t-structures. Whenever a DG category C has a t-structure, we let C ≤ (resp., C ≥ )denote the full subcategory of connective (resp., co-connective) objects. We denote by C ♥ theheart of the t-structure.1.2. Compactness and compact generation.
HE CATEGORY OF D-MODULES ON Bun G c in a (cocomplete) DG category C is called compact if the functorHom C ( c , − ) : C → Vect ♥ commutes with arbitrary direct sums. This is equivalent to the (a priori non-continuous) functor M aps C ( c , − ) : C → Vectbeing continuous, or the functor of ∞ -categoriesMaps C ( c , − ) : C → ∞ -Grpdcommuting with filtered colimits.For a DG category C , we let C c denote the full (but not cocomplete) DG subcategory thatconsists of compact objects.1.2.2. Compact generation.
Let C be a cocomplete DG category. We say that a set of objects c α ∈ C generates C if for every c ∈ C the following implication holds:(1.1) Hom C ( c α , c ) = 0 , ∀ α ⇒ c = 0 . This is known to be equivalent to the following condition: C does not contain a proper fullcocomplete DG subcategory that contains all the objects c α .A cocomplete DG category C is called compactly generated if there exists a set of compactobjects c α that generates C in the above sense.1.2.3. The following observations will be used repeatedly throughout the paper:Let C and C be a pair of DG categories, and let G : C → C be a (not necessarilycontinuous) functor. If G admits a left adjoint functor F : C → C then F is automaticallycontinuous.Let F , G be as above and suppose, in addition, that C is compactly generated. Then G iscontinuous if and only if F preserves compactness (i.e., F ( C c ) ⊂ C c ). This implies the “onlyif” part of the following well-known proposition. Proposition 1.2.4.
Let C be a compactly generated DG category and F : C → C a contin-uous DG functor. Then F has a continuous right adjoint if and only if F ( C c ) ⊂ C c .Proof of the “if” statement. The existence of the not necessarily continuous right adjoint G follows from the Adjoint Functor Theorem, see [Lu1, Corollary 5.5.2.9]. To test that G iscontinuous, it is enough to show that the functors M aps C ( c , G ( − )) : C → Vectare continuous for c ∈ C c . The required continuity follows from the assumption on F . (cid:3) Ind-completions. C be an essentially small (but not cocomplete) DG category. We can functoriallyassign to it a cocomplete DG category, denoted Ind( C ) (and called the ind-completion of C ),equipped with a functor C → Ind( C ) and characterized by the property that restrictiondefines an equivalence(1.2) Funct cont (Ind( C ) , D ) → Funct( C , D )for a cocomplete category D (see [Lu1, Sect. 5.3.5] for the corresponding construction forgeneral ∞ -categories).The category Ind( C ) can be explicitly constructed as Funct(( C ) op , Vect).It is known that the functor C → Ind( C ) is fully faithful, and that its essential imagebelongs to the subcategory Ind( C ) c . It follows formally from (1.2) that the essential image of C generates Ind( C ).1.3.2. Thus, the assignment C Ind( C ) is a way to obtain compactly generated categories.In fact, all cocomplete compactly generated DG categories arise in this way. Namely, we havethe following assertion (see [Lu1, Proposition 5.3.5.11]): Lemma 1.3.3.
Let C be a cocomplete compactly generated DG category. Let F : C → C c bea fully faithful functor, such that its essential image generates C . Then the resulting functor F : Ind( C ) → C , obtained from F via (1.2) , is an equivalence. As a consequence, we obtain:
Corollary 1.3.4.
Let C be a cocomplete compactly generated DG category. Then the tautolog-ical functor Ind( C c ) → C is an equivalence. Karoubi-completions. C be an essentially small (but non-cocomplete) DG category. We say that C is Karoubian if its homotopy category is idempotent-complete.For example, for a cocomplete compactly generated DG category C , the corresponding sub-category C c is Karoubian.1.4.2. Let C → C be a functor between essentially small (but non-cocomplete) DG cate-gories.We say that the above functor realizes C as a Karoubi-completion of C if restrictiondefines an equivalence Funct( C , ′ C ) → Funct( C , ′ C )for any Karoubian ′ C . Clearly, C , if it exists, is defined up to a canonical equivalence.The following is a reformulation of the Thomason-Trobaugh-Neeman localization theorem(see [N, Theorem 2.1] or [BeV, Propostion 1.4.2]): Lemma 1.4.3. (a)
Let C be an essentially small (but not cocomplete) DG category. The canonical functor C → Ind( C ) c realizes Ind( C ) c as a Karoubi-completion of C . (b) Every object of
Ind( C ) c can be realized as a direct summand of one in C ⊂ Ind( C ) c . Lemma 1.4.3 implies that the functor Ho( C ) → Ho( C ) identifies Ho( C ) with theidempotent completion of Ho( C ). HE CATEGORY OF D-MODULES ON Bun G C Ind( C ) and C C c define mutually inverse equivalences between the appropriate ∞ -categories.The two ∞ -categories are as follows. One is DGCat Kar , whose objects are essentially smallKaroubian DG categories and morphisms are exact functors. The other is DGCat comp . gen . cont , pr . comp . ,whose objects are cocomplete compactly generated categories and morphisms are continuousfunctors preserving compactness.1.4.5. Let C be an essentially small (but not cocomplete) DG category. Let S be a subset ofits objects.We say that S Karoubi-generates C if every object in the homotopy category of C can beobtained from objects in S by a finite iteration of operations of taking the cone of a morphism,and passing to a direct summand of an object.By combining Lemmas 1.4.3 with 1.3.3 we obtain: Corollary 1.4.6.
Let C be a cocomplete DG category. Let S ⊂ C c be a subset of objects thatgenerates C . Then S Karoubi-generates C c . Symmetric monoidal structure and duality.
The notion of dual of a DG category.
A DG category C is called dualizable if it is such asan object of the symmetric monoidal category (DGCat cont , ⊗ ). We refer the reader to [DrGa1,Sect. 4.1] for a review of some of the properties of this notion. The most important ones arelisted below.For a dualizable category C we denote by C ∨ its dual. One constructs C ∨ explicitly as(1.3) C ∨ ≃ Funct cont ( C , Vect) . In addition, for any D ∈ DGCat cont , the natural functor C ∨ ⊗ D → Funct cont ( C , D )is an equivalence.1.5.2. If F : C → C is a (continuous) functor between dualizable categories, there exists acanonically defined dual functor F ∨ : C ∨ → C ∨ (the construction follows, e.g., from (1.3)). Theassignment F F ∨ is functorial in F . One has ( F ∨ ) ∨ = F , ( G ◦ F ) ∨ = F ∨ ◦ G ∨ .From here we obtain that if the functors F : C ⇆ C : G are mutually adjoint, then so are the functors G ∨ : C ∨ ⇆ C ∨ : F ∨ . C is compactly generated, then it is dualizable. We have a canonical identification( C ∨ ) c ≃ ( C c ) op . Vice versa, if C and C are two compactly generated categories, then an identification C c ≃ ( C c ) op gives rise to an identification C ∨ ≃ C . Limits of DG categories.
The reason that we work with DG categories rather thanwith triangulated ones is that the limit (i.e., projective limit) of DG categories is well-definedas a DG category (while the corresponding fact for triangulated categories is false).More precisely, the ( ∞ , cont and DGCat admit limits and the forget-ful functor DGCat cont → DGCat commutes with limits (this is essentially [Lu1, Proposition5.5.3.13]) This is important for us because the DG category of D-modules on an algebraic stackis defined as a limit (see Sect. 2.1.1 below).1.6.1. Let i C i , ( i → j ) ( φ i,j ∈ Funct cont ( C i , C j ))be a diagram of DG categories, parameterized by an index category I . The limit C := lim ←− i ∈ I C i is a priori defined by a universal property in DGCat cont : for a DG category D we have afunctorial isomorphism(Funct cont ( D , C )) grpd ≃ lim ←− i ∈ I (Funct cont ( D , C i )) grpd where the in the left-hand side the limit is taken in the ( ∞ , ∞ -Grpd. We remindthat the superscript “grpd” means that we are taking the maximal ∞ -subgroupoid in thecorresponding ∞ -category.1.6.2. Note that [Lu1, Corollary 3.3.3.2 ] provides a more explicit description of C . Namely,objects of C are Cartesian sections, i.e., assignments i ( c i ∈ C i ) , φ i,j ( c i ) α φi,j ≃ c j , equipped with data making α φ i,j coherently associative. In fact, this description followseasily from the above functorial description, by taking D = Vect, and using the factFunct cont (Vect , C ) ≃ C as DG categories.If c := ( c i , α φ i,j ) and e c := ( e c i , e α φ i,j ) are two such objects, then one can upgrade the assign-ment i M aps C i ( c i , e c i )into a homotopy I -diagram in Vect, and M aps C ( c , e c ) ≃ lim ←− i ∈ I M aps C i ( c i , e c i )as objects of Vect.1.6.3. The following observation will be useful in the sequel. Let C = lim ←− i ∈ I C i be as above, andlet ( α ∈ A ) ( c α ∈ C )be a collection of objects of C parameterized by some category A . In particular, for every i ∈ I we obtain a functor ( α ∈ A ) ( c i,α ∈ C i ) . We have:
HE CATEGORY OF D-MODULES ON Bun G Lemma 1.6.4.
For every i , the map from colim −→ α ∈ A c i,α ∈ C i to the i -th component of the object colim −→ α ∈ A c α is an isomorphism. In other words, colimits in a limit of DG categories can be computed component-wise.
Remark . The assertion of Lemma 1.6.4 can be reformulated as saying that the evaluationfunctors ev i : C → C i commute with colimits, i.e., are continuous. This is tautological fromthe definition of C as a limit in the category DGCat cont .1.7. Colimits in
DGCat cont . The goal of the remaining part of Sect. 1 is to provide a cate-gorical framework for Sects. 4.2-4.3. This material is not used in other parts of the article.1.7.1. As was mentioned in Sect. 1.6, limits in DGCat cont are the same as limits in DGCat.However, colimits are different (for example, the colimit taken in DGCat does not have to becocomplete).It is known to experts that under suitable set-theoretical conditions, colimits in DGCat cont always exist. We are unable to find a really satisfactory reference for this fact.On the other hand, in this paper we work only with colimits of those functorsΨ : I → DGCat cont that satisfy the following condition: for every arrow i → j in I the corresponding functor ψ i,j : Ψ( i ) → Ψ( j ) admits a continuous right adjoint . In this case existence of the colimit of Ψis provided by Proposition 1.7.5 below.1.7.2. The setting.
Let I be a small category, and let Ψ : I → DGCat cont be a functor i C i , ( i → j ) ∈ I ψ i,j ∈ Funct cont ( C i , C j ) . Assume that for every arrow i → j in I , the above functor ψ i,j admits a continuous rightadjoint, φ j,i .We can then view the assignment i C i , ( i → j ) ∈ I φ j,i ∈ Funct cont ( C j , C i ) . as a functor Φ : I op → DGCat cont . Remark . Some readers may prefer to assume, in addition, that each DG category C i is compactly generated. As explained in Sect. 1.9 below, this special case of the situation ofSect. 1.7.2 is very easy (Propositions 1.7.5 and 1.8.3 formulated below become obvious, andit is easy to understand “who is who”). Moreover, this case is enough for the applications inSects. 4.2-4.3.1.7.4. The following proposition is a variant of [Lu1, Corollary 5.5.3.4]; a digest of the proofis given in [GL:DG, Lemma 1.3.3]. Proposition 1.7.5.
In the situation of Sect. 1.7.2, the colimit colim −→ i ∈ I C i := colim −→ I Ψ ∈ DGCat cont exists and is canonically equivalent to the limit lim ←− i ∈ I op C i := lim ←− I op Φ ∈ DGCat cont ; the equivalence is uniquely characterized by the condition that for i ∈ I , the evaluation functor ev i : lim ←− i ∈ I op C i → C i is right adjoint to the tautological functor ins i : C i → colim −→ i ∈ I C i , in a way compatible with arrows in I . The above proposition can be reformulated as follows. Let C denote the limit of the DGcategories C i . The claim is that each functor ev i : C → C i admits a left adjoint functor ′ ins i : C i → C , and that the functors ′ ins i : C i → C together with the isomorphisms ′ ins j ◦ ψ i,j ≃ ′ ins i , ( i → j ) ∈ I, that one obtains by adjunction, make C into a colimit of the DG categories C i . Remark . Let I be filtered. In this case one can show (see [GL:DG, Lemma 1.3.6]) thatif an index i ∈ I is such that for every arrow i → i the functor ψ i ,i : C i → C i is fullyfaithful then the functor ins i is fully faithful. If C is compactly generated this follows fromLemma 1.9.5(ii) below.1.8. Colimits and duals. C i are dualizable. Then we can produce yet anotherfunctor Φ ∨ : I → DGCat cont that sends i C ∨ i , ( i → j ) ∈ I ( φ j,i ) ∨ ∈ Funct cont ( C ∨ i , C ∨ j ) . Proposition 1.8.3.
The category lim ←− i ∈ I op C i := lim ←− I op Φ is dualizable, and its dual is given by colim −→ i ∈ I C ∨ i := colim −→ I Φ ∨ . This identification is uniquely characterized by the property that for i ∈ I , we have (1.4) (ins i , Φ ∨ ) ∨ ≃ ev i , Φ , in a way compatible with arrows in I . In formula (1.4), the notation ins i , Ψ ∨ means the functorins i : C ∨ i → colim −→ I Φ ∨ , and the notation ev i , Φ means the functorev i : lim ←− I op Φ → C i . Remark . By adjunction between ins and ev (see Proposition 1.7.5), one gets from (1.4) asimilar isomorphism (ins i , Ψ ) ∨ ≃ ev i , Ψ ∨ . HE CATEGORY OF D-MODULES ON Bun G Colimits of compactly generated categories.
The main goal of this subsection is todemonstrate that the results of Sects. 1.7-1.8 are very easy under the additional assumptionthat each DG category C i is compactly generated.1.9.1. Who is who.
Suppose that in the situation of Sect. 1.7.2 each of the categories C i iscompactly generated, so C i ≃ Ind( C ci )or equivalently,(1.5) C i ≃ Funct(( C ci ) op , Vect) . By Proposition 1.2.4, the assumption that the functor ψ i,j : C i → C j has a continuous rightadjoint just means that ψ i,j ( C ci ) ⊂ C cj (so ψ i,j is the ind-extension of a functor ψ ci,j : C ci → C cj ).Moreover, the right adjoint functor φ j,i : C j → C i is just the restriction functorFunct(( C cj ) op , Vect) → Funct(( C ci ) op , Vect)corresponding to ψ ci,j : C ci → C cj .1.9.2. On Propositions 1.7.5 and 1.8.3 in the compactly generated case.
Set C := colim −→ i ∈ I C i . In our situation the existence of this colimit is clear: in fact,(1.6) C ≃ Ind( colim −→ i ∈ I C ci ) , where the colimit in the right hand side is computed in DGCat.Just as in Sect. 1.9.1, we can rewrite (1.6) as(1.7) C ≃ Funct( colim −→ i ∈ I ( C ci ) op , Vect) . Now the canonical equivalence C ≃ lim ←− i ∈ I op C i from Proposition 1.7.5 becomes obvious: this is just the composition C ≃ Funct( colim −→ i ∈ I ( C ci ) op , Vect) ≃ lim ←− i ∈ I op Funct(( C ci ) op , Vect) ≃ lim ←− i ∈ I op C i , where the first equivalence is (1.7) and the third one comes from (1.5).Proposition 1.8.3 says that C is dualizable and(1.8) C ∨ ≃ colim −→ i ∈ I C ∨ i . This is clear because by formula (1.6) and Sect. 1.5.3, both sides of (1.8) canonically identifywith Ind( colim −→ i ∈ I ( C ci ) op )(the colimit in this formula is computed in DGCat). Corollary 1.9.4.
In the situation of Sect. 1.9.1 the category C is compactly generated. Moreprecisely, objects of C of the form (1.9) ins i ( c ) , i ∈ I, c ∈ C ci are compact and generate C . (cid:3) In addition, one has the following lemma.
Lemma 1.9.5.
Suppose that in the situation of Sect. 1.9.1 the category I is filtered. Then (i) every compact object of C is of the form (1.9) ; (ii) for any i, i ′ ∈ I , c ∈ C ci , and c ′ ∈ C ci ′ the canonical map colim j, α : i → j, β : i ′ → j M aps C j ( ψ i,j ( c ) , ψ i ′ ,j ( c ′ )) → M aps C (ins i ( c ) , ins i ′ ( c ′ )) is an isomorphism.Proof. For statement (ii), see [Roz].Using (ii) and the assumption that I is filtered, it is easy to see that the class of objectsof the form (1.9) is closed under cones and direct summands. So (i) follows from (ii) andCorollary 1.4.6. (cid:3) Preliminaries on the DG category of D-modules on an algebraic stack
In this section we recall some definitions and results from [DrGa1].2.1.
D-modules on prestacks and algebraic stacks. Y be a prestack (always assumed locally of finite type). Recall following [DrGa1,Sect. 6.1] that the category D-mod( Y ) is defined as the limit(2.1) lim ←− S ∈ (Sch affft ) / Y D-mod( S ) , where the limit is taken in the ( ∞ , cont . Here S D-mod( S )is the functor (Sch affft ) op → DGCat cont were for f : S ′ → S the corresponding map D-mod( S ) → D-mod( S ′ ) is f ! .I.e., as was explained in Sect. 1.6.2, informally, an object F ∈ D-mod(Bun G ) is an assignmentfor every S → Y of an object F S ∈ D-mod( S ), and for every f : S ′ → S over Y of an isomorphism f ! ( F S ) ≃ F S ′ .In particular, for F , F ∈ D-mod(Bun G ), the complex M aps ( F , F ) is calculated as lim ←− S ∈ (Sch affft ) / Y M aps D-mod( S ) (( F ) S , ( F ) S ) . This definition has several variants. For example, we can replace the category of affineschemes by that of quasi-compact schemes, or all schemes.
HE CATEGORY OF D-MODULES ON Bun G Y is an Artin stack (see [GL:Stacks, Sect. 4] for our conventionsregarding Artin stacks).In this case, as in [Ga1, Corollary 11.2.3], in the formation of the limit in (2.1), we canreplace the category (Sch affft ) / Y by its non-full subcategory (Sch affft ) / Y , smooth , where we restrictobjects to be those pairs ( S, g : S → Y ) for which the map g is smooth, and 1-morphisms tosmooth maps between affine schemes.As before, we can replace the word “affine” by “quasi-compact”, or just consider all schemes.2.2. D-modules on a quasi-compact algebraic stack.
QCA and locally QCA stacks.
QCA is shorthand for “quasi-compact and with affine automorphism groups”.
Definition 2.2.2.
We say that an algebraic stack Y is locally QCA if the automorphism groupsof its field-valued points are affine. We say that Y is QCA if it is quasi-compact and locallyQCA.
Convention: in this article all stacks will be assumed to be locally QCA.
The reason is clearfrom Theorem 2.2.4 below.2.2.3.
A property of QCA stacks.
The following result is established in [DrGa1, Theorem 8.1.1].
Theorem 2.2.4.
Let Y be a QCA stack. Then the category D-mod( Y ) is compactly generated.Remark . In fact, [DrGa1, Theorem 8.1.1] produces an explicit set of compact generatorsof D-mod( Y ). These are objects induced from coherent sheaves on Y . Remark . Before [DrGa1], the above result was known for algebraic stacks that can berepresented as
Z/G , where Z is a quasi-compact scheme and G is an affine algebraic groupacting on S . Most quasi-compact Artin stacks that appear in practice (e.g., all quasi-compactopen substacks of Bun G ) admit such a representation. More generally, it was known for algebraicstacks that are perfect in the sense of [BFN].2.2.7. Cartesian products.
The following result is established in [DrGa1, Corollary 8.3.4].
Proposition 2.2.8.
Let Y and Y ′ be QCA stacks. Then the natural functor D-mod( Y ) ⊗ D-mod( Y ′ ) → D-mod( Y × Y ′ ) is an equivalence.Remark . In fact, as is remarked in the proof of [DrGa1, Corollary 8.3.4], the assertionof Proposition 2.2.8 is valid for any pair of prestacks Y and Y ′ as long as either D-mod( Y ) orD-mod( Y ′ ) is dualizable (see Sect. 1.5.1).2.2.10. Compactness and coherence.
Let Z be a quasi-compact scheme. An object of D-mod( Z )is said to be coherent if it is a bounded complex whose cohomology sheaves are coherent D-modules.It is known that the (non cocomplete) subcategory D-mod coh ( Z ) that consists of coherentobjects coincides with D-mod( Z ) c (see [DrGa1, Sect. 5.1.17]). Recall from Sect. 1.2.1 that fora DG category C we denote by C c the full subcategory of compact objects.For an algebraic stack Y , an object F ∈ D-mod( Y ) is said to be coherent if f ! ( F ) (or equiva-lently, f ∗ dR ( F )) is coherent for any smooth map f : Z → Y , where Z is a quasi-compact scheme. So by definition, the property of coherence is local for the smooth topology. The full (butnon-cocomplete) subcategory of coherent objects of D-mod( Y ) is denoted by D-mod coh ( Y ). Theorem 2.2.11.
Let Y be a QCA stack. (i) We have the inclusion
D-mod( Y ) c ⊂ D-mod coh ( Y ) . (ii) The above inclusion is an equality if and only if for every geometric point y of Y , thequotient of the automorphism group Aut( y ) by its unipotent radical is finite. This theorem is proved in [DrGa1, Lemma 7.3.3 and Corollary 10.2.7].
Remark . One may wonder how far coherence is from compactness. The answer is pro-vided by the notion of safety , introduced in [DrGa1, Sect. 9.2]. In [DrGa1, Proposition 9.2.3]it is shown that an object of D-mod coh ( Y ) is compact if and only if it is safe. Remark . Note that the notion of coherence of D-modules makes sense for any algebraicstack Y , i.e., it does not have to be quasi-compact: we test it by smooth maps Z → Y , where Z is a quasi-compact scheme. The inclusion of point (i) of Theorem 2.2.11 remains valid inthis context. The proof is very easy: for a map f : Z → Y , the functor f ∗ dR sends compacts tocompacts because it admits a continuous right adjoint, namely f dR , ∗ .2.2.14. Verdier duality.
Let Y be a QCA stack. Accoding to [DrGa1, Sect. 7.3.4], the (non-cocomplete) DG category D-mod coh ( Y ) carries a natural anti-involution D Verdier Y : (D-mod coh ( Y )) op → D-mod coh ( Y ) , which we refer to as Verdier duality .The following key feature of this functor is established in [DrGa1, Corollary 8.4.2]:
Theorem 2.2.15.
The functor D Verdier Y sends the subcategory (D-mod( Y ) c ) op ⊂ (D-mod coh ( Y )) op to D-mod( Y ) c ⊂ D-mod coh ( Y ) . D Verdier Y : (D-mod( Y ) c ) op → D-mod( Y ) c uniquely extends to an equivalence(2.2) D Verdier Y : D-mod( Y ) ∨ ≃ D-mod( Y ) . Alternatively, we can view the Verdier duality functor as follows. By Sect. 1.5.1, the DGcategory Funct cont (D-mod( Y ) , D-mod( Y )) identifies tautologically withD-mod( Y ) ∨ ⊗ D-mod( Y ) . The equivalence (2.2) is characterized by the property that the identity functor on D-mod( Y )corresponds to the object of D-mod( Y ) ⊗ D-mod( Y ) that identifies via Proposition 2.2.8 with(∆ Y ) dR , ∗ ( ω Y ) ∈ D-mod( Y × Y ) . Here ω Y ∈ D-mod( Y ) is the dualizing object and ∆ Y : Y → Y × Y is the diagonal.Let Y , Y be QCA stacks. If F : D-mod( Y ) → D-mod( Y ) is a continuous functor then thedual functor F ∨ : D-mod( Y ) ∨ → D-mod( Y ) ∨ (see Sect. 1.5.2) will be considered, via (2.2), asa functor D-mod( Y ) → D-mod( Y ).We will use the following fact [DrGa1, Proposition 8.4.8]. HE CATEGORY OF D-MODULES ON Bun G Proposition 2.2.17.
For any schematic quasi-compact morphism f : Y −→ Y , the functors f dR , ∗ : D-mod( Y ) → D-mod( Y ) , f ! : D-mod( Y ) → D-mod( Y ) are dual to each other in the sense of Sect. 1.5.2. Non quasi-compact algebraic stacks.
Let Y be now a stack, which is only assumedto be locally QCA. Then every quasi-compact open substack U ⊂ Y is QCA, so the categoryD-mod( U ) is compactly generated by Theorem 2.2.4. However, it is not true, in general, thatthe category D-mod( Y ) is compactly generated. For a counterexample, see Sect. 12.In this subsection we give a description of the subcategory of compact objectsD-mod( Y ) c ⊂ D-mod( Y ) , see Proposition 2.3.7 below.2.3.1. The category
D-mod( Y ) as a limit. The following statement immediately follows fromthe definition of D-mod( Y ), see Sect. 2.1.1. Lemma 2.3.2.
The restriction functor
D-mod( Y ) → lim ←− U ⊂ Y D-mod( U ) , is an equivalence, where the limit is taken over the poset of open quasi-compact substacks of Y . In particular, we obtain that for F , F ∈ D-mod( Y ), the natural map(2.3) M aps D-mod( Y ) ( F , F ) → lim ←− U ⊂ Y M aps D-mod( U ) ( F | U , F | U )is an isomorphism.The following observation will be useful in the sequel: Corollary 2.3.3.
Suppose that a family of objects F α ∈ D-mod( Y ) is locally finite, i.e., forevery quasi-compact open U ⊂ Y the set of α ’s such that F α | U = 0 is finite. Then the map ⊕ α F α → α Π F α is an isomorphism.Proof. Follows immediately from (2.3) and Lemma 1.6.4. (cid:3)
The functors j ∗ and j ! . Let U j ֒ → Y be an open substack. We have a pair of (continuous)adjoint functors j ∗ : D-mod( Y ) ⇄ D-mod( U ) : j ∗ . In particular, the functor j ∗ sends D-mod( Y ) c to D-mod( U ) c .Now, the functor j ∗ has a partially defined left adjoint, denoted j ! . It again follows auto-matically that if for F U ∈ D-mod( U ) c , the object j ! ( F U ) ∈ D-mod( Y ) is defined, then it iscompact.We claim: Lemma 2.3.5.
Let F U ∈ D-mod( U ) be such that j ! ( F U ) is defined. (a) The canonical map (2.4) F U → j ∗ ( j ! ( F U )) is an isomorphism. (b) If j ′ : U ′ ֒ → Y is another open substack, then ( j ′ ) ∗ ( j ! ( F U )) ≃ e j ! ( F U | U ∩ U ′ ) , (where e j : U ∩ U ′ ֒ → U ′ ). In particular, e j ! ( F U | U ∩ U ′ ) is defined.Proof. The functor j ∗ ◦ j ! is the partially defined left adjoint of j ∗ ◦ j ∗ , and the natural trans-formation Id → j ∗ ◦ j ! is obtained by adjunction from the co-unit j ∗ ◦ j ∗ → Id. However, thelatter is an isomorphism since j ∗ is fully faithful.Statement (b) follows similarly. (cid:3) A description of the subcategory
D-mod( Y ) c ⊂ D-mod( Y ) . Proposition 2.3.7.
An object F ∈ D-mod( Y ) is compact if and only if (2.5) F = j ! ( F U ) for some open quasi-compact U j ֒ → Y and some F U ∈ D-mod( U ) c . Formula (2.5) should be understood as follows: the partially defined functor j ! is defined on F U , and the resulting object is isomorphic to F . Remark . By Lemma 2.3.5(a), the object F U can be recovered from F as F | U := j ∗ ( F ).2.3.9. Proof of Proposition 2.3.7.
First, let us give two more reformulations of condition (2.5):
Lemma 2.3.10.
For F ∈ D-mod( Y ) the following conditions are equivalent: (1) F = j ! ( F U ) for some F U ∈ D-mod( U ) . (2) For any F ∈ D-mod( Y ) , supported on Y − U , we have Hom
D-mod( Y ) ( F , F ) = 0 . (3) For any U e j ֒ → U ′ j ′ ֒ → Y , where U ′ is another open quasi-compact substack of Y , we have: F | U ′ ≃ e j ! ( F U ) , in particular, the object e j ! ( F U ) is defined.Proof. By adjunction, (1) ⇔ (2). The implication (1) ⇒ (3) follows from Lemma 2.3.5(b).Let us show that (3) implies (2). By formula (2.3), for any F , F ∈ D-mod( Y ) one has(2.6) M aps D-mod( Y ) ( F , F ) ≃ lim ←− U ′ M aps D-mod( U ′ ) ( F | U ′ , F | U ′ ) . If F is supported on Y − U then all the terms in the RHS are zero, so the LHS is zero. (cid:3) Let us now prove Proposition 2.3.7.
Proof.
As was remarked in Sect. 2.3.4, if (2.5) holds then the compactness of F follows byadjunction.Conversely, suppose F ∈ D-mod( Y ) is compact. Then by Sect. 2.3.4, for every open U ⊂ Y the object F | U ∈ D-mod( U ) is compact. So it remains to show that (2.5) holds for somequasi-compact open U j ֒ → Y . HE CATEGORY OF D-MODULES ON Bun G Assume the contrary. Using the equivalence (1) ⇔ (3) of Lemma 2.3.10, we obtain that forevery quasi-compact open U ⊂ Y there is a quasi-compact open U ′ ⊂ Y containing U such that( j U,U ′ ) ! ( F | U ) = ( F | U ′ ) (here j U,U ′ : U ֒ → U ′ ).Thus, we obtain an increasing sequence of open quasi-compact substacks U i ⊂ Y suchthat ( j U i ,U i +1 ) ! ( F | U i ) = F | U i +1 . Therefore, by Lemma 2.3.10, for each i there exists E i ∈ D-mod( U i +1 ) such that E i | U i = 0 but Hom( F | U i +1 , E i ) = 0.Let V be the union of the U i ’s and let ˜ E i ∈ D-mod( V ) be the direct image of E i under U i ֒ → V . Then(2.7) Hom( F | V , ˜ E i ) = Hom( F | U i +1 , E i ) = 0 . By Corollary 2.3.3,(2.8) Hom( F | V , ⊕ i ˜ E i ) ≃ Y i Hom( F | V , ˜ E i ) . On the other hand, by Sect. 2.3.4, F | V is compact, so Hom( F | V , ⊕ i ˜ E i ) ≃ ⊕ i Hom( F | V , ˜ E i ). Thiscontradicts (2.8) because of (2.7). (cid:3) Truncativeness and co-truncativeness
Until the last subsection of this section we let Y be a QCA stack.3.1. The notion of truncative substack. Z i ֒ → Y be a closed substack, and let Y j ← ֓ U be the complementary open. Considerthe corresponding pairs of adjoint functors i dR , ∗ : D-mod( Z ) ⇄ D-mod( Y ) : i ! , j ∗ : D-mod( Y ) ⇄ D-mod( U ) : j ∗ . Recall that by Theorem 2.2.4, all the categories involved are compactly generated.
Proposition 3.1.2.
The following conditions are equivalent: (i)
The functor i ! sends D-mod( Y ) c to D-mod( Z ) c . (i ′ ) The functor i ! admits a continuous right adjoint. (ii) The functor j ∗ sends D-mod( U ) c to D-mod( Y ) c . (ii ′ ) The functor j ∗ admits a continuous right adjoint. (iii) The functor j ! , left adjoint to j ∗ , is defined on all of D-mod( U ) . (iii ′ ) The functor j ! , left adjoint to j ∗ , is defined on D-mod( U ) c . (iv) The functor i ∗ dR , left adjoint to i dR , ∗ , is defined on all of D-mod( Y ) . (iv ′ ) The functor i ∗ dR , left adjoint to i dR , ∗ , is defined on D-mod( Y ) c . Note that in the situation of (iii) and (iv) if the functors j ! and i ∗ dR are defined they areautomatically continuous by adjunction.To prove the proposition, we need the following lemma. Lemma 3.1.3.
The essential image of
D-mod( Y ) c under j ∗ : D-mod( Y ) → D-mod( U ) Karoubi-generates
D-mod( U ) c .Proof. Since j ∗ has a continuous right adjoint j ∗ , we have j ∗ (D-mod( Y ) c ) ⊂ D-mod( U ) c . Sincethe functor j ∗ is conservative j ∗ (D-mod( Y ) c ) generates D-mod( U ). By Corollary 1.4.6, thisimplies that j ∗ (D-mod( Y ) c ) Karoubi-generates D-mod( U ) c . (cid:3) Proof of Proposition 3.1.2.
Since j ∗ preserves compactness and i dR , ∗ is fully faithful and con-tinuous, the fact that (ii) implies (i) follows from the exact triangle i dR , ∗ ( i ! ( F )) → F → j ∗ ◦ ( j ∗ ( F )) . The implication (i) ⇒ (ii) follows from Lemma 3.1.3 and the same exact triangle.The equivalences (i) ⇔ (i ′ ) and (ii) ⇔ (ii ′ ) follow from (the tautological) Proposition 1.2.4.Let us show that (iii ′ ) ⇔ (iii) ⇔ (ii ′ ). The full subcategory of objects of D-mod( U ) on which j ! is defined is closed under colimits. Since D-mod( U ) is generated by D-mod( U ) c we see that(iii ′ ) ⇔ (iii). By Proposition 2.2.17, the dual of the functor j ∗ = j ! : D-mod( Y ) → D-mod( U )identifies, via the self-duality equivalences D Verdier U : D-mod( U ) ∨ ≃ D-mod( U ) , D Verdier Y : D-mod( Y ) ∨ ≃ D-mod( Y ) , with j ∗ : D-mod( U ) → D-mod( Y ). By duality (see Sect. 1.5.2), the existence of a continuousright adjoint to j ∗ is equivalent to the existence of (the automatically continuous) left adjointof j ∨∗ ≃ j ∗ . I.e., (iii) ⇔ (ii ′ ).Similarly to the above proof of (iii ′ ) ⇔ (iii) ⇔ (ii ′ ), one shows that (iv ′ ) ⇔ (iv) ⇔ (i ′ ). (cid:3) Definition 3.1.5.
A closed substack Z i ֒ → Y is called truncative (resp., an open substack U j ֒ → Y is called co-truncative ) if it satisfies the equivalent conditions of Proposition 3.1.2. Y ) c , D-mod( Z ) c , D-mod( U ) c .First, let Z i ֒ → Y be any closed substack, and let U j ֒ → Y be the complementary open, thenwe have an exact sequence of Karoubian (non-cocomplete) DG categories(3.1) 0 → D-mod( Z ) c ( i dR , ∗ ) c → D-mod( Y ) c ( j ∗ ) c → D-mod( U ) c → . The exactness of (3.1) follows from the fact that the corresponding sequence of the ind-completions 0 → D-mod( Z ) i dR , ∗ → D-mod( Y ) j ∗ → D-mod( U ) → Z ) c ⊂ D-mod( Y ) c is right-admissible , which by definition means that the functor( i dR , ∗ ) c = ( i ! ) c : D-mod( Z ) c → D-mod( Y ) c admits a right adjoint ( i ! ) c : D-mod( Y ) c → D-mod( Z ) c , or equivalently, the functor( j ∗ ) c : D-mod( Y ) c → D-mod( U ) c has a right adjoint ( j ∗ ) c : D-mod( U ) c → D-mod( Y ) c .Similarly, conditions (iii)-(iv) from Proposition 3.1.2 say that the subcategoryD-mod( Z ) c ⊂ D-mod( Y ) c By definition, exactness means that i c ∗ identifies D-mod( Z ) c with a full subcategory of D-mod( Y ) c , and ( j ∗ ) c identifies the Karoubi-completion of the quotient D-mod( Y ) / D-mod( Z ) c with D-mod( U ) c . Synonyms: right-admissible=coreflective, left-admissible=reflective.
HE CATEGORY OF D-MODULES ON Bun G is left-admissible , which by definition means that the functor( i dR , ∗ ) c : D-mod( Z ) c → D-mod( Y ) c admits a left adjoint ( i ∗ dR ) c : D-mod( Y ) c → D-mod( Z ) c , or, equivalently, the functor( j ∗ ) c : D-mod( Y ) c → D-mod( Z ) c has a left adjoint ( j ! ) c : D-mod( Z ) c → D-mod( Y ) c .In our situation left admissibility is equivalent to right admissibility by Verdier duality.Thus if i : Z ֒ → Y is truncative then in addition to (3.1) one has the exact sequences(3.2) 0 → D-mod( U ) c ( j ! ) c → D-mod( Y ) c ( i ∗ dR ) c → D-mod( Z ) c → , (3.3) 0 → D-mod( U ) c ( j ∗ ) c → D-mod( Y ) c ( i ! ) c → D-mod( Z ) c → . It is convenient to arrange the functors between D-mod( Z ) c and D-mod( Y ) c into a sequence(3.4) ( i ∗ dR ) c , ( i dR , ∗ ) c , ( i ! ) c and the functors between D-mod( U ) and D-mod( Y ) into a sequence(3.5) ( j ! ) c , ( j ∗ ) c , ( j ∗ ) c . In each of the sequences, each neighboring pair forms an adjoint pair of functors.3.2.
Some examples of (co)-truncative substacks. j : U ֒ → Y is an open embedding of schemes whichis not a closed embedding then U cannot be co-truncative. Indeed, choose M ∈ Coh( U ) suchthat j ∗ ( M ) is not coherent. Then j ∗ ( ind D-mod( U ) ( M )) ≃ ind D-mod( Y ) ( j ∗ ( M ))is not in D-mod( Y ) c . Here ind D-mod( − ) denotes the induction functor from IndCoh( − ) toD-mod( − ), see [DrGa1, Sect. 5.1.3].3.2.2. Example.
The following example of a co-truncative substack is most important for us:Take Y = A n / G m , where G m acts on A n by dilations. Take U = ( A n − { } ) / G m ≃ P n − . InSect. 5 we will see that U ֒ → Y is co-truncative.3.2.3. The most basic case of the above example is when n = 1. In this case, the co-truncativeness assertion is particularly evident. Namely, let us check that condition (iii ′ ) ofProposition 3.1.2 holds. Indeed, D-mod( U ) ≃ Vect, so it is sufficient to show that j ! ( k ) isdefined, where k is the generator of Vect. This is clear since we are dealing with holonomicD-modules.3.2.4. Here is a generalization of the example of Sect. 3.2.3 in a direction different fromSect. 3.2.2: if Y is any QCA stack that has only finitely many isomorphism classes of k -pointsthen every open substack U ⊂ Y is co-truncative. Indeed, condition (iii ′ ) of Proposition 3.1.2is verified because every object of D-mod( U ) c is holonomic.Examples of such Y include N \ G/B , or any quasi-compact open of Bun G for X of genus 0. The non-standard functors.
Let Z i ֒ → Y be a truncative closed substack and U j ֒ → Y the corresponding co-truncative open substack. Definition 3.3.1.
The functors right adjoint to i ! and j ∗ are denoted by i ? : D-mod( Z ) → D-mod( Y ) , j ? : D-mod( Y ) → D-mod( U ) . Remark . The proof of the equivalences (iii) ⇔ (ii ′ ) and (iv) ⇔ (i ′ ) from Proposition 3.1.2shows that i ? is the dual to i ∗ dR : D-mod( Y ) → D-mod( Z ) and j ? is the dual to j ! : D-mod( U ) → D-mod( Y ) in the sense of Sect. 1.5.2. Recall that these dualities follow from the duality between i ! and i dR , ∗ , and between j ∗ and j ∗ . Remark . Recall that the existence of i ? and/or j ? as a continuous functor is among theequivalent definitions of truncativeness, see Definition 3.1.5 and Proposition 3.1.2(i ′ , ii ′ ).The existence of i ∗ dR and/or j ! as an everywhere defined (and automatically continuous)functor is also among the equivalent definitions of truncativeness, see Proposition 3.1.2(iii, iv).The functors i ? , j ? , i ∗ dR , j ! are called the non-standard functors associated to Z ⊂ Y (or to U ⊂ Y ).The functors i ∗ and j ! are at least, familiar as partially defined functors (e.g., they arealways defined on the holonomic subcategory), but i ? and j ? are quite unfamiliar. On the otherhand, in some situations the non-standard functors identify with certain standard functors, seeExample 3.3.9 and Remark 3.3.10 below.3.3.4. Inventory.
It is convenient to arrange the functors between D-mod( Z ) and D-mod( Y )into a sequence(3.6) i ∗ dR , i dR , ∗ , i ! , i ? and the functors between D-mod( U ) and D-mod( Y ) into a sequence(3.7) j ! , j ∗ , j ∗ , j ? . In each of the sequences, each neighboring pair forms an adjoint pair of functors. The firstand last functors in (3.6) and in (3.7) are non-standard, the other functors are standard. ByRemark 3.3.2, each of the sequences (3.6)-(3.7) is self-dual in the sense of Sect. 1.5.2.3.3.5. We know that the functors i dR , ∗ and j ∗ are fully faithful; equivalently, the adjunctions(3.8) i ∗ dR ◦ i dR , ∗ → Id D-mod( Z ) , Id D-mod( U ) → j ? ◦ j ∗ are isomorphisms (just as are the adjunctions j ∗ ◦ j ∗ → Id D-mod( U ) and Id D-mod( Z ) → i ! ◦ i dR , ∗ ,which involve only the standard functors). Proposition 3.3.6. (i)
The functors i ? and j ! are fully faithful. (ii) The adjunctions i ! ◦ i ? → Id D-mod( Z ) and Id D-mod( U ) → j ∗ ◦ j ! are isomorphisms. Although this proposition is extremely simple, we will give two proofs.
Proof 1.
Statements (i) and (ii) are clearly equivalent, so it suffices to prove (ii).Recall that the adjoint pairs ( i ! , i ? ) and ( j ∗ , j ? ) are dual to the adjoint pairs ( i ∗ dR , i dR , ∗ ) and( j ! , j ∗ ). So statement (ii) follows from the fact that the adjunctions (3.8) are isomorphisms. (cid:3) HE CATEGORY OF D-MODULES ON Bun G Proof 2.
We will deduce statement (i) from the following general lemma, which is part of thecategorical folklore. Lemma 3.3.7.
Let F be a functor between ∞ -categories that admits a left adjoint F L and aright adjoint F R . Then F L is fully faithful if and only if F R is. Let us apply Lemma 3.3.7 to F := j ∗ . Since ( j ∗ ) R = j ∗ is fully faithful, we obtain that( j ∗ ) L = j ! is fully faithful.Let us apply Lemma 3.3.7 to F := i ! . Since ( i ! ) L = i dR , ∗ is fully faithful, we obtain that( i ! ) R = i ? is fully faithful. (cid:3) Z ⊂ Y is truncative, one has canonical exact se-quences of DG categories(3.9) 0 → D-mod( Z ) i dR , ∗ → D-mod( Y ) j ∗ → D-mod( U ) → → D-mod( U ) j ∗ → D-mod( Y ) i ! → D-mod( Z ) → , where the latter is obtained from the former by passing to right adjoints.If Z is truncative one also has exact sequences(3.11) 0 → D-mod( Z ) i ? → D-mod( Y ) j ? → D-mod( U ) → → D-mod( U ) j ! → D-mod( Y ) i ∗ dR → D-mod( Z ) → , where (3.11) is obtained by passing to right adjoints from (3.10), and (3.12) is obtained bypassing to left adjoints from (3.9).In addition, (3.9) and (3.10) are obtained from one another by passing to the dual categoriesand functors. Similarly, (3.12) and (3.11) are obtained from one another by passing to the dualcategories and functors.3.3.9. Example.
Consider the situation of Sect. 3.2.3, i.e., the embedding i : Z ֒ → Y , where Y = A / G m , Z = { } / G m . Let π : Y → Z be the morphism induced by the map A → { } . Letus show that the non-standard functors i ∗ dR : D-mod( Y ) → D-mod( Z ) and i ? : D-mod( Z ) → D-mod( Y ) identify with the following standard functors: i ∗ dR ≃ π dR , ∗ , i ? ≃ π ! ;in other words, ( π dR , ∗ , i dR , ∗ ) and ( i ! , π ! ) are adjoint pairs. By Proposition 2.2.17, π dR , ∗ is dualto π ! and i dR , ∗ is dual to i ! , so it suffices to show that ( π dR , ∗ , i dR , ∗ ) is an adjoint pair. Let usprove that for any M ∈ D-mod( Y ), N ∈ D-mod( Z ) the map π dR , ∗ : Hom( M , i dR , ∗ ( N )) → Hom( π dR , ∗ ( M ) , π dR , ∗ ◦ i dR , ∗ ( N )) = Hom( π dR , ∗ ( M ) , N )is an isomorphism. Lemma 3.3.7 for ∞ -categories immediately follows from the same statement for usual categories. For proofsin the setting of usual categories, see [DT, Lemma 1.3], [KeLa, Proposition 2.3], and the article on adjoint triplesfrom [nLab] (on the other hand, the reader can easily reconstruct the argument because we essentially used itin the proof of Lemma 2.3.5(a)). Note that in the case of triangulated categories and functors (which is enoughfor our purpose) Lemma 3.3.7 is well known. This is clear if M ∈ D-mod( Z ) ⊂ D-mod( Y ). The DG category D-mod( Y ) is generated byD-mod( Z ) and j ! ( k ), where j : pt = Y − Z ֒ → Y is the open embedding. So it remains to considerthe case M = j ! ( k ). Then Hom( M , i dR , ∗ ( N )) = 0 and π dR , ∗ ( M ) = 0 (the latter follows from thefact the de Rham cohomology of A equals k ). Remark . Example 3.3.9 is a “baby case” of Proposition 5.3.2.3.4.
Truncativeness of locally closed substacks.
Let Z i ֒ → Y be a locally closed substack.This means that i becomes a locally closed embedding after any base change Y → Y , where Y is a scheme (in fact, it suffices to verify this condition for just one smooth or flat covering Y → Y ). Definition 3.4.1.
A locally closed substack Z i ֒ → Y is said to be truncative if the functor i ! preserves compactness (or equivalently, has a continuous right adjoint functor i ? ). For instance, any open substack is truncative.3.4.2. Definition 3.4.1 immediately implies that truncativeness is transitive:
Lemma 3.4.3.
Let Y ֒ → Y ֒ → Y be locally closed embeddings. If Y is truncative in Y and Y is truncative in Y , then Y is truncative in Y . (cid:3) As in the case of schemes, every locally closed embedding Z ֒ → Y can be factored (and evencanonically so) as(3.13) Z i ′ ֒ → Y ′ j ֒ → Y , where i ′ is a closed embedding, and j is an open embedding. Namely, Y ′ := Y − (¯ Z − Z ), where¯ Z is the closure of Z in Y (so that Z is open in ¯ Z ). Lemma 3.4.4.
A locally closed substack Z i ֒ → Y is truncative if and only if for some/anyfactorization (3.13) with i ′ being closed and j open, Z in truncative in Y ′ .Proof. The “if” statement follows from Lemma 3.4.3. It remains to show that if the composition(3.13) is truncative then so is Z i ′ ֒ → Y ′ . This follows from the fact that the essential image ofD-mod( Y ) c under j ∗ Karoubi-generates D-mod( Y ′ ) c , see Lemma 3.1.3. (cid:3) Remark . In the case of locally closed substacks the situation with the non-standard func-tors is as follows. By duality (in the sense of Sect. 1.5.2), a locally closed substack Z i ֒ → Y istruncative if and only if the functor i dR , ∗ has a left adjoint functor i ∗ dR (which is automaticallycontinuous).Thus for a truncative locally closed substack we have adjoint pairs of continuous functors( i ∗ dR , i dR , ∗ ) and ( i ! , i ? ) dual to each other. Just as in the case of closed embeddings (seeProposition 3.3.6), the functors i dR , ∗ and i ? are fully faithful; equivalently, the adjunctions i ! ◦ i ? → Id D-mod( Z ) and Id D-mod( Z ) → i dR , ∗ ◦ i ∗ dR are isomorphisms. But if the substack Z isnot closed then i dR , ∗ = i ! , so the functors i dR , ∗ and i ! do not form an adjoint pair.3.5. Truncativeness via coherence.
HE CATEGORY OF D-MODULES ON Bun G Proposition 3.5.2. (a)
A locally closed substack Z i ֒ → Y is truncative if and only if the functor i ! sends D-mod coh ( Y ) to D-mod coh ( Z ) . (b) An open substack U j ֒ → Y is co-truncative if and only if j ∗ sends D-mod coh ( U ) to D-mod coh ( Y ) .Proof. To prove the “if” implications in both (a) and (b) we will use the notion of safety from[DrGa1, Sect. 9.2], and the fact that for a morphism f : Y → Y between QCA stacks, thefunctor f dR , ∗ always preserves safety, and f ! preserves safety if f itself is safe (in particular,when f is schematic); see [DrGa1, Lemma 10.4.2].Thus, the “if” implications follow from the fact that “compactness=coherence+safety”, see[DrGa1, Proposition 9.2.3].To prove the “only if” implication in (a), we will use the following result (see [DrGa1, Lem-ma 9.4.7(a)]): Lemma 3.5.3.
For a QCA stack Y , an object F ∈ D-mod coh ( Y ) and an integer n , there exists F ′ ∈ D-mod( Y ) c and a map F ′ → F , such that its cone lies in D-mod( Y ) < − n . (cid:3) Note that the functor i ! is left t-exact, and has a finite cohomological amplitude, say k . For F ∈ D-mod coh ( Y ), which lies in D-mod( Y ) ≥− m , choose F ′ as in Lemma 3.5.3 with n > k + m .Consider the exact triangle i ! ( F ′ ) → i ! ( F ) → i ! ( F ′′ ) , where F ′′ := Cone( F ′ → F ). By construction, the maps(3.14) τ ≥− m ( i ! ( F ′ )) → τ ≥− m ( i ! ( F )) → i ! ( F )are isomorphisms.By assumption, i ! ( F ′ ) ∈ D-mod( Z ) c ⊂ D-mod coh ( Z ). Note also that the truncation functorspreserve the subcategory D-mod coh ( − ). Hence τ ≥− m ( i ! ( F ′ )) ∈ D-mod coh ( Z ). Hence, (3.14)implies that i ! ( F ) ∈ D-mod coh ( Z ), as desired.The “only if” implication in (b) is proved similarly. (cid:3) Stability of truncativeness.
In this subsection i : Z ֒ → Y denotes a locally closedembedding.3.6.1. Cartesian products.
Lemma 3.6.2.
Suppose that a substack Z i ֒ → Y is truncative. Then for any QCA stack X , thesubstack Z × X ֒ → Y × X is also truncative.Proof. By [DrGa1, Corollary 8.3.4], for a pair of QCA stacks X and X , the natural functorD-mod( X ) ⊗ D-mod( X ) → D-mod( X × X )is an equivalence. So the functor ( i × id X ) ! : D-mod( Y × X ) → D-mod( Z × X ) identifies withthe functor i ! ⊗ Id D-mod( X ) , which clearly preserves compactness. (cid:3) Descent.
Proposition 3.6.4.
Let Z ⊂ Y be a locally closed substack, f : e Y → Y a smooth morphism, and e Z ⊂ Z × Y e Y an open substack such that the resulting morphism f ′ : e Z → Z is surjective. If thelocally closed embedding e i : e Z ֒ → e Y is truncative then so is i : Z ֒ → Y .Proof. By Proposition 3.5.2(a), it suffices to show that i ! sends D-mod coh ( Y ) to D-mod coh ( Z ).The morphism f ′ is smooth and surjective, so it suffices to show that the functor f ′ ! ◦ i ! preservescoherence. But f ′ ! ◦ i ! ≃ e i ! ◦ f ! , and each of the functors e i ! and f ! preserves coherence. (cid:3) Corollary 3.6.5.
Let Z ⊂ Y be a locally closed substack. Suppose that each z ∈ Z has a Zariskineighborhood U ⊂ Y such that Z ∩ U is truncative in U . Then Z is truncative in Y . (cid:3) Remark . The converse to Proposition 3.6.4 is false: truncativeness downstairs does not imply truncativeness upstairs (e.g., consider the embedding pt / G m ֒ → A / G m smoothly cov-ered by pt ֒ → A ). However, the converse to Proposition 3.6.4 does hold for ´etale schematicmorphisms; this follows from Lemma 3.6.9 below. Lemma 3.6.7.
Suppose that in a Cartesian diagram e Z e i −−−−→ e Y f ′ y y f Z i −−−−→ Y f is schematic, proper and surjective, and i a locally closed embedding. If e Z is truncative in e Y then Z is truncative in Y .Proof. First, by [DrGa1, Lemma 5.1.6], the functor f ! is conservative. Hence, the essentialimage of f dR , ∗ generates D-mod( Y ). Hence, by Corollary 1.4.6, the essential image of D-mod( e Y ) c under f dR , ∗ Karoubi-generates D-mod( Y ) c . Therefore, it is sufficient to show that the functor i ! ◦ f dR , ∗ preserves compactness. But i ! ◦ f dR , ∗ ≃ f ′ dR , ∗ ◦ e i ! , the functor e i ! preserves compactnessby assumption, and f ′ dR , ∗ preserves compactness by properness (it has a continuous right adjointgiven by ( f ′ ) ! ). (cid:3) Quasi-finite base change.
Lemma 3.6.9.
Suppose that f : e Y → Y is ´etale and schematic. If a locally closed embedding i : Z ֒ → Y is truncative then so is e i : Z × Y e Y ֒ → e Y .Proof. The functor f dR , ∗ : D-mod( e Y ) → D-mod( Y ) is conservative. So by Corollary 1.4.6, theessential image of D-mod( Y ) c under f ∗ dR ≃ f ! Karoubi-generates D-mod( e Y ) c . So it is enough toshow that e i ! ◦ f ! preserves compactness. However, e i ! ◦ f ! ≃ f ′ ! ◦ i ! . Now, i ! preserves compactnessby assumption, and f ′ ! preserves compactness because it is isomorphic to ( f ′ ) ∗ dR , which is theleft adjoint of a continuous functor, namely, f ′ dR , ∗ . (cid:3) Lemma 3.6.10. If f : e Y ֒ → Y is a locally closed embedding and a locally closed substack Z ֒ → Y is truncative then so is Z × Y e Y ֒ → e Y .Proof. If f is an open embedding the statement holds by Lemma 3.6.9. If f is a closed embed-ding use the fact that an object F ∈ D-mod( e Y ) is compact if and only if f dR , ∗ ( F ) ∈ D-mod( Y )is; this follows from the fact that the functor f dR , ∗ is fully faithful and continuous. (cid:3) HE CATEGORY OF D-MODULES ON Bun G Lemma 3.6.10 for a closed embedding f admits the following generalization. Proposition 3.6.11.
Let f : e Y → Y be a finite schematic morphism. If a locally closedembedding i : Z ֒ → Y is truncative then so is e i : Z × Y e Y ֒ → e Y . To prove the proposition, we need the following lemma.
Lemma 3.6.12.
Let g : X ′ → X be a finite schematic morphism. If F ′ ∈ D-mod( X ′ ) is suchthat g dR , ∗ ( F ′ ) ∈ D-mod( X ) is coherent then F ′ is coherent.Proof. Follows immediately from the fact that the functor g dR , ∗ is t-exact and conservative. (cid:3) Proof of Proposition 3.6.11.
We have to show that the functor e i ! preserves coherence. ApplyingLemma 3.6.12 to the morphism f ′ : Z × Y e Y → Z , we see that it suffices to prove that thecomposition f ′ dR , ∗ ◦ e i ! preserves coherence. But f ′ dR , ∗ ◦ e i ! ≃ i ! ◦ f dR , ∗ and each of the functors i ! and f dR , ∗ preserves coherence. (cid:3) Remark . One can combine Lemma 3.6.9 and Proposition 3.6.11 to the following state-ment: the assertion of Proposition 3.6.11 continues to hold when f is a quasi-finite compactifi-able morphism.3.7. Intersections and unions of truncative substacks.Lemma 3.7.1. If Z and Z are locally closed truncative substacks of Y , then so is Z ∩ Z .Proof. By Lemma 3.6.10, Z ∩ Z is truncative in Z . Now, the assertion follows fromLemma 3.4.3. (cid:3) Proposition 3.7.2.
Suppose that a locally closed substack Z ⊂ Y is equal to the union of(possibly intersecting) locally closed substacks Z i , i = 1 , ..., n . If each Z i is truncative in Y , thenso is Z . First, let us prove the following particular case of Proposition 3.7.2.
Lemma 3.7.3.
Let Z ′ ֒ → Z ֒ → Y be closed embeddings. If Z ′ and Z − Z ′ are truncative in Y then so is Z .Proof. Consider the open substacks Y − Z ⊂ Y − Z ′ ⊂ Y . The fact that Z ′ is truncative in Y means, by definition, that Y − Z ′ is co-truncative in Y . By Lemma 3.4.4, the fact that Z − Z ′ is truncative in Y implies that Z − Z ′ is truncative in Y − Z ′ , i.e., that Y − Z is co-truncative in Y − Z ′ . But the relation of co-truncativeness is transitive: this is clear if one uses property (ii)from Proposition 3.1.2 as a definition of co-truncativeness. So Y − Z is co-truncative in Y , i.e., Z is truncative in Y . (cid:3) Proof of Proposition 3.7.2.
We proceed by induction on n .By Corollary 3.6.5, it suffices to show that each z ∈ Z has a Zariski neighborhood U ⊂ Y such that Z ∩ U is truncative in U . Choose i so that z ∈ Z i . After replacing Z by an openneighborhood of z , one can assume that Z i and Z are closed in Y .Writing Z − Z i as a union of the substacks Z j − ( Z i ∩ Z j ), j = i , and applying the inductionassumption, we see that Z − Z i is truncative in Y − Z i and therefore in Y . It remains to applyLemma 3.7.3 to Z i ֒ → Z ֒ → Y . (cid:3) Truncativeness and co-truncativeness for non quasi-compact stacks.
Now sup-pose that Y is locally QCA (but not necessarily quasi-compact).
Definition 3.8.2. (i)
A locally closed substack Z ֒ → Y is said to be truncative if for every open quasi-compactsubstack ◦ Y ⊂ Y the intersection Z ∩ ◦ Y is truncative in ◦ Y . (ii) An open substack U ⊂ Y is said to be co-truncative if for every open quasi-compact substack ◦ Y ⊂ Y the intersection U ∩ ◦ Y is co-truncative in ◦ Y . Z is truncative if and only if its complementary open isco-truncative.In addition: Lemma 3.8.4.
If open substacks U , U ⊂ Y are co-truncative then so is U ∪ U .Proof. This immediately follows from Lemma 3.7.1. (cid:3) U is co-truncative if and only if the functor j ! , left adjoint to j ∗ , is defined.This formally implies that if i : Z ֒ → Y is truncative, then the functor i ∗ dR , left adjoint to i dR , ∗ , is also defined.3.8.6. Finally, we note: Lemma 3.8.7.
For a co-truncative open quasi-compact substack U j ֒ → Y the functor j ! : D-mod( U ) → D-mod( Y ) is fully faithful.Proof. Follows from Lemma 2.3.5(a). (cid:3) Truncatable stacks
Let Y be an algebraic stack which is locally QCA. In this setting the notions of truncativenessand co-truncativeness were introduced in Sect. 3.8.4.1. The notion of truncatibility.
We will now formulate a condition on Y called “trun-catibility”. According to Proposition 4.1.6 below, it implies that the category D-mod( Y ) iscompactly generated. Definition 4.1.1.
The stack Y is said to be truncatable if it can be covered by open quasi-compact substacks that are co-truncative. Lemma 4.1.3.
A stack Y is truncatable if and only if every open quasi-compact substack iscontained in one which is co-truncative. Equivalently, Y is truncatable if and only if the sub-poset of co-truncative open quasi-compact substacks in Y is cofinal among all open quasi-compactsubstacks. HE CATEGORY OF D-MODULES ON Bun G Notation.
The poset of co-truncative open quasi-compact substacks U ⊂ Y is denoted byCtrnk( Y ); we will often consider this poset as a category. Let Ctrnk( Y ) op denote the oppositeposet (or category). Lemma 3.8.4 implies that Ctrnk( Y ) is filtered .The next statement immediately follows from Lemma 4.1.3. Corollary 4.1.5. If Y is truncatable then the natural restriction functor D-mod( Y ) → lim ←− U ∈ Ctrnk( Y ) op D-mod( U ) is an equivalence. Proposition 4.1.6. If Y is truncatable then the category D-mod( Y ) is compactly generated.Proof. Let U j ֒ → Y be a co-truncative open quasi-compact substack and F U ∈ D-mod( U ) c . ByProposition 2.3.7, the object j ! ( F U ) ∈ D-mod( Y ) (which is well-defined by the co-truncativenessassumption) is compact. It suffices to show that such objects generate D-mod( Y ). In otherwords, we have to show that if F ∈ D-mod( Y ) is right-orthogonal to all such objects, then F = 0.For a given U , the fact that F is right-orthogonal to all j ! ( F U ) as above is equivalent,by adjunction, to the fact that j ∗ ( F ) is right-orthogonal to D-mod( U ) c . Since D-mod( U ) iscompactly generated, this implies that j ∗ ( F ) = 0. By Corollary 4.1.5, this implies that F = 0. (cid:3) G )) from the followingresult: Theorem 4.1.8.
Let G be a connected reductive group and X a smooth complete connectedcurve over k . Let Bun G denote the stack of G -bundles on X . Then Bun G is truncatable. The proof for any connected reductive group G will be given in Sect. 9. But its main idea isthe same as in the easy case G = SL , which is considered separately in Sect. 6.4.1.9. In Sects. 4.2-4.5 below we discuss some general properties of the category D-mod( Y )for a truncatable stack Y .4.2. Presentation as a colimit.
In this subsection we fix Y to be a truncatable locally QCAstack. We will use the notation Ctrnk( Y ) from Sect. 4.1.4.4.2.1. Note that for a morphism U j , ֒ → U in Ctrnk( Y ), the pullback functor φ U ,U := j ∗ , : D-mod( U ) → D-mod( U )admits a left adjoint, ψ U ,U := ( j , ) ! : D-mod( U ) → D-mod( U ).Hence, we are in the situation of Sect. 1.7.2 with I = Ctrnk( Y ). In fact, we are in the morerestrictive (and possibly more understandable) situation of Sects. 1.7.3 and 1.9. Corollary 4.2.3.
The category
D-mod( Y ) is canonically equivalent to colim −→ U ∈ Ctrnk( Y ) D-mod( U ) , where the functor Ctrnk( Y ) → DGCat cont is U D-mod( U ) , ( U j , ֒ → U ) ( j , ) ! . Under this equivalence, for a co-truncative open quasi-compact substack U j ֒ → Y , the functor ins U : D-mod( U ) → colim −→ U ∈ Ctrnk( Y ) D-mod( U ) ≃ D-mod( Y ) , is ( j ) ! .Remark . Note that the assertion of Proposition 2.3.7 for a truncative QCA stack Y followsalso from Lemma 1.9.5(i). Note also that the assertion of Lemma 2.3.5 for U (resp., U and U ′ )co-truncative is a particular case of Remark 1.7.6.4.3. Description of the dual category.
Corollary 4.3.2.
The category
D-mod( Y ) is dualizable. Its dual category is canonically equiv-alent to (4.1) colim −→ U ∈ Ctrnk( Y ) D-mod( U ) , where the functor Ctrnk( Y ) → DGCat cont is (4.2) U D-mod( U ) , ( U j , ֒ → U ) ( j , ) ∗ . Under this equivalence, for a co-truncative open quasi-compact substack U j ֒ → Y , the functor ins U : D-mod( U ) → colim −→ U ∈ Ctrnk( Y ) D-mod( U ) is the dual of restriction functor j ∗ : D-mod( Y ) → D-mod( U ) .Proof. Follows from Proposition 2.2.17. (cid:3)
Notation.
The category (4.1) that appears in Corollary 4.3.2 will be denoted byD-mod( Y ) co . The equivalence of Corollary 4.3.2 will be denoted by(4.3) D Verdier Y : D-mod( Y ) ∨ ≃ D-mod( Y ) co . Note that when Y is quasi-compact, this is the same as the equivalence of (2.2). HE CATEGORY OF D-MODULES ON Bun G Y ) co also as alimit: Corollary 4.3.5.
The category
D-mod( Y ) co is canonically equivalent to lim ←− U ∈ Ctrnk( Y ) op D-mod( U ) , where the functor Ctrnk( Y ) op → DGCat cont is U D-mod( U ) , ( U j , ֒ → U ) j ?1 , . U j ֒ → Y , we havea canonically defined functor D-mod( U ) → D-mod( Y ) co . We denote this functor by j co , ∗ . By construction, in terms of the identifications D Verdier U : D-mod( U ) ∨ ≃ D-mod( U ) and D Verdier Y : D-mod( Y ) ∨ ≃ D-mod( Y ) co , we have ( j co , ∗ ) ∨ ≃ j ∗ . Similarly, from Corollary 4.3.5, we have a canonically defined functor j ? : D-mod( Y ) co → D-mod( U ) , which is the dual of j ! : D-mod( U ) → D-mod( Y ), and the right adjoint of j co , ∗ .4.3.7. We claim: Lemma 4.3.8.
The functor j co , ∗ is fully faithful.Proof. We need to show that the unit of the adjunction Id
D-mod( U ) → j ? ◦ j co , ∗ is an isomorphism.This is obtained by passing to dual functors (see Sect. 1.5.2) in the mapId D-mod( U ) → j ∗ ◦ j ! , which is an isomorphism by Lemma 3.8.7. (cid:3) Remark . Note that Lemma 4.3.8 follows more abstractly from Remark 1.7.6. However,this way to deduce Lemma 4.3.8 is equivalent to the proof given above in view of Remark 4.2.4.4.3.10. We claim that the category D-mod( Y ) co is compactly generated and that its compactobjects are ones of the form j co , ∗ ( F U ) for F U ∈ D-mod( U ) c , where U is a co-truncative quasi-compact open substack of Y .This follows from Proposition 2.3.7 and Sect. 1.5.3.Alternatively, this follows from Corollary 1.9.4 and Lemma 1.9.5(i).4.4. Relation between the category and its dual.
In this subsection we continue to assumethat Y is a truncatable locally QCA stack. cont (D-mod( Y ) co , D-mod( Y ))identifies canonically with(D-mod( Y ) co ) ∨ ⊗ D-mod( Y ) ≃ D-mod( Y ) ⊗ D-mod( Y ) . In addition, by Proposition 2.2.8 and Remark 2.2.9, we haveD-mod( Y ) ⊗ D-mod( Y ) ≃ D-mod( Y × Y ) . Thus, every object Q ∈ D-mod( Y × Y ) defines a functor F Q : D-mod( Y ) co → D-mod( Y ) . The naive functor.
Note that if Y is quasi-compact we have a tautological equivalenceD-mod( Y ) co ≃ D-mod( Y ) . Recall from Sect. 2.2.14 that the corresponding object in D-mod( Y × Y ) is (∆ Y ) dR , ∗ ( ω Y ).For any truncatable Y the functor D-mod( Y ) co → D-mod( Y ) corresponding to(∆ Y ) dR , ∗ ( ω Y ) ∈ D-mod( Y × Y )will be denoted by Ps-Id Y , naive : D-mod( Y ) co → D-mod( Y )(here Ps-Id stands for “pseudo-identity”).Let D Verdier Y , naive : D-mod( Y ) ∨ → D-mod( Y ) denote the compositionD-mod( Y ) ∨ D Verdier Y ≃ D-mod( Y ) co Ps-Id Y , naive −→ D-mod( Y ) . An alternative description.
Here is a tautologically equivalent description of the functorPs-Id Y , naive : D-mod( Y ) co → D-mod( Y ).By definition, to specify a continuous functor F from D-mod( Y ) co to an arbitrary DG category C , is equivalent to specifying a compatible collection of functors F U : D-mod( U ) → C for co-truncative quasi-compact open substacks U ⊂ Y . The compatibility condition reads that for U j , ֒ → U , we must be given a (homotopy-coherent) system of isomorphism F U ≃ F U ◦ ( j , ) ∗ . Taking C = D-mod( Y ), the corresponding functors (Ps-Id Y , naive ) U are j ∗ : D-mod( U ) → D-mod( Y )for U j ֒ → Y .4.4.4. Warning.
For a general truncatable stack Y , the functor Ps-Id Y , naive is not an equiva-lence. In particular, it is not an equivalence for Y = Bun G unless G is solvable.In fact, we have the following assertion: Proposition 4.4.5.
If the functor
Ps-Id Y , naive : D-mod( Y ) co → D-mod( Y ) is an equivalencethen the closure of any quasi-compact open substack of Y is quasi-compact. The converse statement is also true (for tautological reasons).The proof of Proposition 4.4.5 given below is based on the following lemma.
Lemma 4.4.6.
Let Z be a quasi-compact scheme, U a QCA stack, and f : Z → U a morphism.Then for any holonomic D-module F on Z the object f dR , ∗ ( F ) ∈ D-mod( U ) is compact. Let us give two proofs:
HE CATEGORY OF D-MODULES ON Bun G Proof 1.
This follows from the following general observation:
Lemma 4.4.7.
Let F : C → C be a continuous functor between cocomplete DG categories.Let c ∈ C c be such that the partially defined left adjoint F L to F is defined on c . Then F L ( c ) ∈ C is compact. The functor f ! , left adjoint to f ! is defined on holonomic objects. Hence, by the above lemma, f ! ( D Verdier Z ( F )) ∈ D-mod coh ( U ) is compact. By Theorem 2.2.15, D Verdier U ( f ! ( D Verdier Z ( F ))) ≃ f dR , ∗ ( F )is compact, as required. (cid:3) Proof 2.
The object f dR , ∗ ( F ) is holonomic and therefore coherent. Since Z is a scheme, byTheorem 2.2.11(ii), F is safe. By [DrGa1, Lemma 9.4.2] we obtain that f dR , ∗ ( F ) is also safe.Thus, f dR , ∗ ( F ) is coherent and safe = compact. (cid:3) Proof of Proposition 4.4.5.
Suppose that Ps-Id Y , naive is an equivalence. Since Y is truncatable,it is enough to show that the closure of every co-truncative open quasi-compact substack isquasi-compact.By assumption, the functor Ps-Id Y , naive preserves compactness. From Sect. 4.4.3, we obtainthat Ps-Id Y , naive sends a compact object j op , ∗ ( F U ) ∈ D-mod( Y ) op , F U ∈ D-mod( U ) c with U j ֒ → Y co-truncative and quasi-compact, to j ∗ ( F U ) ∈ D-mod( Y ). Thus, we obtain that j ∗ ( F U )needs to be compact for any F U ∈ D-mod( U ) c whenever U is co-truncative.Take F U = f dR , ∗ ( k Z ), where Z is any quasi-compact scheme equipped with a morphism f : Z → U and k Z is the “constant sheaf” on Z . By Proposition 2.3.7, there exists a quasi-compact open substack V ⊂ Y such that the ∗ -stalk of j dR , ∗ ( F U ) = ( j ◦ f ) dR , ∗ ( k Z ) over any pointof Y − V is zero. This means that the closure of the image of j ◦ f : Z → Y is contained in V andtherefore quasi-compact. Taking f surjective we see that the closure of U is quasi-compact. (cid:3) A better functor.
Following [Ga3, Sect. 6], we definePs-Id Y , ! : D-mod( Y ) co → D-mod( Y )to be the functor corresponding in terms of Sect. 4.4.1 to the object(∆ Y ) ! ( k Y ) ∈ D-mod( Y × Y ) , where k Y ∈ D-mod( Y ) is the “constant sheaf” on Y . (The above object is well-defined because k Y is holonomic.)Let D Verdier Y , ! : D-mod( Y ) ∨ → D-mod( Y ) denote the compositionD-mod( Y ) ∨ D Verdier Y ≃ D-mod( Y ) co Ps-Id Y , ! −→ D-mod( Y ) . Y is smooth of dimension n , and that the diagonal map∆ Y : Y → Y × Y is separated. In this case we have an isomorphism k Y ≃ ω Y [ − n ] , and a natural transformation (∆ Y ) ! → (∆ Y ) dR , ∗ , which together define a natural transformation(4.4) Ps-Id Y , ! → Ps-Id Y , naive [ − n ] . Remark . If Y is separated (i.e., if ∆ Y is proper) then (4.4) is an isomorphism. However,most stacks are not separated. Thus Ps-Id Y , ! is usually different from Ps-Id Y , naive (even for Y smooth and quasi-compact).4.4.11. Here is a basic feature of the functor Ps-Id Y , ! : D-mod( Y ) co → D-mod( Y ). Lemma 4.4.12.
Let U j ֒ → Y be a co-truncative quasi-compact open substack. Then there existsa canonical isomorphism of functors D-mod( U ) → D-mod( Y ) : Ps-Id Y , ! ◦ j co , ∗ ≃ j ! ◦ Ps-Id U, ! . Proof.
Define ∆ U, Y : U → U × Y by ∆ U, Y ( u ) := ( u, j ( u )). It is easy to check that both functorsPs-Id Y , ! ◦ j co , ∗ and j ! ◦ Ps-Id U, ! correspond to the object (∆ U, Y ) ! ( k U ) ∈ D-mod( U × Y ) via theequivalenceD-mod( U × Y ) ≃ D-mod( U ) ⊗ D-mod( Y ) ≃ D-mod( U ) ∨ ⊗ D-mod( Y ) ≃ Funct cont (D-mod( U ) , D-mod( Y )) . (cid:3) The meaning of this lemma is that the functor
Ps-Id Y , ! sends objects that are ∗ -extensionsfrom a co-truncative quasi-compact open substack in D-mod( Y ) op to objects in D-mod( Y ) thatare !-extensions (from the same open).4.4.13. Self-duality.
Both functors D Verdier Y , naive : D-mod( Y ) ∨ → D-mod( Y ) , D Verdier Y , ! : D-mod( Y ) ∨ → D-mod( Y )are canonically self-dual because the corresponding objects(∆ Y ) dR , ∗ ( ω Y ) , (∆ Y ) ! ( k Y ) ∈ D-mod( Y × Y )are equivariant with respect to the action of the symmetric group S on Y × Y .4.5. Miraculous stacks.
Definition 4.5.2.
A truncatable stack Y is called miraculous if the functor Ps-Id Y , ! : D-mod( Y ) co → D-mod( Y ) is an equivalence. Clearly this happens if and only if the functor D Verdier Y , ! : D-mod( Y ) ∨ → D-mod( Y ) is anequivalence. A separated locally QCA stack has to be a Deligne-Mumford stack. Indeed, if ∆ Y is proper and affine thenit is finite, and in characteristic 0 this means that Y is Deligne-Mumford. HE CATEGORY OF D-MODULES ON Bun G Lemma 4.5.4.
A separated quasi-compact scheme Z is a miraculous stack if and only if Z hasthe following “cohomological smoothness” property: k Z and ω Z are locally isomorphic up to ashift.Proof. In our situation Ps-Id Z, ! : D-mod( Z ) → D-mod( Z ) is the functor M M ! ⊗ k Z .Applying the functor to skyscrapers, we see that if Ps-Id Z, ! is an equivalence then each !-stalkof k Z is isomorphic to k up to a shift. It is well known that this implies that k Z and ω Z arelocally isomorphic up to a shift. (cid:3) It is also easy to produce an example of a smooth quasi-compact algebraic stack Y which isnot miraculous: it suffices to take Y to be the non-separated scheme equal to A with a doublepoint 0. We refer the reader to [Ga3, Sect. 5.3.5], where this example is analyzed (one easilyshows that in this case the functor Ps-Id Y , ! does not preserve compactness).4.5.5. A basic example of a miraculous stack is Y := A n / G m ; see [Ga3, Corollary 5.3.4].In addition, the following theorem is proved in [Ga2]: Theorem 4.5.6.
Let G be a reductive group. Then the stack Bun G is miraculous. This theorem is equivalent to each quasi-compact co-truncative substack of Bun G beingmiraculous. The equivalence follows from the next lemma. Lemma 4.5.7.
A truncatable stack Y is miraculous if and only if every quasi-compact co-truncative open substack U ⊂ Y is.Proof. The “if” statement follows from Lemma 4.4.12 and the descriptions of D-mod( Y ) andD-mod( Y ) co as colimits (see Corollary 4.2.3 and Sect. 4.3.3).Let us prove the “only if” statement. Suppose that Y is miraculous and j : U ֒ → Y is aquasi-compact co-truncative open substack. The functor j ! has a left inverse (namely, j ! = j ∗ ).The functor j co , ∗ : D-mod( U ) co → D-mod( Y ) co also has a left inverse (see the first proof ofLemma 4.3.8). So Lemma 4.4.12 implies that Ps-Id U, ! has a left inverse.We obtain that the functor D Verdier U, ! : D-mod( U ) ∨ → D-mod( U ) has a left inverse. By self-duality of D Verdier U, ! (see Sect. 4.4.13), this implies that it has a right inverse as well. So D Verdier U, ! is an equivalence. (cid:3) Contractive substacks
In its simplest form, the contraction principle says that the substack { } / G m ֒ → A n / G m istruncative (here G m acts on A n by homotheties). In this section we will prove a generalizationof this fact, see Proposition 5.1.2.In Sect. 5.3 we explicitly describe the non-standard functors i ∗ dR and i ? in the setting of Propo-sition 5.1.2.We say that a substack of a stack is contractive if it locally satisfies the conditions of Propo-sition 5.1.2; for a precise definition, see Sect. 5.2.1.The upshot of this section is that “contractiveness” ⇒ “truncativeness.”5.1. The contraction principle. p : W → S betweenschemes. Assume that the monoid A (with respect to multiplication) acts on W over S (sothat the action of A on S is trivial). Assume also that the endomorphism of W correspondingto 0 ∈ A admits a factorization W p → S ι → W, where ι is a section of p : W → S . (Informally, we can say that the action of G m ⊂ A “contracts” W onto the closed subscheme ι ( S ).)Set Y := W/ G m , Z := S/ G m = S × (pt / G m ). Proposition 5.1.2.
Under the above circumstances, the closed substack Z i ֒ → Y is truncative. The rest of this subsection is devoted to the proof of Proposition 5.1.2.5.1.3. Without loss of generality, we can assume that S is quasi-compact.We have W = Spec S ( A ) , where A = L n A n is a quasi-coherent sheaf of non-negatively graded O S -algebras with A = O S .The section ι corresponds to the projection A → A = O S .For n ∈ N , let A ( n ) ⊂ A be the O S -subalgebra generated by A n . Choose n so that A is finite over A ( n ) (if A is generated by A m , . . . , A m r then one can take n to be the leastcommon multiple of m ,. . . , m r ). Set W ′ := Spec( A ( n ) ), then the morphism f : W → W ′ is finite. Moreover, the embedding ι ( S ) ֒ → f − ( f ( ι ( S ))) induces an isomorphism between thecorresponding reduced schemes. So by Proposition 3.6.11, it suffices to prove the propositionfor W ′ instead of W .5.1.4. Thus, we can assume that A is generated by A n . Moreover, since the proposition tobe proved is local with respect to S (see Corollary 3.6.5), we can assume that A n is a quotientof a locally free O S -module E . Let V denote the vector bundle over S corresponding to E ∗ (inother words, V is the spectrum of the symmetric algebra of E ). Then W = Spec( A ) identifieswith a closed conical subscheme in V .5.1.5. Thus by Lemma 3.6.10 (for closed embeddings) it suffices to consider the case where W is a vector bundle V over S equipped with an A -action obtained from the standard one bycomposing it with the homomorphism(5.1) A −→ A , λ λ n . (here n is some positive integer). In this situation we have to prove that / G m ⊂ V / G m istruncative, where ⊂ V is the zero section.5.1.6. Let e V f → V be the blow-up of V along . Set e := f − ( ). Since f is proper andsurjective, by Lemma 3.6.7, in order to prove the truncativeness of / G m ⊂ V / G m , it sufficesto show that e / G m is truncative in e V / G m .Note that e V is a line bundle over P ( V ) and e is its zero section. So we see that it suffices toprove the statement from Sect. 5.1.5 for line bundles over arbitrary bases. Moreover, since thestatement is local, it suffices to consider the trivial line bundle over an arbitrary quasi-compactscheme. HE CATEGORY OF D-MODULES ON Bun G S the substack S × ( { } / G m ) ⊂ S × ( A / G m )is truncative (here we assume that λ ∈ G m acts on A as multiplication by λ n for some n ∈ N ).By Lemma 3.6.2, we can assume that S = Spec( k ). In this case the statement follows from thefact that the number of G m -orbits in A is finite (see Sect. 3.2.4). (cid:3) Contractiveness. Z ′ of a stack Y ′ is contractive if there exists acommutative diagram(5.2) Z (cid:15) (cid:15) / / Y (cid:15) (cid:15) Z ′ / / Y ′ such that(i) the upper row of (5.2) is as in Proposition 5.1.2;(ii) the morphism Z → Z ′ × Y ′ Y is an open embedding;(iii) the vertical arrows of (5.2) are smooth and the left one is surjective.In other words, a substack is contractive if it locally satisfies the conditions of Proposi-tion 5.1.2.5.2.2. We obtain: Corollary 5.2.3.
A contractive substack is truncative.Proof.
With no loss of generality, we can assume that Y is quasi-compact. Now combine Propo-sition 5.1.2 and 3.6.4. (cid:3) Note that the above definition of contractive substack makes sense without the characteris-tic 0 assumption.
Remark . We are not sure that the notion of contractiveness is really good. But it isconvenient for the purposes of this article.5.3.
An adjointness result. W → S ι → W be as in Proposition 5.1.2 (in particular, A acts on W ).Consider the corresponding morphisms i : S/ G m ֒ → W/ G m and π : W/ G m → S/ G m . ByProposition 5.1.2, the functor i dR , ∗ has a continuous left adjoint (denoted by i ∗ dR ) and i ! has acontinuous right adjoint (denoted by i ? ).The next proposition identifies the non-standard functors i ∗ dR : D-mod( W/ G m ) → D-mod( S/ G m ) and i ? : D-mod( S/ G m ) → D-mod( W/ G m )with certain standard functors. Namely, i ∗ dR ≃ π dR , ∗ , i ? = π ! . Proposition 5.3.2.
The functors π dR , ∗ : D-mod( W/ G m ) ⇄ D-mod( S/ G m ) : i dR , ∗ and i ! : D-mod( W/ G m ) ⇄ D-mod( S/ G m ) : π ! form adjoint pairs with the adjunctions π dR , ∗ ◦ i dR , ∗ ∼ −→ Id D-mod( S/ G m ) and i ! ◦ π ! ∼ −→ Id D-mod( S/ G m ) coming from the isomorphism π ◦ i ∼ −→ Id S/ G m . Note that a simple particular case of Proposition 5.3.2 was proved in Sect. 3.3.9.
Remark . Proposition 5.3.2 clearly implies Proposition 5.1.2.Proposition 5.3.2 is well known (at least, in the setting of constructible sheaves instead of D-modules). It goes back to the works by Verdier [Ve, Lemma 6.1] and Springer [Sp, Proposition 1];see also [KL, Lemma A.7] and [Br, Lemma 6].The reader can easily prove Proposition 5.3.2 by slightly modifying the argument fromSect. 5.1 (which is based on blowing up and properness).On the other hand, in Appendix C we give a complete proof of a “stacky” generalization ofProposition 5.3.2 (see Theorem C.5.3 and Corollary C.5.4). The approach from Appendix C isclose to [Sp] (there are no blow-ups, no properness arguments, and we work with the monoid A rather than with the scheme or stack on which A acts).5.4. A general lemma on contractiveness.
The reader may prefer to skip this subsectionon the first pass. Its main result (Lemma 5.4.3) will not be used until Proposition 11.3.7(b).5.4.1. The notion of contractive substack was defined in Sect. 5.2.1. This notion is clearlylocal in the following sense:Let f : Y ′ → Y be a smooth surjective morphism of algebraic stacks and Z ⊂ Y a locallyclosed substack. If f − ( Z ) is contractive in Y ′ then Z is contractive in Y .5.4.2. As before, we consider A as a monoid with respect to multiplication. It contains G m as a subgroup.We have: Lemma 5.4.3.
Let π : W → S be an affine schematic morphism of algebraic stacks. Supposethat the monoid A acts on W by S -endomorphisms (i.e., over S , with the action of A on S being trivial). Assume that (i) The S -endomorphism of W corresponding to ∈ A equals i ◦ π for some section i : S → W ; (ii) The action of G m on W viewed as a stack over pt (rather than over S ) is isomorphic to thetrivial action.Then the substack S i ֒ → W is contractive.Remark . Condition (i) implies that the action of G m on W viewed as a stack over S is nontrivial unless π : W → S is an isomorphism. This does not contradict (ii): we aredealing with stacks, and the functor from the groupoid of S -endomorphisms of W to that of k -endomorphisms of W is not fully faithful.Before proving Lemma 5.4.3, let us consider two examples. HE CATEGORY OF D-MODULES ON Bun G Example . Let p : W → S be as in Sect. 5.1.1. Set W := W/ G m , S := S/ G m = S × (pt / G m ) . Then the conditions of Lemma 5.4.3 hold for the morphism π : W → S . The conclusion ofLemma 5.4.3 holds tautologically. Example . Let π : W → S be an affine schematic morphism of algebraic stacks. Supposethat an action of A on W by S -endomorphisms satisfies condition (i) of Lemma 5.4.3. Set W ′ := W / G m and S ′ := S / G m = S × (pt / G m ). Then the morphism π ′ : W ′ → S ′ satisfies bothconditions of Lemma 5.4.3. The conclusion of Lemma 5.4.3 is clear because after a smoothsurjective base change S → S we get the situation of Example 5.4.5.5.4.7. To prove Lemma 5.4.3, we need the following assertion: Lemma 5.4.8.
Let ϕ : Y → Y ′ and ψ : Y ′ → Y be morphisms between algebraic stacks such that ψ ◦ ϕ ≃ Id Y . Suppose that ϕ is smooth and surjective. Then: (a) The maps { locally closed substacks of Y ′ } → { locally closed substacks of Y } , Z ′ ϕ − ( Z ′ ) , { locally closed substacks of Y } → { locally closed substacks of Y ′ } , Z ψ − ( Z ) are mutually inverse bijections; (b) A locally closed substack Z ⊂ Y is contractive if and only if the corresponding substack Z ′ ⊂ Y ′ is.Remark . Since ϕ and ψ ◦ ϕ are smooth and surjective ψ has the same properties. Proof.
The maps from statement (a) are clearly injective. Since ψ ◦ ϕ ≃ Id Y one has ϕ − ( ψ − ( Z )) = Z . Statement (a) follows. To prove (b), use statement (a), Remark 5.4.9, andthe locality of strong contractiveness, see Sect. 5.4.1. (cid:3) Proof of Lemma 5.4.3.
Define π ′ : W ′ → S ′ as in Example 5.4.6, then the corresponding em-bedding i ′ : S ′ ֒ → W ′ is contractive. We have a Cartesian square S i −−−−→ W y ϕ y S ′ i ′ −−−−→ W ′ So by Lemma 5.4.8, it remains to show that the morphism ϕ : W → W ′ admits a left inverse ψ : W ′ → W . But W ′ is the quotient of W by a trivial action of G m . Choosing a trivializationof this action we can identify W ′ with W × (pt / G m ) and take ψ to be the projection W × (pt / G m ) → W . (cid:3) The case of SL In this section we will give a proof of Theorem 4.1.8 in the case G = SL , which will be theprototype of the argument in general.6.1. The substack
Bun ( ≤ n ) G . By the way, this implies that Y ′ is the classifying space of a group over Y . n ≥
0, let Bun ( ≤ n ) G ⊂ Bun G be the open substack consisting of vectorbundles that do not admit line sub-bundles of degree > n . It is easy to see that the substacksBun ( ≤ n ) G are quasi-compact and that their union is all of Bun G .Let g be the genus of X . We will show that for n ≥ max( g − , ( ≤ n ) G is co-truncative.6.1.2. Let Bun ( n ) G be the locally closed substackBun ( n ) G := Bun ( ≤ n ) G − Bun ( ≤ n − G , endowed, say, with the reduced structure.By Proposition 3.7.2, it suffices to show that if n > max( g − ,
0) then Bun ( n ) G is a truncativesubstack of Bun ( ≤ n ) G . We will do this by combining Propositions 3.6.4 and 5.1.2.6.1.3. Note, however, that if n is small relative to the genus of X , then the stratum Bun ( n ) G is not truncative. Indeed, one can choose X and n so that Bun ( n ) G has a non-empty intersectionwith an open substack Bun G that is actually a scheme ; then apply Sect. 3.2.1.6.2. Reducing to a contracting situation. n , let Bun nB be the stack classifying short exact sequences(6.1) 0 → L − → M → L → , where M ∈ Bun SL , and L is a line bundle of degree − n . Let p n : Bun nB → Bun G denote thenatural projection. If n > p n equals Bun ( n ) G . Lemma 6.2.2.
Suppose that n > max( g − , . Then the morphism p − n : Bun − nB → Bun G issmooth and its image contains Bun ( n ) G .Proof. A point x ∈ Bun − nB corresponds to an exact sequence (6.1) with deg L = n . The cokernelof the differential of p − n at x equals H ( X, L ⊗ ), which is zero because deg L ⊗ = 2 n > g − p − n is smooth.A point y ∈ Bun ( n ) G corresponds to an SL -bundle M that can be represented as an extension(6.1) with deg L = − n . Such an extension splits because 2 n > g −
2. So M is also an extensionof L − by L . Hence, y is in the image of p − n . (cid:3) n > max( g − ,
0) thenthe substack(6.2) Bun ( n ) G × Bun G Bun − nB ⊂ Bun − nB is truncative.6.3. Applying the contraction principle. nGL (1) denote the stack of line bundles on X of degree n . Note that we have acanonical isomorphism Bun nGL (1) ≃ Bun ( n ) G × Bun G Bun − nB that sends a line bundle L ∈ Bun nGL (1) to0 → L − → L − ⊕ L → L → . HE CATEGORY OF D-MODULES ON Bun G n denote the coarse moduli scheme corresponding to Bun nGL (1) . We have a vectorbundle V on Bun nGL (1) whose fiber over L ∈ Bun nGL (1) equals Ext( L , L − ).Choose a section s : Pic n → Bun nGL (1) of the morphism Bun n → Pic n (e.g., choose x ∈ X and identify Pic n with the stack of line bundles of degree n trivialized over x ).Set V ′ = s ∗ ( V ). Let ⊂ V ′ denote the zero section. Then Bun − nB identifies with the quotientstack V ′ / G m and the substackBun ( n ) G × Bun G Bun − nB ≃ Bun nGL (1) ֒ → Bun − nB identifies with / G m . Hence, the substack (6.2) is truncative by Proposition 5.1.2. (cid:3) Recollections from reduction theory
The goal of this section is to prepare for the proof of Theorem 4.1.8 by recalling the Harder-Narasimhan-Shatz stratification of Bun G .With future applications in mind, when defining these open substacks, we will remove theassumption that our ground field is of characteristic 0, unless we explicitly specify otherwise.Thus, we let G be a connected reductive group over any algebraically closed field k .7.1. Notation related to G . B ⊂ G .Conjugacy classes of parabolics are then in bijection with the set of parabolics that contain B ,called the standard parabolics. From now on, by a parabolic we will mean a standard parabolic,unless explicitly stated otherwise.For a parabolic P we will denote by U ( P ) its unipotent radical.We denote by Γ G the set of vertices of the Dynkin diagram of G . Parabolics in G are inbijection with subsets of Γ G . For a parabolic P with Levi quotient M we let Γ M ⊂ Γ G denotethe corresponding subset; it identifies with the set of vertices of the Dynkin diagram of M .7.1.2. Let Λ G denote the coweight lattice of G and Λ Q G := Q ⊗ Z Λ G . Let Λ + G ⊂ Λ G denote themonoid of dominant coweights and Λ posG ⊂ Λ G the monoid generated by positive simple coroots.Let Λ + , Q G , Λ pos, Q G ⊂ Λ Q G be the corresponding rational cones.Let ˇ α i , i ∈ Γ G , be the simple roots; we have:Λ + , Q G = { λ ∈ Λ Q G | h λ, ˇ α i i ≥ i ∈ Γ G } . P be a parabolic of G and M its Levi quotient. Let Z ( M ) be the neutral connectedcomponent of the center of M , then Λ Z ( M ) ⊂ Λ G . Set Λ Q G,P := Λ Q Z ( M ) ⊂ Λ Q G . Explicitly,Λ Q G,P = { λ ∈ Λ Q G | h λ, ˇ α i i = 0 for i ∈ Γ M } . Note that Λ Q G,G = Λ Q Z ( G ) and Λ Q G,B = Λ Q G . + , Q G,P := Λ + , Q G ∩ Λ Q G,P and(7.1) Λ ++ , Q G,P := { λ ∈ Λ Q G | h λ, ˇ α i i = 0 for i ∈ Γ M and h λ, ˇ α i i > i / ∈ Γ M } . In other words, Λ ++ , Q G,P is the set of those elements of Λ + , Q G,P that are regular (i.e., lie off the wallsof Λ + , Q G,P ). Clearly(7.2) Λ + , Q G = G P Λ ++ , Q G,P , where the union is taken over the conjugacy classes of parabolics.7.1.5. Note also that the inclusion Λ Q G,P ֒ → Λ Q G canonically splits as a direct summand: thecorresponding projector pr P : Λ Q G → Λ Q G,P is defined so thatker (pr P ) = M i ∈ Γ M Q · α i . We can also view the map Λ Q G → Λ Q G,P as follows: it comes from the mapΛ G ≃ Λ M → Λ M/ [ M,M ] and the isomorphism Λ Q Z ( M ) ∼ −→ Λ Q M/ [ M,M ] induced by the isogeny Z ( M ) → M/ [ M, M ].7.1.6. We introduce the partial order on Λ Q G by λ ≤ G λ ⇔ λ − λ ∈ Λ pos, Q G . The following useful observation is due to S. Schieder:
Lemma 7.1.7.
For a parabolic P , the projection pr P is order-preserving. For a proof, see [Sch, Proposition 3.1.2(a)].7.2.
The degree of a bundle.
Fix a connected smooth complete curve X . For any algebraicgroup H let Bun H denote the stack of H -bundles on X .7.2.1. One has a canonical isomorphism deg : π (Bun G m ) ∼ −→ Z . Accordingly, for any torus T one has a canonical isomorphism deg T : π (Bun T ) ∼ −→ Λ T .7.2.2. Let e G be any connected affine algebraic group and let e G tor be its maximal quotienttorus. The composition π (Bun e G ) → π (Bun e G tor ) deg e G tor −→ Λ e G tor will be denoted by deg e G .If e G = G is reductive then G tor = G/ [ G, G ], and the map Z ( G ) → G tor is an isogeny, soΛ Q G tor ≃ Λ Q Z ( G ) . Therefore one has a locally constant map deg G : Bun G → Λ Q Z ( G ) . Its fibersare not necessarily connected but have finitely many connected components; this follows fromRemark 7.2.4 below. HE CATEGORY OF D-MODULES ON Bun G P be a parabolic subgroup of a reductive group G , and let M be the Leviquotient of P .Then by Sects. 7.1.3 and 7.2.2, one has the locally constant maps deg M : Bun M → Λ Q G,P and therefore deg P : Bun P → Λ Q G,P .The preimage of λ ∈ Λ Q G,P in Bun M (resp. Bun P ) is denoted by Bun λM (resp. Bun λP ).It is easy to see that Bun λM and Bun λP are empty unless λ belongs to a certain finitelygenerated subgroup A G,P ⊂ Λ Q G,P such that A G,P ⊗ Q = Λ Q G,P ; namely, A G,P = pr P (Λ G ),where pr P : Λ Q G → Λ Q G,P is as in Sect. 7.1.5.7.2.4.
Remark.
Let e G be any connected affine algebraic group and e G red its maximal reductivequotient. Define π ( e G ) to be the quotient of Λ e G red by the subgroup generated by coroots. It iswell known that there is a unique bijection π (Bun e G ) ∼ −→ π ( e G ) such that the diagram π (Bun e B ) (cid:15) (cid:15) / / Λ e T ≃ Λ e G red (cid:15) (cid:15) π (Bun e G ) / / π ( e G )commutes. Here e B is a Borel subgroup of e G and e T is the maximal quotient torus of e B .7.3. Semistability. G ad denote the quotient of G by its center andΥ G : Λ Q G → Λ Q G ad , the projection.Let p P : Bun P → Bun G be the natural morphism. Recall that a G -bundle P G ∈ Bun G iscalled semi-stable if for every parabolic P such that P G = p P ( P P ) with P P ∈ Bun µP we haveΥ G ( µ ) ≤ G ad . In fact, semi-stability can be tested just using reductions to the Borel:
Lemma 7.3.2. A G -bundle P G is semi-stable if and only if for every reduction P B of P G tothe Borel B with P B ∈ Bun µB , we have Υ G ( µ ) ≤ G ad .Proof. This follows from Lemma 7.1.7 and the fact that every M -bundle admits a reduction tothe Borel of M . (cid:3) It is known that semi-stable bundles form an open substack Bun ssG ⊂ Bun G , whose intersec-tion with each connected component of Bun G is quasi-compact.7.3.3. More generally, for θ ∈ Λ + , Q G and a G -bundle P G , we say that P G has Harder-Narasimhancoweight ≤ G θ if for every parabolic P such that P G = p P ( P P ) with P P ∈ Bun µP we have µ ≤ G θ. As in Lemma 7.3.2, it suffices to check this condition for P = B . G -bundles having Harder-Narasimhan coweight ≤ G θ form an open sub-stack of Bun G . The argument repeats the proof of the fact that Bun ssG is open, given in [Sch,Proposition 6.1.6].We denote the above open substack by Bun ( ≤ θ ) G and sometimes by Bun ( ≤ G θ ) G . It lies in the(finite) union of connected components of Bun G corresponding to the image of θ underΛ Q G → Λ Q G,G ≃ Λ Q Z ( G ) . Furthermore, θ ≤ G θ ⇒ Bun ( ≤ θ ) G ⊂ Bun ( ≤ θ ) G , and [ θ ∈ Λ + , Q G Bun ( ≤ θ ) G = Bun G . Finally, we have:
Proposition 7.3.5.
The open substack
Bun ( ≤ θ ) G is quasi-compact. We will give two proofs:
Proof 1.
With no loss of generality, we can assume that G is of adjoint type. We will use therelative compactification p B : Bun B → Bun G of the map p B : Bun B → Bun G , see Sect. 7.5.5.For each connected component ′ Bun G ⊂ Bun G choose a coweight λ ∈ − Λ + G such that themap p B : Bun λB → Bun G lands in ′ Bun G and is smooth (for smoothness, it is enough to take λ so that h λ, ˇ α i i < − (2 g −
2) for each simple root ˇ α i ). Then the map p B : Bun λB → ′ Bun G issurjective.It is a basic property of Bun B (see [Sch, Sect. 6.1.4]) that p B (Bun λB ) = [ µ ∈ Λ posG p B (Bun λ + µB ) . Therefore ′ Bun G ∩ Bun ( ≤ θ ) G = [ µ ∈ Λ posG , λ + µ ≤ G θ p B (Bun λ + µB ) . However, the set { µ ∈ Λ posG | λ + µ ≤ G θ } is finite. Hence, ′ Bun G ∩ Bun ( ≤ θ ) G is contained in the image of finitely many quasi-compactstacks Bun λ + µB , and hence is itself quasi-compact. (cid:3) The second proof will be given after Corollary 7.4.5.7.3.6. By definition, for λ ∈ Λ Q G,G = Λ Q Z ( G ) Bun ssG ∩ Bun λG = Bun ( ≤ λ ) G and Bun ssG = [ λ ∈ Λ Q G,G
Bun ( ≤ λ ) G . HE CATEGORY OF D-MODULES ON Bun G P ⊂ G with Levi quotient M we have the corresponding opensubstack Bun ssM ⊂ Bun M ; let Bun ssP denote the pre-image of Bun ssM in Bun P .For λ ∈ Λ Q G,P we letBun λ,ssM := Bun ssM ∩ Bun λM = Bun ( ≤ M λ ) M , Bun λ,ssP := Bun ssP ∩ Bun λP . The Harder-Narasimhan-Shatz stratification of
Bun G . This stratification was de-fined in [HN, Sh1, Sh2] in the case G = GL ( n ). For any reductive G it was defined in[R1, R2, R3] and [Beh, Beh1].7.4.1. We give the following definition: Definition 7.4.2.
A schematic morphism of algebraic stacks f : X → X is an almost-isomorphism if f is finite and each geometric fiber of f has a single point. The next theorem is a basic result of reduction theory.
Theorem 7.4.3. (1)
Let λ ∈ Λ + , Q G and let P ⊂ G be the unique parabolic such that λ belongs to the set Λ ++ , Q G,P defined by (7.1) . Then p P : Bun P → Bun G induces an almost-isomorphism between Bun λ,ssP and a quasi-compact locally closed reduced substack
Bun ( λ ) G ⊂ Bun G . (1 ′ ) If k has characteristic then the morphism Bun λ,ssP → Bun ( λ ) G is an isomorphism. (2) The substacks
Bun ( λ ) G , λ ∈ Λ + , Q G , are pairwise non-intersecting, and every geometric pointof Bun G belongs to exactly one Bun ( λ ) G . (3) Let P ′ ⊂ G be a parabolic and let λ ′ be any (not necessarily dominant) element of Λ Q G,P ′ .If p P ′ (Bun λ ′ P ′ ) ∩ Bun ( λ ) G = ∅ then λ ′ ≤ G λ . Statements (1), (1 ′ ), (2), and a slightly weaker version of (3) are due to K. Behrend [Beh,Beh1]. A complete proof of the theorem was given by S. Schieder, see [Sch, Theorem 2.3.3]. InSect. 7.5 we give a sketch of the proof from [Sch].7.4.4. We apply Theorem 7.4.3 to obtain the following more explicit description of the opensubstacks Bun ( ≤ θ ) G : Corollary 7.4.5.
For θ ∈ Λ + , Q G we have: (7.3) Bun ( ≤ θ ) G = [ λ, λ ≤ G θ Bun ( λ ) G , and the set (7.4) { λ ∈ Λ + , Q G | λ ≤ G θ and Bun ( λ ) G = ∅} is finite.Proof. The fact that Bun ( λ ) G ∩ Bun ( ≤ θ ) G = ∅ ⇒ λ ≤ G θ follows from the definition of Bun ( ≤ θ ) G .The inclusion Bun ( λ ) G ⊂ Bun ( ≤ θ ) G for λ ≤ G θ follows from Theorem 7.4.3(3).This proves (7.3) in view of Theorem 7.4.3(2). The finiteness of the set (7.4) follows fromthe fact that Bun ( λ ) G = ∅ ⇒ λ ∈ [ P A G,P , see the end of Sect. 7.2.3. (cid:3) As a corollary, we obtain a 2nd proof of Proposition 7.3.5:
Proof 2 (of Proposition 7.3.5).
Follows from Corollary 7.4.5 and the fact that each Bun ( λ ) G isquasi-compact. (cid:3) As another corollary of Corollary 7.4.5 we obtain:
Corollary 7.4.6.
We have: (7.5) Bun ( θ ) G = Bun ( ≤ θ ) G − [ θ ′ ,θ = θ ′ ≤ G θ Bun ( ≤ θ ′ ) G . Remark . We could a priori define the locally closed substacks Bun ( θ ) G by formula (7.5).However, without the interpretation of Bun ( θ ) G via Theorem 7.4.3, it would not be clear thatthese locally closed substacks are pairwise non-intersecting.7.4.8. The Harder-Narasimhan map.
Let | Bun G ( k ) | denote the set of isomorphism classes of G -bundles on X (or equivalently, of objects of the groupoid Bun G ( k )). We equip | Bun G ( k ) | with the Zariski topology.By Theorem 7.4.3(2), for every F ∈ Bun G ( k ) there exists a unique λ ∈ Λ + , Q G such that F ∈ Bun ( λ ) G ( k ). This λ is called the Harder-Narasimhan coweight of F and denoted byHN( F ). Thus we have a map(7.6) HN : | Bun G ( k ) | → Λ + , Q G . Lemma 7.4.9.
The map (7.6) has the following properties. (i)
It is upper-semicontinuous, i.e., for each λ ∈ Λ + , Q G the preimage of the subset (7.7) { λ ∈ Λ + , Q G | λ ≤ λ } is open. (ii) The image of the map (7.6) is discrete in the real vector space Λ R G := Λ G ⊗ R . (iii) A subset S ⊂ | Bun G ( k ) | is quasi-compact if and only if HN( S ) is bounded in Λ R G .Proof. Follows from Corollary 7.4.5 and the fact that the substacks Bun ( ≤ θ ) G are open andquasi-compact. (cid:3) The quasi-compactness of Bun ( λ ) G relied on the fact that the open substack Bun λ,ssM is quasi-compact,which in itself is a particular case of Proposition 7.3.5. By Corollary 7.4.6, this agrees with the usage of the words “Harder-Narasimhan coweight” in Sect. 7.3.4.
HE CATEGORY OF D-MODULES ON Bun G + , Q G with the order topology , i.e., the one whose base is formed bysubsets of the form (7.7). Then statement (i) of Lemma 7.4.9 can be reformulated as follows:the map (7.6) is continuous .Now it is clear that if a subset of Λ + , Q G is locally closed then so is its preimage in Bun G . Notethat for a subset of Λ + , Q G it is easy to understand whether it is open, closed, or locally closed,see Lemma A.1.1 from Appendix A. Thus we obtain: Corollary 7.4.11.
Let S ⊂ Λ + , Q G be a subset. Consider the corresponding subset Bun ( S ) G := [ λ ∈ S Bun ( λ ) G ⊂ Bun G . (a) If S has the property that λ ∈ S, λ ≤ G λ ⇒ λ ∈ S , then Bun ( S ) G is closed in Bun G . (b) If S has the property that λ ∈ S, λ ≤ G λ ⇒ λ ∈ S , then Bun ( S ) G is open in Bun G . (c) If S has the property that λ , λ ∈ S, λ ≤ G λ ≤ G λ ⇒ λ ∈ S , then Bun ( S ) G is locally closedin Bun G . In cases (a) and (c) of the lemma we will regard Bun ( S ) G as a substack of Bun G with thereduced structure.7.5. On the proof of Theorem 7.4.3.
Let us make some remarks regarding the proof ofTheorem 7.4.3. For a full proof along these lines see [Sch].7.5.1. For a G -bundle P G let λ be a maximal element in Λ Q G , with respect to the ≤ G orderrelation, such that there exists a parabolic P and P P ∈ Bun λP such that P G = p P ( P P ). Oneshows using Lemma 7.1.7 that the maximality assumption on λ implies that λ ∈ Λ + , Q G and that P P ∈ Bun λ,ssP . For details see [Sch, Sect. 6.2].7.5.2. Using Bruhat decomposition, one shows (see [Sch, Theorem 4.5.1]) that if P ′ is anotherparabolic and P P ′ ∈ Bun λ ′ P ′ such that P G = p P ′ ( P P ′ ), then λ ′ ≤ G λ , and the equality takes placeif and only if P ′ ⊂ P and P P is induced from P P ′ via the above inclusion.7.5.3. We obtain that the set of maximal elements λ as in Sect. 7.5.1 contains a single element.Moreover, the set of parabolics as in Sect. 7.5.1 also contains a unique maximal element P ;namely, one for which λ ∈ Λ ++ , Q G,P .7.5.4. This establishes points (2) and (3) of the theorem, modulo the fact that Bun ( λ ) G is locallyclosed, and not just constructible. The “a” is italicized because we do not yet know that such a maximal element is unique, although we willeventually show that it is. λ and P be as in Sect. 7.5.3. To prove point (1), one uses the relative compactifi-cation p P : Bun P → Bun G of the map Bun P → Bun G defined in [BG, Sect. 1.3.2] under the assumption that [ G, G ] issimply connected and in [Sch, Sect. 7] for an arbitrary reductive G . Since p P is proper, theimages of Bun λP and Bun λP − Bun λ,ssP in Bun G are both closed. Using Sect. 7.5.2, one showsthat the latter does not intersect Bun ( λ ) G . This implies that p defines a finite map from Bun λ,ssP to a locally closed substack of Bun G . It is bijective at the level of k -points by Sect. 7.5.2. See[Sch, Sect. 6.2.2] for details. ′ ) is equivalent to the fact that the map Bun λ,ssP → Bun G is a monomorphism (on S -points for any scheme S ). This is proved (see [Sch, Prop. 5.2.1])using the fact that in characteristic 0, a homomorphism of reductive groups G → G thatsends Z ( G ) to Z ( G ) sends Bun ssG to Bun ssG , see [Sch, Prop. 5.2.1] for details. (We will usea similar argument in the proof of Proposition 9.2.2(a) given in Sect. 10).8. Complements to reduction theory: P -admissible sets In this section we fix a parabolic P ⊂ G . Let M be the corresponding Levi.Our goal is to prove Proposition 8.3.3, which allows us to produce locally closed substacksof Bun G from locally closed substacks of Bun M .8.1. Some elementary geometry.
Instead of reading the proofs of Lemmas 8.1.2 and 8.1.4below, the reader may prefer to check the statements in the rank 2 case by drawing the picture,and believe that the statements are true in general.8.1.1. Recall that according to the definitions from Sect. 7.1.2, we have Λ Q G = Λ Q M and Λ + , Q G ⊂ Λ + , Q M . Lemma 8.1.2.
Let λ, λ ′ ∈ Λ Q G = Λ Q M and λ ′ ≤ M λ . Then (a) h λ ′ , ˇ α i i ≥ h λ , ˇ α i i for i Γ M ; (b) If λ ∈ Λ + , Q G and λ ′ ∈ Λ + , Q M then λ ′ ∈ Λ + , Q G .Proof. Statement (a) follows from the inequality h α j , ˇ α i i ≤ i = j .To prove (b), we have to show that h λ ′ , ˇ α i i ≥ i ∈ Γ G . If i ∈ Γ M this followsfrom the assumption that λ ′ ∈ Λ + , Q M . If i Γ M this follows from (a) and the assumption that λ ∈ Λ + , Q G . (cid:3) Q G,P ⊂ Λ Q G and the projectorpr P : Λ Q G → Λ Q G,P . Lemma 8.1.4. If λ ∈ Λ + , Q G then (8.1) pr P ( λ ) ≤ M λ, (8.2) h pr P ( λ ) , ˇ α i i ≥ h λ , ˇ α i i for i Γ M . The latter part of the argument will actually be carried out in a slightly more general situation in the proofof Proposition 8.3.3.
HE CATEGORY OF D-MODULES ON Bun G Proof.
On the one hand, for i ∈ Γ M , one has h λ − pr P ( λ ) , ˇ α i i = h λ , ˇ α i i ≥
0. On the otherhand, λ − pr P ( λ ) belongs to the subspace generated by the coroots of M . Thus λ − pr P ( λ ) is inthe dominant cone of the root system of M . The latter is contained in Λ pos, Q M , so we get (8.1).The inequality (8.2) follows from (8.1) by Lemma 8.1.2(a). (cid:3) P -admissible subsets of Λ + , Q G . S be a subset of Λ + , Q G , and let P be a parabolic. Definition 8.2.2.
We say that S is P - admissible if the following three properties hold: (8.3) There exists µ ∈ Λ Q G,P such that S ⊂ pr − P ( µ ) ∩ Λ + , Q G . (8.4) If λ ∈ S and λ ∈ Λ + , Q G , λ ≤ M λ then λ ∈ S. (8.5) ∀ λ ∈ S, ∀ i ∈ Γ G − Γ M we have h λ , ˇ α i i > . Remark . If S = ∅ is P -admissible and pr P ( S ) = µ ∈ Λ Q G,P then µ ∈ Λ ++ , Q G,P ⊂ Λ + , Q G , where, as before,Λ ++ , Q G,P := { λ ∈ Λ Q G | h λ, ˇ α i i = 0 for i ∈ Γ M and h λ, ˇ α i i > i / ∈ Γ M } . This follows from (8.2) and (8.5).8.2.4.
Examples. (i) The subset of pr − P ( µ ) ∩ Λ + , Q G consisting of elements satisfying (8.5), is P -admissible.(ii) If λ ∈ Λ + , Q G is such that h λ , ˇ α i i > i Γ M then the set S = { λ ′ ∈ Λ + , Q M | λ ′ ≤ M λ } is P -admissible by Lemma 8.1.2.(iii) If µ ∈ Λ ++ , Q G,P then the one-element set { µ } is P -admissible and moreover, it is the smallestnon-empty P -admissible subset of pr − P ( µ ) ∩ Λ + , Q G . This follows from (8.1).8.2.5. Let S ⊂ Λ + , Q G be P -admissible subset. Note that we can also regard S as a subsetof Λ + , Q M . Lemma 8.2.6.
The subset S ⊂ Λ + , Q M is open.Proof. Follows from Lemma 8.1.2(b). (cid:3)
Reduction theory and P -admissible subsets. S ⊂ Λ + , Q G be a P -admissible subset.By Corollary 7.4.11, the subset(8.6) Bun ( S ) G := [ λ ∈ S Bun ( λ ) G ⊂ Bun G is locally closed (and thus we can regard it as a substack with the reduced structure).By Lemma 8.2.6 and Corollary 7.4.11, the subsetBun ( S ) M := [ λ ∈ S Bun ( λ ) M ⊂ Bun M is open. Set Bun ( S ) P := Bun P × Bun M Bun ( S ) M . P -admissible subset S has one element, see Example 8.2.4(iii). Proposition 8.3.3.
Let S ⊂ Λ + , Q G be a P -admissible subset. Then the restriction of p P :Bun P → Bun G to Bun ( S ) P defines an almost-isomorphism (8.7) Bun ( S ) P → Bun ( S ) G . Remark . Later we will show that if k has characteristic 0 then the map (8.7) is, in fact,an isomorphism (see Lemma 10.2.1 and Remark 10.2.2 below).The rest of this section is devoted to the proof of Proposition 8.3.3.8.3.5. First, let us prove that the map (8.7) maps Bun ( S ) P to Bun ( S ) G and is bijective at thelevel of k -points. To this end, it suffices to show that if λ is an element of S ⊂ Λ + , Q G ⊂ Λ + , Q M then the map p P sends Bun P × Bun M Bun ( λ ) M bijectively to Bun ( λ ) G .Let M ′ be the Levi of G such thatΓ M ′ = { i ∈ Γ G | h λ, ˇ α i i = 0 } . By condition (8.5), we have Γ M ′ ⊂ Γ M . Let P ′ ⊂ P be the corresponding parabolic. Set P ′ := P ′ /U ( P ); this is a parabolic in M whose Levi quotient identifies with M ′ . We have λ ∈ Λ ++ , Q G,P ′ ⊂ Λ ++ , Q M,P ′ . By the definition of Bun ( λ ) M , we have a bijectionBun P ′ × Bun M ′ Bun λ,ssM ′ → Bun ( λ ) M . Hence, it is enough to show that the mapBun P × Bun M Bun P ′ × Bun M ′ Bun λ,ssM ′ → Bun G defines a bijection onto Bun ( λ ) G . HE CATEGORY OF D-MODULES ON Bun G However, Bun P × Bun M Bun P ′ × Bun M ′ Bun λ,ssM ′ ≃ Bun P ′ × Bun M ′ Bun λ,ssM ′ , and the required assertion follows from the definition of Bun ( λ ) G .8.3.6. To finish the proof of the proposition, we have to show that the map (8.7) is finite. Wealready know that it is bijective, so it suffices to show the map (8.7) is proper. This will bedone by generalizing the argument in Sect. 7.5.5 using the stack Bun P .Let µ ∈ Λ Q G,P be such that S ⊂ pr − P ( µ ) ∩ Λ + , Q G . Consider the map ¯p P : Bun µP → Bun G . This map is proper. So to prove properness of (8.7), it is enough to show that(8.8) ¯p P (Bun µP − Bun ( S ) P ) ∩ Bun ( S ) G = ∅ . We have Bun µP − Bun ( S ) P = (Bun µP − Bun µP ) ∪ (Bun µP − Bun ( S ) P ) , so to prove (8.8), it suffices to show that(8.9) ¯p P (Bun µP − Bun µP ) ∩ Bun ( λ ) G = ∅ for all λ ∈ S and(8.10) p P (Bun µP − Bun ( S ) P ) ∩ Bun ( λ ) G = ∅ for all λ ∈ S. ¯p P (Bun µP − Bun µP ) = [ µ ′ ∈ Λ Q G,P µ ′ − µ ∈ Λ pos G − p P (Bun µ ′ P ) , which follows from [Sch, Sect. 6.1.4]. This equality shows that if (8.9) were false we would have(8.11) p P (Bun µ ′ P ) ∩ Bun ( λ ) G = ∅ for some µ ′ ∈ Λ Q G,P such that(8.12) µ ≤ G µ ′ = µ. However, (8.11) implies, by Theorem 7.4.3(3), that µ ′ ≤ G λ . So by Lemma 7.1.7, µ ′ ≤ G pr P ( λ ) = µ, which contradicts (8.12).8.3.8. To prove (8.10), we have to show that if λ ′ ∈ Λ + , Q M is such that(8.13) p P (Bun µP × Bun M Bun ( λ ′ ) M ) ∩ Bun ( λ ) G = ∅ then λ ′ ∈ S .If (8.13) holds then λ ′ ≤ G λ by Theorem 7.4.3(3). Since pr P ( λ ) = µ = pr P ( λ ′ ) this impliesthat λ ′ ≤ M λ .Since λ ′ ∈ Λ + , Q M and λ ′ ≤ M λ we get λ ′ ∈ Λ + , Q G by Lemma 8.1.2(b). Since λ ∈ S , λ ′ ∈ Λ + , Q G ,and λ ′ ≤ M λ we get λ ′ ∈ S by the admissibility of S . (cid:3) Proof of the main theorem
The main result of this section.
We wish to prove Theorem 4.1.8 (=Theorem 0.2.5),which says that Bun G is truncatable.9.1.1. For each θ ∈ Λ + , Q G , we have the quasi-compact open substack Bun ( ≤ θ ) G ⊂ Bun G , seeSect. 7.3.4 and formula (7.3). The substacks Bun ( ≤ θ ) G cover Bun G .So Theorem 4.1.8 is a consequence of the following fact: Theorem 9.1.2.
The substack
Bun ( ≤ θ ) G is co-truncative if for every simple root ˇ α i one has (9.1) h θ , ˇ α i i ≥ g − , where g is the genus of X . In this section we will prove Theorem 9.1.2 modulo a key geometric assertion, Proposi-tion 9.2.2.
Remark . In Theorem 9.1.2 we assume that the ground field k has characteristic 0 (becausethe notion of truncativeness is defined in terms of D-modules). However, Proposition 9.2.2 (ofwhich Theorem 9.1.2 is an easy consequence) is valid over any k . Below follow some remarks on the proof of Theorem 9.1.2.9.1.4.
The main difficulty.
In Sect. 6 we already proved Theorem 9.1.2 for G = SL . The proofin the general case is more or less similar.However, one has to keep in mind the following. If G = SL we saw that all but finitely manyHarder-Narasimhan-Shatz strata Bun ( λ ) G are truncative. This is false already for G = SL × SL .Indeed, the stratum of the form Bun ( n ) SL × Bun ( m ) SL , with n small relative to the genus of X ,is not truncative in Bun SL × Bun SL = Bun SL × SL because Bun ( n ) SL is not truncative inBun SL , see Sect. 6.1.3.For any G , it turns out that Bun ( λ ) G is truncative if λ is “deep inside” the interior of someface of the cone Λ + , Q G ; the problem arises if λ is close to the boundary of the open face of Λ + , Q G containing λ .9.1.5. Resolving the difficulty.
We prove that certain unions of the strata Bun ( µ ) G are truncative(see Corollary 9.2.7). In particular, we show that if λ ∈ Λ + , Q G and(9.2) S λ := { µ ∈ Λ + , Q G | λ − µ ∈ X i ∈ Γ G,λ Q ≥ · α i } , where Γ G,λ := { i ∈ Γ G | h λ , ˇ α i i ≤ g − } then [ µ ∈ S λ Bun ( µ ) G is a truncative locally closed substack of Bun G . To finish the proof of Theorem 9.1.2, we showthat if θ satisfies (9.1) then the set { λ ∈ Λ + , Q G | λ G θ } can be represented as a union of subsets of the form (9.2). We have in mind future applications to the ℓ -adic derived category on Bun G , and this category makessense in any characteristic. HE CATEGORY OF D-MODULES ON Bun G The rank 2 case is representative enough.
All the difficulties of the proof of Theorem 9.1.2appear already if G is a semi-simple group of rank 2. On the other hand, in this case variouscombinatorial-geometric statements (e.g., the above statement at the end of Sect. 9.1.5) becomeobvious once you draw a picture.9.1.7. In Appendix B we give a variant of the proof of Theorem 9.1.2, which has some advan-tages compared with the one from Sect. 9.3. The relation between the two proofs is explainedin Sects. 9.4 and B.4.9.2. A key proposition.
Proposition 9.2.2.
There exists an assignment i ∈ Γ G c i ∈ Q ≥ such that for any parabolic P and any P -admissible subset S ⊂ Λ + , Q G satisfying the condition (9.3) ∀ λ ∈ S, ∀ i ∈ Γ G − Γ M h λ , ˇ α i i > c i (where as usual, M is the Levi quotient of P ) the following properties hold: (a) The morphism
Bun ( S ) P → Bun ( S ) G induced by p P : Bun P → Bun G is an isomorphism; (b) The locally closed substack
Bun ( S ) G ⊂ Bun G is contractive in the sense of Sect. 5.2.1.When char k = 0 we can take c i = max(0 , g − . c i and prove property (a) for these numbers (and, in fact, for smaller ones). InSect. 11 we will prove property (b). Remark . It is only property (b) that will be needed for the proof of Theorem 9.1.2.Property (a) will be used for the proof of property (b).
Remark . Note that the assertion of point (a) of Proposition 9.2.2 differs from that ofProposition 8.3.3 only slightly: the former asserts “isomorphism”, while the latter “almost-isomorphism”.9.2.6. We now specialize to the case of char k = 0, in which case we have the theory of D-modules and of truncative substacks (see Definition 3.4.1).We have: Corollary 9.2.7.
Let S ⊂ Λ + , Q G be a P -admissible subset such that (9.4) ∀ λ ∈ S, ∀ i ∈ Γ G − Γ M we have h λ , ˇ α i i > g − . Then the locally closed substack
Bun ( S ) G ⊂ Bun G is truncative.Proof. Since S is admissible, the condition h λ , ˇ α i i > g − h λ , ˇ α i i > max(0 , g − c i = max(0 , g − (cid:3) Proof of Theorem 9.1.2. ∀ i ∈ Γ G we have h θ , ˇ α i i ≥ g − θ ′ ≥ G θ then the substack Bun ( ≤ θ ′ ) G − Bun ( ≤ θ ) G ⊂ Bun G is truncative.If g = 0 then any locally closed substack of Bun G is truncative (see Sect. 3.2.4). So we canand will assume that g ≥ ( ≤ θ ′ ) G − Bun ( ≤ θ ) G by finitely many truncativesubstacks.Let λ ∈ Λ + , Q G be such that λ G θ , i.e.,(9.6) θ − λ Λ pos, Q G := X i ∈ Γ G Q ≥ · α i . It suffices to construct for each such λ a subset S λ ⊂ Λ + , Q G containing λ such that the substackBun ( S λ ) G ⊂ Bun G is truncative and(9.7) Bun ( ≤ θ ) G ∩ Bun ( S λ ) G = ∅ . Here is the construction. Let P be the parabolic whose Levi quotient, M , corresponds tothe following subset of Γ G :(9.8) Γ M = { i ∈ Γ G | h λ , ˇ α i i ≤ g − } . Now define(9.9) S λ := { λ ′ ∈ Λ + , Q G | λ ′ ≤ M λ } . Note that by (9.8), for each i ∈ Γ G − Γ M we have h λ , ˇ α i i > g −
2, which implies that h λ , ˇ α i i > g ≥ S λ is P -admissibleand satisfies the condition of Corollary 9.2.7. Hence, the substack Bun ( S λ ) G ⊂ Bun G is truncative.Therefore, to prove Theorem 9.1.2 it remains to check (9.7).9.3.3. Proof of equality (9.7) . We need the following lemma.
Lemma 9.3.4.
Let ν = P i ∈ Γ G a i · α i , a i ∈ Q . Assume that a i ≥ for i Γ M and h ν , ˇ α i i ≥ for i ∈ Γ M . Then a i ≥ for all i ∈ Γ G .Proof. Set ν M := P i ∈ Γ M a i · α i . We have to show that ν M ∈ Λ pos, Q M . The assumptions of thelemma and the inequality h α j , ˇ α i i ≤ i = j imply that h ν M , ˇ α i i ≥ i ∈ Γ M . Thus ν M belongs to the dominant cone of the root system of M and therefore to Λ pos, Q M . (cid:3) We are now ready to prove (9.7). It suffices to prove the following
Lemma 9.3.5.
There is no λ ′ ∈ Λ + , Q G such that λ ′ ≤ G θ and λ ′ ≤ M λ .Proof. Suppose that such λ ′ exists. Then θ − λ has the form P i ∈ Γ G c i · α i , where c i ≥ i Γ M .By (9.5) and (9.8), h θ − λ , ˇ α i i ≥ i ∈ Γ M . Hence, by Lemma 9.3.4, θ − λ ∈ Λ pos, Q G , contraryto the assumption (9.6). (cid:3) Thus we have proved Theorem 9.1.2.
HE CATEGORY OF D-MODULES ON Bun G A variant of the proof.
In the above proof of Theorem 9.1.2 we used substacks of theform Bun ( S λ ) G , λ ∈ Λ + , Q G , where S λ is defined by (9.8) -(9.9). Instead of considering all substacksof this form, one could consider only maximal ones among them; one can show that they forma stratification of Bun G all of whose strata are truncative. This somewhat “cleaner” picture isexplained in Appendix B. 10. The estimates
In this section we produce the numbers c i mentioned in Proposition 9.2.2 and prove Propo-sition 9.2.2(a) for these numbers (and, in fact, for smaller ones).10.1. The vanishing of H and H . T ⊂ B . This allows us to view the Leviquotient M of a (standard) parabolic P as a subgroup M ⊂ P (the unique splitting thatcontains T ).Recall that for a parabolic P we denote by U ( P ) its unipotent radical. We will use thefollowing notation for Lie algebras: g := Lie ( G ), p := Lie ( P ), n ( P ) := Lie ( U ( P )).For an algebraic group H , a principal H -bundle F H and an H -representation V , we denoteby V F H the associated vector bundle.10.1.2. The main result of this section is: Proposition 10.1.3.
There exists a collection of numbers c ′ i , ∈ Q , c ′′ i ∈ Q ≥ , i ∈ Γ G such that for any quadruple ( P, M, λ, F M ) , where P is a parabolic, M the corresponding Levi, λ ∈ Λ + , Q G and F M ∈ Bun ( λ ) M , the followingstatements hold: (10.1) if ∀ i ∈ Γ G − Γ M we have h λ , ˇ α i i > c ′ i then H ( X, n ( P ) F M ) = 0;(10.2) if ∀ i ∈ Γ G − Γ M we have h λ , ˇ α i i > c ′′ i then H ( X, ( g / p ) F M ) = 0 . If char k = 0 then one can take c ′ i = 2 g − , c ′′ i = 0 . Proposition 10.1.3 will be proved in Sect. 10.4. For a discussion of the case char k >
0, seeSect. 10.5.10.2.
The numbers c i : proof of Proposition 9.2.2(a). In this subsection we will assumeProposition 10.1.3.Let c ′ i and c ′′ i be as in Proposition 10.1.3. Set(10.3) c i := max( c ′ i , c ′′ i ) . Eventually we will show that Proposition 9.2.2 holds for the numbers c i defined by (10.3). ForProposition 9.2.2(a) this follows from the next lemma (which is slightly sharper than Proposi-tion 9.2.2(a) because the numbers c i are replaced by c ′′ i ≤ c i ). Lemma 10.2.1.
Let c ′′ i be as in Proposition 10.1.3. Let S ⊂ Λ + , Q G be a P -admissible subsetsuch that (10.4) ∀ λ ∈ S, ∀ i ∈ Γ G − Γ M h λ , ˇ α i i > c ′′ i . Then the morphism
Bun ( S ) P → Bun ( S ) G induced by p P : Bun P → Bun G is an isomorphism.Remark . The lemma implies that if char k = 0 then the morphism Bun ( S ) P → Bun ( S ) G is an isomorphism for any P -admissible S ⊂ Λ + , Q G . Indeed, if char k = 0 one can take c ′′ i = 0(see the last sentence of Proposition 10.1.3); on the other hand, for c ′′ i = 0 the inequality (10.4)holds by the definition of P -admissibility, see Definition 8.2.2. Proof of Lemma 10.2.1.
By Proposition 8.3.3, it suffices to show that for any y ∈ Bun P ( k ), thetangent space at y to the fiber of p P : Bun P → Bun G over p P ( y ) is zero.The tangent space in question identifies with H ( X, ( g / p ) F P ), where F P is the P -bundlecorresponding to y .Note that the vector bundle ( g / p ) F M can be identified with the associated graded of ( g / p ) F P with respect to a (canonically defined) filtration on the latter. Hence, (10.2) implies that H ( X, ( g / p ) F P ) = 0. (cid:3) The notion of strangeness. e G be a reductive group and V a finite-dimensional e G -module on which Z ( e G ) actsby a character ˇ µ . Lemma 10.3.2. (i)
There exists a number c ∈ Q such that for every F e G ∈ Bun ss e G the degree of any line sub-bundleof V F e G is ≤ h deg e G ( F e G ) , ˇ µ i + c . (ii) If char k = 0 one can take c = 0 .Proof. Statement (i) follows from the fact that the intersection of Bun ss e G with every connectedcomponent of Bun e G is quasi-compact, and that under the action of Bun Z ( e G ) on Bun e G , thenumber of orbits of π (Bun Z ( e G ) ) on π (Bun e G ) is finite.Statement (ii) follows from the fact that if char k = 0 then for every F e G ∈ Bun ss e G the vectorbundle V F e G is semistable of slope h deg e G ( F e G ) , ˇ µ i (a proof of this fact can be found in [RR, Sect.3]; for references to other proofs see the introduction to [RR]). (cid:3) Definition 10.3.4.
The strangeness strng X ( e G, V ) is the smallest number c ∈ Q having theproperty from Lemma 10.3.2(i). One always has strng X ( e G, V ) ≥ e G -bundle is semi-stable). If char k = 0then strng X ( e G, V ) = 0.
Remark . As before, let e G be a reductive group, V a finite-dimensional e G -module onwhich Z ( e G ) acts by a character ˇ µ , and F ∈ Bun ss e G . Then H ( X, V F e G ) = 0 if h deg e G ( F e G ) , ˇ µ i < − strng X ( e G, V ) ,H ( X, V F e G ) = 0 if h deg e G ( F e G ) , ˇ µ i > g − X ( e G, V ∗ ) . In particular, if char k = 0 then(10.5) H ( X, V F ) = 0 if h deg e G ( F e G ) , ˇ µ i < , H ( X, V F ) = 0 if h deg e G ( F e G ) , ˇ µ i > g − . HE CATEGORY OF D-MODULES ON Bun G Proof of Proposition 10.1.3. P ′ ⊂ G be a parabolic and M ′ ⊂ P ′ the corre-sponding Levi (see our conventions in Sect. 10.1.1); in particular M ′ ⊃ T .Given a root ˇ α of G which is not a root of M ′ , define an M ′ -submodule V M ′ , ˇ α ⊂ g by(10.6) V M ′ , ˇ α := M ˇ β, ˇ β − ˇ α ∈ R ( M ′ ) g ˇ β , where R ( M ′ ) is the root lattice of M ′ .The coefficient of ˇ α i in a root ˇ α will be denoted by coeff i (ˇ α ).10.4.2. We are going to formulate a slightly more precise version of Proposition 10.1.3.Let i ∈ Γ G c ′ i , c ′′ i ∈ Q be numbers satisfying the following inequalities:For every Levi subgroup M ′ ⊂ G , every i ∈ Γ G − Γ M ′ , and every root ˇ α of G such thatcoeff i (ˇ α ) >
0, we have:(10.7) coeff i (ˇ α ) · c ′ i ≥ g − X ( M ′ , ( V M ′ , ˇ α ) ∗ ) . (10.8) coeff i (ˇ α ) · c ′′ i ≥ strng X ( M ′ , V M ′ , − ˇ α ) . Remark.
In the characteristic 0 case we can take c ′ i = 2 g − c ′′ i = 0: indeed, in thiscase the numbers strng X from formulas (10.7)-(10.8) are zero.10.4.4. Here is the promised version of Proposition 10.1.3. Proposition 10.4.5.
Let c ′ i , c ′′ i be numbers satisfying the conditions from Sect. 10.4.2. Let P ⊂ G be a parabolic, M be the corresponding Levi, λ ∈ Λ + , Q G ⊂ Λ + , Q M and F M ∈ Bun ( λ ) M . (a) If h λ , ˇ α i i > c ′ i for all i ∈ Γ G − Γ M then H ( X, n ( P ) F M ) = 0 . (b) If h λ , ˇ α i i > c ′′ i for all i ∈ Γ G − Γ M then H ( X, ( g / p ) F M ) = 0 .Proof. Let P λ be the parabolic of M corresponding to the subset { i ∈ Γ M | h λ , ˇ α i i = 0 } ⊂ Γ M . Let M λ be the corresponding Levi. The fact that F M ∈ Bun ( λ ) M means that F M admits areduction to P λ such that the corresponding M λ -bundle F M λ is semi-stable of degree λ .Let us prove statement (a). Note that the vector bundle n ( P ) F M has a canonical filtra-tion with the associated graded identified with n ( P ) F Mλ . Hence, it suffices to show that H ( X, n ( P ) F Mλ ) = 0.By Remark 10.3.5, it suffices to prove that h λ , ˇ α i > g − X ( M λ , ( V M λ , ˇ α ) ∗ )for any positive root ˇ α of G which is not a root of M .Let i ∈ Γ G − Γ M be such that coeff i (ˇ α ) >
0. Since λ is dominant for G and h λ , ˇ α i i > c ′ i wehave h λ , ˇ α i ≥ coeff i (ˇ α ) · h λ , ˇ α i i > coeff i (ˇ α ) · c ′ i . Now use (10.7) for M ′ = M λ (this is possiblebecause i Γ M and therefore i Γ M λ ).The proof of statement (b) is similar. (cid:3) Remark . If k has characteristic p > upper bounds for the strangeness of the relevant representations. Itwould be interesting to obtain such bounds.10.5. Remarks on the positive characteristic case. V and V be finite-dimensional vector spaces over a field of any characteristic. Let G i denote the algebraic group GL ( V i ). Then the ( G × G )-modules Hom( V , V ) and V ⊗ V have strangeness 0. This immediately follows from the definition of semi-stability. Corollary 10.5.2.
Suppose that G ad ≃ P GL ( d ) × . . . × P GL ( d n ) . Then all inequalities (10.7) - (10.8) hold for c ′ i = 2 g − , c ′′ i = 0 (without any assumption on char k ). k = 2. Let V be a 2-dimensional vector space over k . If g > GL ( V ) in the symmetric square Sym ( V ) has strangeness g − >
0. Thisfollows from Sect. 10.5.1, combined with the exact sequence0 → V (2) → Sym ( V ) → V ⊗ V and the equality strng X ( V (2) ) = g −
1, which is proved, e.g., in [JRXY, Sect. 4.5].10.5.4. The assertion in Sect. 10.5.3 implies that if g > k = 2 and G = Sp (2 n ), n ≥ do not hold for c ′ i = 2 g − c ′′ i is as follows:J. Heinloth [He] proved that if G is a classical group over a field of odd characteristic thenall inequalities (10.8) hold for c ′′ i = 0. He also showed that if char k = 2 this is still true if G ad is a product of groups of type A and C .On the other hand, according to [He, P], some of the inequalities (10.8) do not hold ifchar k = 2 and G ad has one of the following types: G , B n ( n ≥ D n ( n ≥ Constructing the contraction
The goal of this section is to prove point (b) of Proposition 9.2.2 for the numbers c i definedin formula (10.3) from Sect. 10.2.11.1. Morphisms between
Bun P , Bun P − , Bun M , and Bun G . In this subsection and thenext we recall some well known facts that will be used in the proof of Proposition 9.2.2(b).11.1.1. From now on we fix a (standard) parabolic P . Let P − be the parabolic opposite to P such that P − ⊃ T . Note that P − is not a standard parabolic! Namely, P − is the uniqueparabolic such that P ∩ P − = M , when the latter is viewed a subgroup of P , see Sect. 10.1.1. Lemma 11.1.2.
The morphism
Bun M → Bun P − × Bun G Bun P is an open embedding.Proof. An M -bundle on X is the same as a G -bundle F G equipped with an M -structure, i.e.,a section of ( G/M ) F G .The assertion follows from the fact that the morphism G/M → G/P − × G/P is an openembedding. (cid:3) A good upper bound should have the form c ( p, G, ˇ α ) · ( g − c ( p, G, ˇ α ) should be independentof X and small enough to explain the phenomenon in Sect. 10.5.1 below. HE CATEGORY OF D-MODULES ON Bun G U i ⊂ Bun M as follows:(11.1) U := { F M ∈ Bun M | H ( X, ( g / p ) F M ) = 0 } , (11.2) U := { F M ∈ Bun M | H ( X, n ( P ) F M ) = 0 } . Proposition 11.1.4.
We have: (a)
The morphism ι P : Bun M → Bun P induces a smooth surjective morphism U → Bun P × Bun M U . (b) The morphism p P − : Bun P − → Bun G is smooth when restricted to the open substack Bun P − × Bun M U ⊂ Bun P − . (c) The morphism q P − : Bun P − → Bun M is schematic, affine, and smooth over U ⊂ Bun M . (d) The morphism ι P − : Bun M → Bun P − defines a closed embedding U → Bun P − × Bun M U . Proof.
Let F M ∈ U ( k ), i.e., F M is an M -torsor on X such that H ( X, n ( P ) F M ) = 0. Usingan appropriate filtration on U ( P ) one deduces from this that H ( X, U ( P ) F M ) = 0, i.e., every U ( P ) F M -torsor on X is trivial. This implies the surjectivity part of statement (a).To prove the smoothness part of (a), it suffices to check that the differential of the morphism ι P : Bun M → Bun P at any point F M ∈ U ( k ) ⊂ Bun M ( k ) is surjective. Its cokernel equals H ( X, n ( P ) F M ), which is zero by the definition of U , see formula (11.2).Note also that the smoothness part of (a) will follow from (b): to see this, decompose themorphism ι P : Bun M → Bun P asBun M → Bun P − × Bun G Bun P → Bun P and use Lemma 11.1.2.To prove (b), we have to show that the differential of p P − : Bun P − → Bun G at any k -point y of Bun P − × Bun M U is surjective. Its cokernel equals H ( X, ( g / p − ) F P − ), where F P − is the P − -bundle corresponding to y . Let F M be the corresponding M -bundle. We have H ( X, n ( P ) F M ) = 0 by the definition of U , see formula (11.2). Now, the associated graded of( g / p − ) F P − with respect to a (canonically defined) filtration identifies with ( g / p − ) F M , and theassertion follows from the fact that the composition n ( P ) → g → g / p − is an isomorphism of M -modules.To prove (c), consider the filtration U ( P − ) = U ⊃ U ⊃ . . . , where U m is the subgroup generated by the root subgroups corresponding to the roots ˇ α of G such that X i Γ M coeff i (ˇ α ) ≤ − m. (Here coeff i (ˇ α ) denotes the coefficient of ˇ α i in ˇ α .) Note that each quotient U m /U m +1 is avector group (i.e., a product of finitely many copies of G a ). To prove (c), it suffices to checkthat for each m the morphism(11.3) (Bun P − /U m ) × Bun M U → (Bun P − /U m +1 ) × Bun M U is schematic, affine and smooth. In fact, it is a torsor over a certain vector bundle.To see this, note that by (11.1), for each F M ∈ U we have H ( X, ( U m /U m +1 ) F M ) = 0 , so the stack of ( U m /U m +1 ) F M -torsors on X is a scheme; namely, it is the vector space H ( X, ( U m /U m +1 ) F M ). As F M varies, these vector spaces form a vector bundle on U . Let ξ denote its pullback to (Bun P − /U m ) × Bun M U , then the morphism (11.3) is a ξ -torsor.Point (d) follows from point (c) since the map U → Bun P − × Bun M U is a section of the mapBun P − × Bun M U → U , and the latter is schematic and separated. (cid:3) The action of A on Bun P − . Z ( M ) denote the center of M . Choose a homomorphism µ : G m → Z ( M ) suchthat h µ , ˇ α i i > i Γ M . Then the action of G m on P − defined by(11.4) ρ t ( x ) := µ ( t ) − · x · µ ( t ) , t ∈ G m , x ∈ P − extends to an action of the multiplicative monoid A on P − such that the endomorphism ρ ∈ End( P ) equals the composition P − ։ M ֒ → P − .11.2.2. The above action of A on P − induces an A -action on Bun P − . Equip M and Bun M with the trivial A -action. The projection P − → M is A -equivariant, so the correspondingmorphism q P − : Bun P − → Bun M has a canonical A -equivariant structure.Remark . The above description of ρ implies that the morphism : Bun P − → Bun P − corresponding to 0 ∈ A equals the composition(11.5) Bun P − q P − −→ Bun
M ι P − −→ Bun P − . Remark . The action of G m on Bun P − is trivial: this follows from formula (11.4), whichsays that the automorphisms ρ t ∈ Aut( P − ), t ∈ G m , are inner. Moreover, formula (11.4)provides a canonical trivialization of this action. Remark . Despite the previous remark, it is not true that the action of G m on each fiber of the morphism Bun P − → Bun M is trivial. (Note that although G m acts on Bun P − byautomorphisms over Bun M , the trivialization of the G m -action on Bun P − provided by (11.4)is not over Bun M .) Remark . It is not hard to show that the trivialization of the G m -action on Bun P − defined in Remark 11.2.4 yields an action of the monoidal stack A / G m on Bun P − . Theproof is straightforward; it uses the formula ρ t ( µ ( s )) = µ ( s ) , t ∈ A , s ∈ G m , For any scheme S , the groupoid ( A / G m )( S ) is the groupoid of line bundles over S equipped with a section,so ( A / G m )( S ) is a monoidal category with respect to ⊗ . In this sense A / G m is a monoidal stack. HE CATEGORY OF D-MODULES ON Bun G which follows from (11.4).11.3. Proof of Proposition 9.2.2(b). c i , i ∈ Γ G , be as in formula (10.3) from Sect. 10.2. Let S ⊂ Λ + , Q G bea P -admissible subset, and assume that S satisfies (9.3), i.e., ∀ λ ∈ S, ∀ i ∈ Γ G − Γ M we have h λ , ˇ α i i > c i . We have to prove that the locally closed substack Bun ( S ) G ⊂ Bun G is contractive in the sense ofSect. 5.2.1.11.3.2. Recall that c i := max( c ′ i , c ′′ i ), where c ′ i and c ′′ i are as in Proposition 10.1.3. So for all λ ∈ S and i ∈ Γ G − Γ M we have(11.6) h λ , ˇ α i i > c ′ i , (11.7) h λ , ˇ α i i > c ′′ i . By (11.6) and the assumption on the numbers c ′ i (see Proposition 10.1.3), we have(11.8) Bun ( S ) M ⊂ U := { F M ∈ Bun M | H ( X, ( n ( P )) F ) = 0 } . Similarly, (11.7) implies that(11.9) Bun ( S ) M ⊂ U := { F M ∈ Bun M | H ( X, ( g / p ) F M ) = 0 } . ( S ) P − ⊂ Bun P − denote the preimage of the open substack Bun ( S ) M ⊂ Bun M . Theembeddings M ֒ → P ֒ → G and M ֒ → P − ֒ → G induce a commutative diagram(11.10) Bun ( S ) Mι ( S ) P (cid:15) (cid:15) ι ( S ) P − / / Bun ( S ) P − p ( S ) P − (cid:15) (cid:15) Bun ( S ) P p ( S ) P / / Bun G We summarize the properties of the maps in the above diagram in the following lemma:
Lemma 11.3.4. (i)
The morphism
Bun ( S ) M → Bun ( S ) P × Bun G Bun ( S ) P − defined by diagram (11.10) is an open embed-ding. (ii) The morphism ι ( S ) P : Bun ( S ) M → Bun ( S ) P is surjective and smooth. (iii) The morphism p ( S ) P − : Bun ( S ) P − → Bun G is smooth. (iv) The morphism p ( S ) P induces an isomorphism Bun ( S ) P → Bun ( S ) G . (v) The morphism ι ( S ) P − : Bun ( S ) M → Bun ( S ) P − is a closed embedding. Proof.
Statement (i) follows from Lemma 11.1.2.By (11.8), statements (ii) and (iii) follow from Proposition 11.1.4 points (a) and (b), respec-tively.Statement (iv) holds by Proposition 9.2.2(a). By (11.9), statement (v) follows from Propo-sition 11.1.4(d). (cid:3)
Remark . One can show, using [Sch, Proposition 4.4.4], that the map in point (i) ofLemma 11.3.4 is an isomorphism for any P -admissible set S (i.e., S does not even have tosatisfy (9.3).)11.3.6. Our goal is to prove that the locally closed substack Bun ( S ) G ⊂ Bun G is contractive.By the definition of contractiveness (see Sect. 5.2.1), this follows from Lemma 11.3.4 and thenext statement: Proposition 11.3.7.
Let S be as in Proposition 9.2.2. Then the substack Im( ι ( S ) P − ) ⊂ Bun ( S ) P − from Lemma 11.3.4(v) is contractive.Proof. Equip Bun P − with the A -action from Sect. 11.2 corresponding to some µ : G m → Z ( M ).The open substack Bun ( S ) P − ⊂ Bun P − is A -stable, so we obtain an A -action on Bun ( S ) P − .We apply Lemma 5.4.3 to the canonical morphism q ( S ) P − : Bun ( S ) P − → Bun ( S ) M and the above A -action on Bun ( S ) P − . We only have to check that the conditions of the lemma hold.By (11.9) and Proposition 11.1.4(c), the morphism q ( S ) P − : Bun ( S ) P − → Bun ( S ) M is schematic andaffine. Conditions (i)-(ii) from Lemma 5.4.3 hold by Remarks 11.2.3-11.2.4. (cid:3) Counterexamples
The goal of this section is to show that the property of being truncatable is a purely “stacky”phenomenon, i.e., that it “typically” fails for non quasi-compact schemes.12.1.
Formulation of the theorem.Theorem 12.1.1.
Let Y be an irreducible smooth scheme of dimension n , such that for some(or, equivalently, any) non-empty quasi-compact open U ⊂ Y the set (12.1) { y ∈ Y − U | dim y ( Y − U ) = dim Y − } is not quasi-compact. Then D-mod( Y ) is not compactly generated. The theorem will be proved in Sect. 12.2 below. Here are two examples of schemes Y satisfying the condition of Theorem 12.1.1. Example . Let I be an infinite set and let Y be the non-separated curve that one obtainsfrom A × I by gluing together the open subschemes ( A − { } ) × { i } , i ∈ I (in other words, Y is the affine line with the point 0 repeated I times). Example . Let X be a smooth surface and x ∈ X a point. Set U = X − { x } . Let X be the blow-up of X at x . Let x ∈ X be a point on the exceptional divisor. We havean open embedding U = X − { x } ֒ → X − { x } = U such that U − U is a divisor. We can now apply the same process for ( X , x ) instead of( X , x ). Thus we obtain a sequence of schemes U ֒ → U ֒ → U ֒ → ... HE CATEGORY OF D-MODULES ON Bun G Then Y := S i U i satisfies the condition of Theorem 12.1.1. Note that Y is separated if X is.12.2. Proof of Theorem 12.1.1.
We will use facts from Sect. 2.2.10 about the relation be-tween compactness and coherence (in the easier case of smooth schemes).12.2.1. Let Y be a smooth scheme, Z ⊂ Y a non-empty divisor, and Y − Z = U j ֒ → Y be thecomplementary open embedding. Lemma 12.2.2.
Suppose that N ∈ D-mod( Y ) is coherent and j ∗ ◦ j ∗ ( N ) = N . Then thesingular support SS ( N ) ⊂ T ∗ ( Y ) is not equal to T ∗ ( Y ) .Proof. We can assume that Y is affine and Z is smooth. Since j ∗ is t-exact we can also assumethat N is in D-mod( N ) ♥ . Suppose that SS ( N ) = T ∗ ( Y ). Then there exists an injective map D Y ֒ → N , where D Y is the D-module of differential operators on Y . We obtain an injectivemap j ∗ ◦ j ∗ ( D Y ) ֒ → j ∗ ◦ j ∗ ( N ) = N . But N is coherent while j ∗ ◦ j ∗ ( D Y ) is not. (cid:3) Y be as in Theorem 12.1.1 and M ∈ D-mod( Y ) a compact object. Note that byRemark 2.2.13, M is automatically coherent.We claim: Lemma 12.2.4. SS ( M ) = T ∗ ( Y ) .Proof. By Proposition 2.3.7, there exists a quasi-compact open U j ֒ → Y such that M = j ! ( j ∗ ( M ))or equivalently, D Verdier Y ( M ) = j ∗ ◦ j ∗ ( D Verdier Y ( M )) . We can assume that U = ∅ (otherwise M = 0 and SS ( M ) = ∅ ). Then the set (12.1) isnon-empty, so after shrinking Y we can assume that the set Z := Y − U is a non-empty divisor.Applying Lemma 12.2.2 to N = D Verdier Y ( M ) we get SS ( D Verdier Y ( M )) = T ∗ ( Y ). Finally, SS ( M ) = SS ( D Verdier Y ( M )). (cid:3) C is denoted by C c . Lemma 12.2.6.
Let A ⊂ D-mod( Y ) be the DG subcategory generated by D-mod( Y ) c . If M ∈ A is coherent then SS ( M ) = T ∗ ( Y ) .Proof. Let U j ֒ → Y be a non-empty quasi-compact open subset.Let C ⊂ D-mod( U ) be the full DG subcategory of D-mod( U ) generated by j ∗ (D-mod( Y ) c ).Since j ∗ (D-mod( Y ) c ) ⊂ D-mod( U ) c , we have C c = C ∩ D-mod( U ) c , and by Corollary 1.4.6, the latter is Karoubi-generated by j ∗ (D-mod( Y ) c ).This observation, combined with Lemma 12.2.4 and the fact that T ∗ ( U ) is dense in T ∗ ( Y ),implies that for any N ∈ C c , SS ( N ) = T ∗ ( U ) . Now, j ∗ ( M ) ∈ C ∩ D-mod coh ( U ), and since U is quasi-compact, we have D-mod coh ( U ) =D-mod( U ) c . Hence, j ∗ ( M ) ∈ C c , implying the assertion of the lemma. (cid:3) Corollary 12.2.7.
The DG category A from Lemma 12.2.6 does not contain D Y . Theorem 12.1.1 clearly follows from Corollary 12.2.7.
Appendix A. Preordered sets as topological spaces
The material in this section is standard.A.1.
Definition of the topology.
Given a preordered set X we equip it with the followingtopology: a subset U ⊂ X is said to be open if for every x ∈ U one has { y ∈ X | y ≤ x } ⊂ U . Lemma A.1.1. (i)
A subset F ⊂ X is closed if and only if for every x ∈ F one has { y ∈ X | y ≥ x } ⊂ F . (ii) A subset Z ⊂ X is locally closed if and only if (A.1) ∀ x , x ∈ Z { y ∈ X | x ≤ y ≤ x } ⊂ Z. (iii) For every subset Y ⊂ X the topology on Y corresponding to the induced preordering on Y is induced by the topology on X .Proof. We will only prove (ii). Condition (A.1) holds for locally closed subsets because it holdsfor open and closed ones. Conversely, if Z satisfies (A.1) then Z has the following representationas F ∩ U with F closed and U open: F := ¯ Z = { x ∈ X | ∃ z ∈ Z : z ≤ x } , U := { x ∈ X | ∃ z ∈ Z : z ≥ x } . (cid:3) A.2.
Continuous maps.
The following is also easy to see:
Lemma A.2.1.
Let
X, X ′ be preordered sets equipped with the above topology. Then a map f : X → X ′ is continuous if and only if it is monotone, i.e., x ≤ x ⇒ f ( x ) ≤ f ( x ) . (cid:3) Appendix B. The Langlands retraction and coarsenings of theHarder-Narsimhan-Shatz stratification
In Sect. B.1 we recall the definition of the Langlands retraction L : Λ Q G → Λ + , Q G .Using this retraction, we define in Sect. B.2 a coarsening of the usual Harder-Narasimhan-Shatz stratification of Bun G depending on the choice of η ∈ Λ + , Q G (the usual stratification itselfcorresponds to η = 0).In Sect. B.3 we show that if η is “deep inside” Λ + , Q G then all the strata of the correspondingstratification are contractive (and therefore truncative if char k = 0). Combined with Proposi-tion 9.2.2 this immediately implies Theorem 9.1.2 (see Sect. B.3.5 below).In Sect. B.4 we explain the relation between the two proofs of Theorem 9.1.2.B.1. Recollections on the Langlands retraction.
Equip Λ Q G with the ≤ G ordering. Thefollowing notion goes back to [La, Sect. 4]. Definition B.1.1.
The
Langlands retraction L : Λ Q G → Λ + , Q G is defined as follows: for λ ∈ Λ Q G ,let L ( λ ) be the least element of the set { µ ∈ Λ + , Q G | µ ≥ G λ } in the sense of the ≤ G ordering. B.1.2. The existence of the least element is not obvious; it was proved by R.P. Langlands in[La, Sect. 4]. The material from [La, Sect. 4] is known under the name of “Langlands’ geometriclemmas”. We give a short review of it in [Dr]. In particular, we give there two proofs of theexistence of the least element: J. Carmona’s “metric” proof (see [Dr, Sections 2-3]) and anotherone (see Sect. 4 of [Dr], including Example 4.3).
HE CATEGORY OF D-MODULES ON Bun G B.1.3. It is clear that the map L : Λ Q G → Λ + , Q G is an order-preserving retraction. The followingdescription of the fibers of L is given in [La, Sect. 4]; see also [Dr, Cor. 5.3(iii)]. Lemma B.1.4.
For any λ ∈ Λ + , Q G one has L − ( λ ) = λ + X i ∈ I λ Q ≤ · α i , where I λ := { i ∈ Γ G | h λ, α i i = 0 } . B.2.
The η -stratification. B.2.1.
The η -shifted Langlands retraction. Let η ∈ Λ + , Q G . The map(B.1) L + η : Λ + , Q G → ( η + Λ + , Q G ) , L + η ( λ ) := L ( λ − η ) + η is an order-preserving retraction (this follows from a similar property of L ). By definition,(B.2) ∀ λ ′ ∈ Λ + , Q G , ∀ λ ∈ ( η + Λ + , Q G ) we have L + η ( λ ′ ) ≤ G λ ⇔ λ ′ ≤ G λ. B.2.2.
The η -stratification of Bun G . In Sect. 7.4.8 we defined the Harder-Narasimhan mapHN : | Bun G ( k ) | → Λ + , Q G and formulated three properties of it, see Lemma 7.4.9 (i-iii). Sincethe map L + η : Λ + , Q G → ( η + Λ + , Q G ) is order-preserving, the map(B.3) HN η : | Bun G ( k ) | → ( η + Λ + , Q G ) , HN η := L + η ◦ HNhas the same three properties. So the fibers of the map (B.3) form a stratification of Bun G withquasi-compact strata. We call it the η -stratification of Bun G ; the corresponding strata are(B.4) Bun ( λ ) η G := [ λ ′ ∈ ( L + η ) − ( λ ) Bun ( λ ′ ) G , λ ∈ ( η + Λ + , Q G ) . It is clear that the η -stratification is coarser than the Harder-Narsimhan-Shatz stratification(the word ”coarser” is understood in the non-strict sense).B.2.3. Open substacks associated to the η -stratification. Recall that for each λ ∈ Λ + , Q G the opensubstack Bun ( ≤ λ ) G ⊂ Bun G is the union of the strataBun ( λ ′ ) G , λ ′ ≤ G λ. If one considers similar unions of the strata of the η -stratification then one gets “essentially”the same class of open substacks of Bun G ; more precisely, we claim that for each λ ∈ ( η + Λ + , Q G )one has [ λ ′ ∈ ( η +Λ + , Q G ) , λ ′ ≤ G λ Bun ( λ ′ ) η G = Bun ( ≤ λ ) G . This follows from (B.2) and (B.4).B.2.4.
Changing η . If η ′ ∈ ( η + Λ + , Q G ) then L + η ′ ◦ L + η = L + η ′ , so the η ′ -stratification is coarserthan the η -stratification. If η ′ and η have the same image in Λ + , Q G ad then L + η ′ = L + η , so the η ′ -stratification and the η -stratification are the same. B.2.5. ( L + η ) − ( λ ) as a P -admissible set. Let λ ∈ ( η + Λ + , Q G ). Let P be the parabolic whoseLevi quotient, M , corresponds to the following subset of Γ G :(B.5) Γ M = { i ∈ Γ G | h λ − η , ˇ α i i = 0 } . Equivalently,(B.6) Γ G − Γ M = { i ∈ Γ G | h λ , ˇ α i i > h η , ˇ α i i} . By Lemma B.1.4, the subset T λ := ( L + η ) − ( λ ) ⊂ Λ + , Q G has the following description in termsof the ≤ M ordering:(B.7) T λ = { λ ′ ∈ Λ + , Q G | λ ′ ≤ M λ } . So by (B.6) and Lemma 8.1.2(a), the set T λ is P -admissible in the sense of Definition 8.2.2.Moreover, by (B.4) and (B.7), the stratum Bun ( λ ) η G is equal to the locally closed substackBun ( T λ ) G defined in Sect. 8.3.1 by formula (8.6).B.3. The case where η is “deep inside” Λ + , Q G . B.3.1.
Contractiveness of the strata.
Suppose now that(B.8) h η , ˇ α i i ≥ c i for all i ∈ Γ G , where the numbers c i ∈ Q ≥ are as in Proposition 9.2.2. Proposition B.3.2.
Under these conditions, all strata of the η -stratification are contractive.Proof. Let λ ∈ ( η + Λ + , Q G ). By Sect. B.2.5, Bun ( λ ) η G = Bun ( T λ ) G , where T λ ⊂ Λ + , Q G is the P -admissible set defined by (B.7). So by Proposition 9.2.2(b), it suffices to check that for all λ ′ ∈ T λ and i ∈ Γ G − Γ M one has h λ ′ , ˇ α i i > c i . If λ ′ = λ this is clear from (B.6) and (B.8).The general case follows by Lemma 8.1.2(a). (cid:3) B.3.3.
The characteristic 0 case.
Now assume that char k = 0. Then by Proposition 9.2.2, onecan take c i = max(0 , g − g is the genus of X . In this situation condition (B.8) canbe rewritten as(B.9) η ∈ ( η + Λ + , Q G ) , where η := max(0 , g − · ρ (as usual, ρ denotes the half-sum of positive coroots). E.g., one can take η = η .In characteristic 0 we have the notion of truncativeness and the fact that contractivenessimplies truncativeness, see Corollary 5.2.3. Thus we get part (i) of the following Corollary B.3.4.
Suppose that char k = 0 and (B.10) h η , ˇ α i i ≥ max(0 , g − for all i ∈ Γ G . Then: (i) all strata of the η -stratification are truncative; (ii) the open strata of the η -stratification are co-truncative;Proof. We have already proved (i). The complement of an open stratum is a union of strata,so statement (ii) follows from Proposition 3.7.2. (cid:3)
HE CATEGORY OF D-MODULES ON Bun G B.3.5.
Proof of Theorem 9.1.2.
We have to show that the substack Bun ( ≤ η ) G ⊂ Bun G is co-truncative if h η , ˇ α i i ≥ g − i ∈ Γ G . If g = 0 then any open substack of Bun G isco-truncative by Sect. 3.2.4. So we can assume that (B.10) holds. Then the statement followsfrom Corollary B.3.4(ii) because Bun ( ≤ η ) G is an open stratum of the η -stratification; namely, itis the stratum corresponding to η ∈ ( η + Λ + , Q G ). (cid:3) B.4.
Relation between the two proofs of Theorem 9.1.2.
Suppose that char k = 0 and g ≥ ( S λ ) G ⊂ Bun G , λ ∈ Λ + , Q G ,where(B.11) S λ := { λ ′ ∈ Λ + , Q G | λ ′ − λ ∈ X i ∈ I Q ≤ · α i } , I := { i ∈ Γ G | h λ, α i i ≤ g − } . These substacks are related to the strata of the η -stratification, where η is as in (B.9). Therelation is as follows. The stratum of the η -stratification corresponding to λ ∈ ( η + Λ + , Q G )equals Bun ( S λ ) G . On the other hand, for any λ ∈ Λ + , Q G the stack Bun ( S λ ) G is a locally closedsubstack of the stratum of the η -stratification corresponding to L + η ( λ ). (The proof of thesefacts is left to the reader.) Appendix C. A stacky contraction principle
The main goal of this appendix is to prove Theorem C.5.3 and Corollaries C.5.4-C.5.5.Corollary C.5.5 is a “contraction principle”, which is slightly more general than Proposi-tion 5.1.2. Theorem C.5.3 and Corollary C.5.4 are generalizations of the classical adjunctionfrom Proposition 5.3.2.Convention: throughout this appendix algebras are always associative but not necessarilyunital; coalgebras are coassociative but not necessarily counital.C.1.
Idempotent algebras in monoidal categories.
The notions of algebra and coalgebramake sense in any monoidal category.
Definition C.1.1.
An algebra A in a monoidal category is said to be idempotent if the multi-plication morphism A ⊗ A → A is an isomorphism.Remark C.1.2 . The dual notion of idempotent coalgebra is, in fact, equivalent to that ofidempotent algebra: an isomorphism m : A ⊗ A → A is an algebra structure if and only if m − : A → A ⊗ A is a coalgebra structure.In any monoidal category M the unit object M has a canonical structure of idempotentalgebra.Here is another example. Any monoid M can be considered as a monoidal category (with M as the set of objects and no morphisms other than the identities). In particular, this appliesto { , } as a monoid with respect to multiplication. Clearly 0 is an idempotent algebra in themonoidal category { , } . Remark
C.1.3 . The category of idempotent algebras in a monoidal category M is equivalent tothe category of monoidal functors F : { , } → M ; namely, the idempotent algebra correspond-ing to F is F (0).Let C be a category. By an idempotent functor C → C we mean an idempotent algebra inthe monoidal category of functors C → C . One can think of idempotent functors in terms ofthe two mutually inverse constructions below. Construction 1.
Suppose we have categories A and C , functors A i −→ C π −→ A , and anisomorphism f : π ◦ i ∼ −→ Id A . Set := i ◦ π . Then : C → C is an idempotent functor: theisomorphism ◦ ∼ −→ is the composition ◦ = ( i ◦ π ) ◦ ( i ◦ π ) = i ◦ ( π ◦ i ) ◦ π ∼ −→ i ◦ Id A ◦ π = i ◦ π = . Construction 2.
Let C be a category equipped with an idempotent functor : C → C . Equiv-alently, C carries an action of the monoid { , } (see Remark C.1.3). Let C be the categoryof { , } -equivariant functors { } → C . Then the { , } -equivariant maps { } ֒ → { , } → { } induce functors C π ←− C i ←− C with π ◦ i = Id C . Equivalently, one can think of C asthe category of -modules c in C such that the morphism · c → c is an isomorphism; then i : C → C is the forgetful functor and π : C → C is the “free module” functor.It is easy to check that the above Constructions 1 and 2 are mutually inverse. Remark
C.1.4 . In the situation of Construction 1, the algebra is unital (or equivalently, is amonad) if and only if ( π, i ) is an adjoint pair of functors with f : π ◦ i ∼ −→ Id A being one ofthe adjunctions (in this case the unit of is the other adjunction). Similarly, is a counitalcoalgebra (or equivalently, a comonad) if and only if ( i, π ) is an adjoint pair.C.2. The monodromic subcategory.
Let Y be a QCA stack equipped with a G m -action.Then one has the quotient stack Y / G m and the canonical morphism p : Y → Y / G m . Definition C.2.1.
The monodromic subcategory D-mod( Y ) µ ⊂ D-mod( Y ) is the subcategorygenerated by the essential image of p ! : D-mod( Y / G m ) → D-mod( Y ) (or equivalently, by theessential image of p ∗ ). (A more precise name for D-mod( Y ) µ would be “unipotently monodromic subcategory.”) Lemma C.2.2.
If the G m -action on Y is trivial then D-mod( Y ) µ = D-mod( Y ) .Proof. A trivialization of the G m -action on Y identifies Y / G m with Y × (pt / G m ) and themorphism p : Y → Y / G m with the canonical morphism Y = Y × pt → Y × (pt / G m ). (cid:3) C.3.
Recollections on the renormalized direct image.
Let π : Y → Y be a morphismof QCA stacks. The renormalized direct image functor π N : D-mod( Y ) → D-mod( Y )is defined in [DrGa1, Sect. 9.3] to be the functor dual to π ! : D-mod( Y ) → D-mod( Y ) (dual inthe sense of Sects. 1.5.2 and 2.2.16 of this article). By definition, π N is continuous. One also has anot necessarily continuous de Rham direct image functor π dR , ∗ : D-mod( Y ) → D-mod( Y ), see[DrGa1, Sect. 7.4]. If π dR , ∗ is continuous then one has a canonical isomorphism π N ∼ −→ π dR , ∗ ,see [DrGa1, Corollary 9.3.8]. For instance, this happens if the fibers of π are algebraic spaces,see [DrGa1, Corollary 10.2.5].C.4. Formulation of the theorem.
Let Y be a QCA stack equipped with an action of themultiplicative monoid A . Let ∈ Mor( Y , Y ) denote the endomorphism of Y correspond-ing to 0 ∈ A . One has continuous functors ! , N : D-mod( Y ) → D-mod( Y ) (the func-tor dR , ∗ is continuous only if it equals N ). By Remark C.1.3, is an idempotent alge-bra in the monoidal category Mor( Y , Y ). So the functors N and ! are idempotent algebrasin the monoidal category Funct cont (D-mod( Y ) , D-mod( Y )) and also in the monoidal category This notion makes sense because is an algebra in the monoidal category of functors C → C . This is a “baby case” of the theory of retracts and idempotents in ∞ -categories from [Lu1, Sect. 4.4.5]. HE CATEGORY OF D-MODULES ON Bun G Funct cont (D-mod( Y ) µ , D-mod( Y ) µ ) (here D-mod( Y ) µ ⊂ D-mod( Y ) is the monodromic subcate-gory, see Sect. C.2). By Remark C.1.2, one can also consider N and ! as idempotent coalgebras. Theorem C.4.1.
The algebra N ∈ Funct cont (D-mod( Y ) µ , D-mod( Y ) µ ) is unital. The coalgebra ! ∈ Funct cont (D-mod( Y ) µ , D-mod( Y ) µ ) is counital. A proof will be given in Sect. C.6-C.8. A slightly different proof will be sketched in Sect. C.9.
Corollary C.4.2.
If the G m -action on Y is trivial then the algebra N ∈ Funct cont (D-mod( Y ) , D-mod( Y )) is unital and the coalgebra ! ∈ Funct cont (D-mod( Y ) , D-mod( Y )) is counital.Proof. Use Theorem C.4.1 and Lemma C.2.2. (cid:3)
C.5.
Reformulation in terms of adjunctions.
Let Y be as in Sect. C.4. In particular,the submonoid { , } ⊂ A acts on Y . Define Y to be the stack of { , } -equivariant maps { } → Y . Equivalently, for any test scheme S , the groupoid Y ( S ) is obtained from Y ( S ) usingConstruction 2 from Sect. C.1. It is clear that the stack Y is QCA.The { , } -equivariant maps { , } → { } ֒ → { , } induce morphisms(C.1) Y i ←− Y π ←− Y , π ◦ i = Id Y , i ◦ π = . The A -action on Y induces an A -action on the diagram (C.1). The A -action on Y is canon-ically trivial (this follows from the identity λ · A ). So D-mod( Y ) µ = D-mod( Y ). Example
C.5.1 . Let Y be the stack Bun P − equipped with the A -action from Sect 11.2. Then Y = Bun M and diagram (C.1) identifies with diagram (11.5). Remark
C.5.2 . Since π ◦ i = Id Y the morphism i is representable (i.e., its fibers are algebraicspaces rather than stacks). So the renormalized direct image functor i N equals the “usual”direct image i dR , ∗ .By Remarks C.1.4 and C.5.2, one can reformulate Theorem C.4.1 and Corollary C.4.2 asfollows. Theorem C.5.3.
The functors (C.2) π N : D-mod( Y ) µ ⇄ D-mod( Y ) : i dR , ∗ , i ! : D-mod( Y ) µ ⇄ D-mod( Y ) : π ! form adjoint pairs with the adjunctions π N ◦ i dR , ∗ ∼ −→ Id D-mod( Y ) and Id D-mod( Y ) ∼ −→ i ! ◦ π ! coming from the isomorphism π ◦ i ∼ −→ Id Y . Corollary C.5.4.
If the G m -action on Y is trivial then the functors (C.3) π N : D-mod( Y ) ⇄ D-mod( Y ) : i dR , ∗ , i ! : D-mod( Y ) ⇄ D-mod( Y ) : π ! form adjoint pairs with the adjunctions π N ◦ i dR , ∗ ∼ −→ Id D-mod( Y ) and Id D-mod( Y ) ∼ −→ i ! ◦ π ! coming from the isomorphism π ◦ i ∼ −→ Id Y . Example C.5.1 shows that i is not necessarily a monomorphism. Simpler example: let G be an affinealgebraic group equipped with an A -action and set Y := pt /G , then Y = pt /G , where G ⊂ G is thesubgroup of A -fixed points. Corollary C.5.5.
Suppose that the G m -action on Y is trivial and the morphism i : Y → Y is a composition of an almost-isomorphism Y → Z and a locally closed embedding Z ֒ → Y .Then the substack Z ⊂ Y is truncative.Remark C.5.6 . The assumption of Corollary C.5.5 is equivalent to the following one: ( Y ) ⊂ Z and | Z : Z → Z is an almost-isomorphism. Remark
C.5.7 . Corollary C.5.5 holds even if Y is locally QCA (but not necessarily quasi-compact). To show this, we can assume that Y is quasi-compact (otherwise replace Y by Y × Y S , where S is any quasi-compact scheme equipped with a smooth morphism to Y ). Then Y is contained in a quasi-compact open substack U ⊂ Y . Since the G m -action on Y is trivial U is A -stable. Applying Corollary C.5.5 to U we see that Y is truncative in U and thereforein Y .C.6. The key lemma.
Similarly to the notion of monoidal groupoid, there is a notion ofmonoidal stack. Of course, any algebraic group or the multiplicative monoid A are examplesof monoidal stacks. In the proof of Theorem C.4.1 we will use the monoidal stack A / G m . (Ifyou wish, S -points of A / G m can be interpreted as line bundles over S equipped with a section;this is a monoidal category with respect to ⊗ ).Let G be a monoidal QCA stack over k . Then D-mod( G ) is a monoidal category with respectto the convolution(C.4) M ∗ N := m N ( M ⊠ N ) , M, N ∈ D-mod( G ) , where m : G × G → G is the multiplication map.For any g ∈ G ( k ) define g ∈ D-mod( G ) to be the direct image of k ∈ D-mod(pt) under themap g : pt → G (this is a kind of “delta-function” at g ). The assignment g g is a monoidalfunctor G ( k ) → D-mod( G ). In particular, 1 ∈ D-mod( G ) is the unit object.If f : G → G is a morphism of monoidal stacks then f N : D-mod( G ) → D-mod( G ) is amonoidal functor. If f is only a morphism of semigroups then f N is a semigroupal functor,so f N (1) ∈ D-mod( G ) is an idempotent algebra.Applying this to 0 : pt → A / G m we see that 0 ∈ D-mod( A / G m ) is an idempotent algebra. Lemma C.6.1.
The algebra ∈ D-mod( A / G m ) is unital.Proof. Consider the morphisms { } / G m i ֒ → A / G m π −→ { } / G m induced by the morphisms { } ֒ → A → { } . Set C := D-mod( { } / G m ); this is a monoidal category because { } / G m is amonoidal stack. We have a monoidal functor π dR , ∗ : D-mod( A / G m ) → C and a semigroupalfunctor i dR , ∗ : C → D-mod( A / G m ) with π dR , ∗ ◦ i dR , ∗ = Id C . By definition, 0 = i dR , ∗ ( C ),where C is the unit object of C .Let us now construct the unit e : 1 → π dR , ∗ , i dR , ∗ ) is anadjoint pair of functors (this is the “baby case” of Theorem C.5.3). So M aps (1 ,
0) = M aps (1 , i dR , ∗ ( C )) = M aps ( π dR , ∗ (1) , C ) = M aps ( C , C );more precisely, the map π dR , ∗ : M aps (1 , → M aps ( C , C ) is an isomorphism. Define e : 1 → π dR , ∗ ( e ) equals id : C ∼ −→ C .Let us show that e is indeed a unit. Let f : 0 → e ∗ id : 0 = 1 ∗ → ∗ ∗ →
0. We have to prove that f = id . See Definition 7.4.2. There exists a precedent of the usage of “semigroupal” in the literature; this word means “monoidal, butwithout asking that the unit map to the unit.”
HE CATEGORY OF D-MODULES ON Bun G To do this, it suffices to show that π dR , ∗ ( f ) equals the identity. This is clear because π dR , ∗ isa monoidal functor and π dR , ∗ ( e ) equals id : C ∼ −→ C . (cid:3) C.7.
Proof of a particular case of Theorem C.4.1.
The following statement is a particularcase of Theorem C.4.1 and of Corollary C.4.2.
Lemma C.7.1.
Let Y be a QCA stack equipped with an action of the monoidal stack A / G m . Then the algebra N ∈ Funct cont (D-mod( Y ) , D-mod( Y )) is unital and the coalgebra ! ∈ Funct cont (D-mod( Y ) , D-mod( Y )) is counital. This lemma is an immediate consequence of Lemma C.6.1 and the following general consid-erations.Suppose that a monoidal QCA stack G acts on a QCA stack Y . Then the monoidal category G ( k ) acts on D-mod( Y ) on the left by g g N , g ∈ G ( k ). One also has the right action g g ! . Each of these two actions extend to an action of D-mod( G ). Namely, the left action isdefined by(C.5) M ∗ N := a N ( M ⊠ N ) , M ∈ D-mod( G ) , N ∈ D-mod( Y ) , where a : G × Y → Y is the action map. One can get the right action of D-mod( G ) onD-mod( Y ) from the left one using the equivalence D-mod( Y ) ∨ ≃ D-mod( Y ) that comes fromVerdier duality, see (2.2). One can also define the right action explicitly by(C.6) N ∗ M := ( p Y ) N (cid:18) p ! G ( M ) ! ⊗ a ! ( N ) (cid:19) , M ∈ D-mod( G ) , N ∈ D-mod( Y ) , where p G : G × Y → G and p Y : G × Y → Y are the projections.Now Lemma C.7.1 is clear. It immediately implies the following statement. Corollary C.7.2.
Let Y be a QCA stack equipped with an action of the monoidal stack A / G m .Then the functors (C.7) π N : D-mod( Y ) ⇄ D-mod( Y ) : i dR , ∗ , i ! : D-mod( Y ) ⇄ D-mod( Y ) : π ! form adjoint pairs with the adjunctions π N ◦ i dR , ∗ ∼ −→ Id D-mod( Y ) and Id D-mod( Y ) ∼ −→ i ! ◦ π ! coming from the isomorphism π ◦ i ∼ −→ Id Y . C.8.
Proof of Theorems C.4.1 and C.5.3.
We will deduce them from Corollary C.7.2.First, let us make some general remarks.If Z is an algebraic stack equipped with a morphism ψ : Z → B G m then D-mod( Z ) isequipped with the following action of the tensor category (D-mod( B G m ) , ! ⊗ ): M ⊗ F := ψ ! ( M ) ! ⊗ F , M ∈ D-mod( B G m ) , F ∈ D-mod( Z ) . If f : Z → Z is a morphism of QCA stacks over B G m then the functors f N and f ! arecompatible with the above action of D-mod( G m ). Lemma C.8.1.
Suppose we have a Cartesian diagram of QCA stacks e Z −−−−→ pt p y y ϕ Z ψ −−−−→ B G m Usually it does not commute with the left action. E.g., if g ∈ G is invertible then g ! = ( g − ) N does nothave to commute with g ′ N , g ′ ∈ G . Then one has a canonical isomorphism (C.8) M aps ( p ! ( F ) , p ! ( F )) = M aps ( F , A ⊗ F ) , F , F ∈ D-mod( Z ) , where A := ϕ dR , ∗ ( k )[ − and A ⊗ F := ψ ! ( A ) ! ⊗ F .Proof. p ! ( F ) = p ∗ dR ( F )[2], so M aps ( p ! ( F ) , p ! ( F )) = M aps ( F , p dR , ∗ ◦ p ! ( F )[ − M aps ( F , A ⊗ F ) . (cid:3) Now let us prove the assertion of Theorem C.5.3 concerning the pair ( π N , i dR , ∗ ). The proof ofthe other assertion of Theorem C.5.3 is similar, and Theorem C.4.1 follows from Theorem C.5.3.We have to show that for any e F ∈ D-mod( Y ) µ and e F ∈ D-mod( Y ) = D-mod( Y ) µ thecanonical map(C.9) M aps ( e F , i dR , ∗ ( e F )) → M aps ( π N ( e F ) , π N ◦ i dR , ∗ ( e F )) = M aps ( π N ( e F ) , e F )is an isomorphism. By the definition of the monodromic subcategory, we can assume that e F = p ! ( F ) for some F ∈ D-mod( Y / G m ). Since the action of G m on Y is trivial, we can alsoassume that e F = ( p ) ! ( F ) (here p : Y → Y / G m ). Applying Lemma C.8.1 for Z = Y / G m , e Z = Y and for Z = Y / G m , e Z = Y we get M aps ( e F , i dR , ∗ ( e F )) = M aps ( F , A ⊗ i ′ dR , ∗ ( F )) , i ′ : Y / G m → Y / G m , M aps ( π N ( e F ) , e F ) = M aps ( π ′ N ( F ) , A ⊗ F ) , π ′ : Y / G m → Y / G m . The map (C.9) is a particular case of the canonical map(C.10) M aps ( F , M ⊗ i ′ dR , ∗ ( F )) → M aps ( π ′ N ( F ) , M ⊗ F )which is defined for any M ∈ D-mod( B G m ). Applying Corollary C.7.2 to the action of A / G m on Y / G m we see that the map (C.10) is an isomorphism if M = ω B G m . This implies that(C.10) is an isomorphism for any M ∈ D-mod coh ( B G m ) (because by connectedness of G m ,D-mod coh ( B G m ) is the smallest non-cocomplete triangulated subcategory of D-mod( B G m )containing ω B G m ). In particular, (C.10) is an isomorphism for M = A , and we are done.C.9. Sketch of another approach to Theorems C.4.1 and C.5.3.
In Sect. C.8 we deducedTheorems C.4.1 and C.5.3 from Corollary C.7.2, which relies on the study of D-mod( A / G m )(see Lemma C.6.1). Here we sketch a slightly different approach, which is based on the studyof D-mod( A ) µ and does not rely on Corollary C.7.2. Proposition C.9.1.
The subcategories
D-mod( G m ) µ ⊂ D-mod( G m ) and D-mod( A ) µ ⊂ D-mod( A ) are closed under convolution. Moreover, they are monoidal categories. The unitobject of D-mod( G m ) µ equals µ , where is the unit object of D-mod( G m ) and M M µ is the “monodromization” functor D-mod( G m ) → D-mod( G m ) µ , i.e., the functor right adjointto the embedding D-mod( G m ) µ ֒ → D-mod( G m ) . The unit object of D-mod( A ) µ has a similardescription and can also be described as j ∗ (1 µ ) , where j : G m ֒ → A is the embedding. (cid:3) The adjunction mentioned in the proposition defines a canonical morphism ε : 1 µ → HE CATEGORY OF D-MODULES ON Bun G Remark
C.9.2 . Let Γ dR ( G m , − ) denote the de Rham cohomology functor D-mod( G m ) → Vect.The pair (1 µ , ε ) is uniquely characterized by the following properties: 1 µ ∈ D-mod( G m ) µ andthe map Γ dR ( G m , µ ) → Γ dR ( G m ,
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