CHAPTER III.4. AN APPLICATION: CRYSTALS

Transcription

1 CHAPTER III.4. AN APPLICATION: CRYSTALS Contents Introduction Let s do D-modules! D-modules via crystals What else is done in this chapter? 4 1. Crystals on prestacks and inf-schemes The de Rham functor and crystals: recollections Crystals Crystals and (ind)-nil-schemes The functor of de Rham direct image Crystals on ind-nil-schemes as extended from schemes Properties of the category of crystals on (ind)-nil-schemes Crystals as a functor out of the category of correspondences Correspondences and the de Rham functor The multiplicative structure of the functor of crystals Inducing crystals The functor of induction Induction on ind-inf-schemes Relative crystals Comparison with the classical theory of D-modules Left D-modules and left crystals Right D-modules and right crystals Passage between left and right D-modules/crystals Proof of Theorem Identification of functors 27 Introduction In this Chapter we will establish one of the goals indicated in the Introduction to Part III: we will show that inf-schemes give a common framework for ind-coherent sheaves and D-modules. In particular, we will show that the induction and forgetful functors (0.1) ind X : IndCoh(X) D-mod(X) : oblv X interact with the direct and inverse image functors in the expected way. Date: February 27,

2 2 AN APPLICATION: CRYSTALS 0.1. Let s do D-modules! The usual definition of the category of D-modules on a smooth affine scheme X is as the category Diff X -mod, where Diff X is the (classical) ring of Grothendieck operations. This approach to D-modules is very explicit, and is indispensable for concrete applications (e.g. to define regular D-modules and study the notion of holonomicity). However, this approach is not particularly convenient for setting up the theory from the point of view of higher category theory. Here are some typical issues that become painful in this approach One often encounters the question of how to define the category of D-modules on a singular scheme X? The usual answer is that we first assume that X is affine, and choose an embedding X Y, where Y is smooth. Now, define D-mod(X) to be D-mod(Y ) X, i.e., the full subcategory of D-mod(Y ) consisting of objects with set-theoretic support on X. Then, using Kashiwara s lemma, one shows that this construction is canonically independent of the choice of Y. For general X, one considers an affine Zariski cover and glues the corresponding categories. Note, however, that the words choose an embedding X Y mean that in the very definition, we appeal to resolutions. From the homotopical point of view, this exacts a substantial price and is too cumbersome to be convenient Another example is the definition of the direct image functor. For a morphism f : X Y between smooth affine schemes, one introduces an explicit object which defines the desired functor Diff X,Y : (Diff Y Diff op X )-mod, Diff X -mod Diff Y -mod. When X and Y are not necessarily smooth, one again embeds this situation into one where X and Y are smooth. When X and Y are non-affine, this is performed locally on X and Y. All of this can be made to work for an individual morphism: we can prove the proper adjunction between pullbacks and pushforwards, and the base change isomorphism. However, it is not clear how to establish the full functoriality of the category D-mod in this way; namely, as a functor out of the category of correspondences Another layer of complexity (=homotopical nuisance) arises when one wants to construct D-modules together with the adjoint pair (0.1) D-modules via crystals. In this book, we take a different approach to the theory of D-modules. We define the category of D-modules as crystals, establish all the needed functorialities, and then in the case of smooth schemes and morphisms between them identify the resulting categories and functors with the classical ones from the theory of D-modules.

3 AN APPLICATION: CRYSTALS By definition, for a laft prestack Z, the category of crystals on Z is where Z dr is the de Rham prestack of Z. Crys(Z) := IndCoh(Z dr ), Let f : Z 1 Z 2 be a map of laft prestacks. Then!-pullback on IndCoh defines a functor This is the pullback functor for crystals. f! dr : Crys(Z 2 ) Crys(Z 1 ) Assume now that f is ind-nil-schematic, which means that the corresponding morphism red Z 1 red Z 2 is ind-schematics. Then one (easily) sees that the resulting morphism (f dr ) : (Z 1 ) dr (Z 2 ) dr is ind-inf-schematic. Now, using [Chapter III.3, Sect. 4], we define the functor f dr, : Crys(Z 1 ) Crys(Z 2 ) to be the functor (f dr ). This is the de Rham direct image functor Taking Z 1 = Z (so that red Z is an ind-scheme) and Z 2 = pt, we obtain the functor of de Rham sections Γ dr (Z, ) : Crys(Z) Vect. Moreover, the above constructions automatically extend to the data of a functor out of a suitable (, 2)-category of correspondences. Namely, we consider the category PreStk laft equipped with the following classes of functors: horizontal maps are all maps in PreStk laft ; vertical maps are those maps f that red f is ind-schematic (we call them ind-nil-schematic); admissible maps are those vertical maps that are also ind-proper. One shows that the assignment Z Z dr defines a functor indnilsch & ind-proper Corr(PreStk laft ) indnilsch;all Composing with the functor IndCoh indinfsch & ind-proper (PreStklaft ) indinfsch;all we obtain a functor Crys indnilsch & ind-proper Corr(PreStklaft ) indnilsch;all indinfsch & ind-proper Corr(PreStk laft ) indinfsch;all. indinfsch & ind-proper : Corr(PreStk laft ) indinfsch;all DGCat 2 cont -Cat, indnilsch & ind-proper : Corr(PreStk laft ) indnilsch;all DGCat 2 cont -Cat. The above functor Crys indnilsch & ind-proper Corr(PreStklaft ) is the desired expression of functoriality indnilsch;all of the assignment Z Crys(Z) Now suppose that Z PreStk laft admits deformation theory. One shows that in the case the tautological map p dr,z : Z Z dr is an inf-schematic nil-isomorphism. Hence, by [Chapter III.3, Prop ], the functor admits the left adjoint. Thus, we obtain the desired adjoint pair: p! dr,z : Crys(Z) IndCoh(Z) ind dr,z : IndCoh(Z) Crys(Z) : oblv dr,z.

4 4 AN APPLICATION: CRYSTALS But what does this have to do with D-modules? The basic observation, essentially due to Grothendieck 1, is that for a smooth scheme X, the category Crys(X), together with the forgetful functor Ψ X oblv dr,x : Crys(X) QCoh(X), is canonically equivalent to the category of right D-modules, together with its tautological forgetful functor to QCoh(X). We describe this identification in Sect. 4 of this Chapter. We also show that the functors on the category of crystals (direct and inverse image for a map f : X Y ) described above map to the corresponding functors for D-modules under this identification. This is thus our ansatz to the construction of the theory of D-modules: instead of developing the theory of D-modules directly, we develop the theory of crystals, and then identity it with D-modules when D-modules are conveniently defined; namely, in the case of smooth schemes What else is done in this chapter? In Sect. 1 we introduce the category of crystals Crys(Z), where Z PreStk laft. The key observation here is the following: let f : Z 1 Z 2 be a map between prestacks such that the induced map red Z 1 red Z 2 is (ind)-schematic. Then we show that the resulting map is (ind)-inf-schematic. (Z 1 ) dr (Z 2 ) dr This observation, along with the fact that pushforward is defined on IndCoh for (ind)-nilschematic morphisms, is what makes the theory work. I.e., this is the framework that allows to treat the de Rham pushfoward (in particular, de Rham (co)homology) on the same footing as the O-module pushforward (in its IndCoh variant). We then establish some properties, expected from the theory of D-modules: (i) For a closed embedding i : Y Z, the functor i dr, : IndCoh(Y) IndCoh(Z) is fully faithful; (ii) If Z is an (ind)-nil-scheme, the category Crys(Z) is compactly generated and has a reasonably behaved t-structure In Sect. 2 we apply the results of [Chapter III.3, Sect. 5 and 6] and construct Crys as a functor out of the category of correspondences. We show that when evaluated on ind-nil-schemes, this gives rise to the operation of Verdier duality. 1 We learned it from A. Beilinson.

5 AN APPLICATION: CRYSTALS In Sect. 3 we study the functor of forgetting the crystal structure: oblv dr,z : Crys(Z) IndCoh(Z), which, in our framework, is just the pullback functor for the morphism p dr,z : Z Z dr. The key observation is that if Z admits deformation-theory, then the map p dr,z is infschematic. Hence, in this case the functor oblv dr,z admits a left adjoint, given by (p dr,z ) IndCoh. This left adjoint, denoted ind dr,z, is the functor of induction from ind-coherent sheaves to crystals. When Z = X is a smooth affine scheme, under the identification the functor ind dr,z corresponds to Crys(X) (Diff op X )-mod, F F O X Diff X. We show that if Z is an ind-scheme, then the morphism p dr,z is ind-schematic. We use this fact to deduce that the functor ind dr,z is t-exact In Sect. 3.3, we develop the theory of crystals relative to a given prestack Y. Namely, for Z over Y, we set Z/Y dr := Z dr Y dr Y and we set /Y Crys(Z) := IndCoh(Z/Y dr). When Z = X and Y = Y are smooth affine schemes, and the map X Y is smooth, category /Y Crys(Z) identifies with (Diff X/Y ) op -mod, where Diff X/Y is the (classical) ring of vertical differential operators (i.e., the subring of Diff X consisting of elements that commute with functions on Y ). If Z admits deformation theory relative to Y, then the morphism p/y dr,z : Z Z/Y dr is again inf-schematic, and hence the forgetful functor admits a left adjoint, given by (p/y dr,z) IndCoh. (p/y dr,z)! : /Y Crys(Z) IndCoh(Z) In Sect. 4 we show how to identify the theory of crystals with D-modules in the case of smooth schemes. Our exposition here is not self contained: we make frequent references to [GaRo2]. We first consider the case of left D-modules, and we show that the category Crys l (X) of left crystals on a smooth affine scheme X, defined as QCoh(X dr ), identifies with Diff X -mod. We then show that the category of right crystals (i.e., the usual category of crystals) identifies with (Diff X ) op -mod. Crys r (X) := Crys(X) := IndCoh(X dr )

6 6 AN APPLICATION: CRYSTALS Next, we show that the functor Υ XdR : QCoh(X dr ) IndCoh(X dr ) that identifies Crys l (X) with Crys r (X) corresponds under the above equivalences (0.2) Crys l (X) Diff X -mod and Crys r (X) (Diff X ) op -mod with the functor Crys l (X) Crys r (X), M M det(t (X))[dim(X)]. Finally, we show that for a map between f : X Y between smooth schemes, under the identifications (0.2), the functor f,l : Diff Y -mod Diff X -mod from the theory of D-modules corresponds to pullback and the functor f dr : QCoh(Y dr ) QCoh(X dr ), f D-mod, : Diff X -mod Diff Y -mod from the theory of D-modules corresponds to push-forward f dr, : QCoh(X dr ) QCoh(Y dr ). 1. Crystals on prestacks and inf-schemes In this section we will reap the fruits of the work done in [Chapter III.3]. Namely, we will show how the theory of IndCoh gives rise to the theory of crystals The de Rham functor and crystals: recollections. The category Crys(X) of crystals on a prestack X is defined to be IndCoh on the corresponding prestack X dr. In this subsection we recall the functor X X dr and study its basic properties For Z PreStk, we denote by Z dr the corresponding de Rham prestack, defined as for S Sch aff. Maps(S, Z dr ) := Maps( red S, Z), For a morphism f : Z 1 Z 2, let f dr : Z 1 dr Z2 dr between derham prestacks. denote the corresponding morphism Note that the functor dr commutes both with limits and colimits. Also, note that Z dr ( red Z) dr. So, if a morphism f : Z 1 Z 2 is a nil-isomorphism (i.e., red Z 1 red Z 1 is an isomorphism), then (Z 1 ) dr (Z 2 ) dr is an isomorhism.

7 AN APPLICATION: CRYSTALS We claim: Proposition The functor dr takes PreStk laft to PreStk laft. Proof. Let Z be an object of PreStk laft. We need to show that Z dr satisfies: It is convergent; For every n, the truncation n Z belongs to n PreStk lft. The convergence of Z dr is obvious. To show that n Z n PreStk lft, it suffices to show that Z dr takes filtered limits in Sch aff to colimits in Spc. However, this follows from the fact that the functor S red S, Sch aff red Sch aff preserves filtered limits, and the fact that red Z red PreStk lft Crystals. In this subsection we introduce the category of crystals Composing the functor dr : PreStk laft PreStk laft with we obtain a functor denoted by IndCoh! PreStk laft : (PreStk laft ) op DGCat cont, Crys! PreStk laft : (PreStk laft ) op DGCat cont. This is the functor which is denoted Crys r PreStk laft in [GaRo2, Sect 2.3.2] For Z PreStk laft we shall denote the value of Crys! PreStk laft on Z by Crys(Z). For a morphism f : Z 1 Z 2 in PreStk laft, we shall denote by f dr! the resulting functor Crys(Z 2 ) Crys(Z 1 ). Note that if a morphism f : Z 1 Z 2 is a nil-isomorphism, then is an equivalence. f! dr : Crys(Z 2 ) Crys(Z 1 ) For Z PreStk, we let p dr,z denote the tautological projection: Z Z dr. The map p dr,z gives rise to a natural transformation of functors oblv dr : Crys! PreStk laft IndCoh! PreStk laft. For a map f : Z 1 Z 2, we have a commutative square of functors: Crys(Z 1 ) f! dr oblv dr,z1 IndCoh(Z1 ) f! Crys(Z 2 ) oblv dr,z2 IndCoh(Z2 ).

8 8 AN APPLICATION: CRYSTALS Finally, we make the following observation: Proposition For Z PreStk laft, the functor Crys(Z) lim Crys(Z) Z (C /Z ) op is an equivalence, where C is any of the following categories: red Sch aff ft, cl Sch aff ft, < Sch aff ft, Sch aff aft, red Sch ft, cl Sch ft, < Sch ft, Sch aft. Proof. It is enough to show that the functor dr : PreStk laft PreStk laft is isomorphic to the left Kan extension of its restriction to C PreStk laft for C as above. It is sufficient to consider the case of C = red Sch aff ft. First, we note that the functor dr commutes with colimits. This implies that dr is isomorphic to the left Kan extension of its restriction to < Sch aff aft. Hence, it suffices to show that the functor dr : Sch aff aft PreStk laft is isomorphic to the left Kan extension of its restriction to red Sch aff ft. In other words, we have to show that given Z Sch aff aft, S Sch aff aft and a map red S Z, the category of its factorizations as red S Z Z with Z red Sch aff, is contractible. However, the latter is obvious as the above category has a final object, namely, Z := red Z Crystals and (ind)-nil-schemes. In this subsection we introduce the class of prestacks that we call (ind)-nil-schemes, and study the category of crystals on such prestacks. (Ind)-nilschemes play the same role vis-à-vis Crys as (ind)-inf-schemes do for IndCoh Consider the full subcategories and indnilsch laft := PreStk laft red PreStk lft red indsch PreStk laft nilsch laft := PreStk laft red PreStk lft red Sch PreStk laft, where PreStk laft red PreStk lft is the functor Z red Z. In other words, Z belongs to indnilsch laft (resp., nilsch laft ) if and only if red Z is a reduced ind-scheme (resp., scheme). For example, we have infsch laft nilsch laft and indinfsch laft indnilsch laft. We shall refer to objects of indnilsch laft (resp., nilsch laft ) as ind-nil-schemes (resp., nilschemes).

9 AN APPLICATION: CRYSTALS We claim: Lemma The functor dr takes objects of indnilsch laft (resp., nilsch laft ) to indinfsch laft (resp., infsch laft ). Proof. We have red (Z dr ) = red Z. Now, we claim that for any Z PreStk, the corresponding Z dr admits deformation theory. In fact, it admits an -connective deformation theory: all of its cotangent spaces are zero Recall from [Chapter III.2, Definitions 1.6.5(a), 1.6.7(c) and (c)], the notions of (ind)-schematic and (ind)-proper maps of prestacks, as well as (ind)-closed embeddings of prestacks. Definition (a) We shall say that a map of prestacks is (ind)-nil-schematic if the map of the corresponding reduced prestacks is (ind)-schematic. (b) We shall say that a map of prestacks is an nil-closed-embedding(ind)-nil-closed embedding if the map of the corresponding reduced prestacks is an ind-closed embedding. Recall the notion of an (ind)-inf-schematic map of prestacks, see [Chapter III.2, Definitions 3.1.5]. We have: Corollary The functor dr takes (ind)-nil-schematic maps in PreStk laft to (ind)-infschematic maps. Proof. For a map of prestacks f : Z 1 Z 2 and S (Sch aff aft) /(Z2) dr, the Cartesian product identifies with S (Z 2) dr (Z 1 ) dr S S dr ( red S Z 2 Z 1 ) dr. Now, we use Lemma and the fact that the subactegory indinfsch laft is preserved by finite limits We claim: Lemma Let f : Z 1 Z 2 be an ind-nil-proper map in PreStk laft. Then: (a) The functor f dr, : Crys(Z 1 ) Crys(Z 2 ), left adjoint to f dr!, is well-defined, and satisfies base change with respect to!-pullbacks. (b) If f is an ind-nil-closed embedding, then f dr, is fully faithful. Proof. Point (a) follows from Corollary and [Chapter III.3, Proposition 3.2.4]. To prove point (b), we need to show that the unit of the adjunction is an isomorphism. Id Crys(Z1) f! dr f dr,

10 10 AN APPLICATION: CRYSTALS Consider the Cartesian square: p 1 Z 1 Z 1 Z1 Z 2 p 2 Z 1 Z 2. The above unit of the adjunction equals the composite map Id Crys(Z1) (p 2 ) dr, ( Z1 ) dr, ( Z1 )! dr (p 1 )! dr (p 2 ) dr, (p 1 )! dr f! dr f dr,, where Z1 is the diagonal map Z 1 Z 1 Z 2 Z 1, and second arrow is the co-unit of the (( Z1 ) dr,, ( Z1 )! dr )-adjunction. Now, by base change, (p 2 ) dr, (p 1 )! dr f! dr f dr, is an isomorphism. Hence, it is enough to show that (p 2 ) dr, ( Z1 ) dr, ( Z1 )! dr (p 1 )! dr (p 2 ) dr, (p 1 )! dr is an isomorphism as well. However, the map ( Z1 ) dr, ( Z1 )! dr Id Crys(Z1 Z 2 Z 1) is an isomorphism, since ( Z1 )! dr is an equivalence (because the map Z 1 is a nil-isomorphism) The functor of de Rham direct image. In this subsection we develop the functor of de Rham direct image (a.k.a., pushforward) for crystals Recall the functor IndCoh indinfschlaft : indinfsch laft DGCat cont, that sends a morphism f to the functor f IndCoh, see [Chapter III.3, Sect. 4.3]. Precomposing it with the functor we obtain a functor dr : indnilsch laft indinfsch laft (1.1) Crys indnilschlaft : indnilsch laft DGCat cont For a morphism f : Z 1 Z 2 in indnilsch laft we shall denote the resulting functor by f dr,. Crys(Z 1 ) Crys(Z 2 ) In other words, f dr, = (f dr ) IndCoh.

11 AN APPLICATION: CRYSTALS From [Chapter III.3, Corollary 5.2.3], we obtain: Corollary The restriction of the functor Crys indnilschlaft to the 1-full subcategory (indnilsch laft ) ind-proper indnilsch laft is obtained by passing to left adjoints from the restriction functor Crys! indnilsch laft to ((indnilsch laft ) ind-proper ) op (indnilsch laft ) op. Remark Note that we have used the notation f dr, when f is ind-proper earlier (in Lemma 1.3.8), to denote the left adjoint of f dr!. The above corollary implies that the notations are consistent Crystals on ind-nil-schemes as extended from schemes. The material of this subsection will not be used in the sequel and is included for the sake of completeness. We show that the theory of Crys on ind-nil-schemes can be obtained by extending the same theory on schemes Consider the category red Sch ft, and consider the functors and Crys! red Sch ft : ( red Sch ft ) op DGCat cont Crysred Sch ft : red Sch ft DGCat cont. From Proposition we obtain: Corollary The natural map is an isomorphism. Crys! indnilsch laft RKE ( red Sch ft ) op (indnilsch laft ) op(crys! red Sch ft ) We are going to prove the following: Proposition The natural map is an isomorphism. LKEred Sch ft indnilsch laft (Crysred Sch ft ) Crys indnilschlaft The rest of this subsection is devoted to the proof of this proposition Consider the 1-full subcategory of indnilsch laft equal to (indnilsch laft ) nil-closed = PreStk laft red PreStk lft ( red indsch laft ) closed. I.e., we restrict 1-morphisms to be nil-closed maps. It is enough to show that the map in Proposition becomes an isomorphism when restricted to the above subcategory. This follows by [Chapter III.3, Corollary 4.1.4] from Proposition and the following statement: Proposition (a) The map ( (RKE( ) ) red Schft) op (indnilschlaft) (Crys! op red Sch ft ) ((indnilschlaft ) nil-closed ) op is an isomorphism. RKE (( red Sch ft ) closed ) op ((indnilsch laft ) nil-closed ) op(crys! red Sch ft (( red Sch ft ) closed ) op)

12 12 AN APPLICATION: CRYSTALS (b) The map LKE ( red Sch ft ) closed (indnilsch laft ) (Crysred nil-closed Sch ft ( red Sch ft ) closed ) ( LKEred Sch ft indnilsch (Crysred laft Sch ft ) ) (indnilschlaft ) nil-closed is an isomorphism. Proof. Follows from the fact that for Z indnilsch laft, the category {f : Z Z, Z red Sch ft, f is nil-closed} is cofinal in {f : Z Z, Z red Sch ft }, by [Chapter III.2, Corollary 1.7.5(b)] 1.6. Properties of the category of crystals on (ind)-nil-schemes. In this subsection we study properties of the category Crys(Z) on a given object Z indnilsch laft We claim: Proposition The functor Crys(Z) lim Crys(Z) f:z Z is an equivalence, where the limit is taken over the index -category {f : Z Z, Z red Sch ft, f is nil-closed}. For every f : Z Z as above, the corresponding functor is fully faithful. f dr, : Crys(Z) Crys(Z) Proof. The first assertion follows from Proposition for C = red Sch aff ft Corollary 1.7.5(b)]. The second assertion follows from Lemma 1.3.8(b). and [Chapter III.2, Compact generation. From [Chapter III.3, Corollary 3.2.2] and, we obtain: Corollary The category Crys(Z) is compactly generated. From Proposition 1.6.2, combined with [DrGa2, Corollary and Lemma 1.9.5], we have the following more explicit description of the subcategory Crys(Z) c Crys(Z). Corollary Compact objects of Crys(Z) are those that can be obtained as f dr, (M), M Crys(Z) c, Z red Sch ft and f is a nil-closed map Z Z.

13 AN APPLICATION: CRYSTALS t-structure. According to [Chapter III.3, Sect. 3.4], the category Crys(Z) carries a canonical t-structure. It is characterized by the following property: M Crys(Z) 0 oblv dr,z (M) IndCoh(Z) 0. In addition, from [Chapter III.3, Corollary 3.4.4], we obtain: Corollary (a) An object M Crys(Z) lies in Crys(Z) 0 if and only if for every nil-closed map f : Z Z with Z red Sch ft we have f! dr(m) Crys(Z) 0. (b) The category Crys(Z) 0 is generated under colimits by the essential images of Crys(Z) 0 for f : Z Z with Z red Sch ft and f nil-closed. 2. Crystals as a functor out of the category of correspondences In this section we extend the formalism of crystals to a functor out of the category of correspondences Correspondences and the de Rham functor. In this subsection we show that the de Rham functor turns (ind)-nil-schematic morphisms into (ind)-inf-schematic ones Recall that the functor dr commutes with Cartesian products. Combining this observation with Lemma 1.3.6, we obtain that dr gives rise to a functor of (, 2)-categories: Corr(dR) ind-proper indnilsch;all : Corr(PreStk laft) indnilsch & ind-proper indnilsch;all indinfsch & ind-proper Corr(PreStk laft ) indinfsch;all. Hence, from [Chapter III.3, Theorem and Proposition 5.5.3], we obtain: Theorem There exists a canonically defined functor Crys indnilsch & ind-proper Corr(PreStklaft ) indnilsch;all equipped with an isomorphism Furthermore, the restriction indnilsch & ind-proper : Corr(PreStk laft ) indnilsch;all DGCat 2 cont -Cat, Crys indnilsch & ind-proper Corr(PreStklaft ) (PreStklaft ) op Crys! PreStk indnilsch;all laft. Crys Corr(PreStklaft ) indnilsch;all := Crys indnilsch & ind-proper Corr(PreStklaft ) Corr(PreStklaft ) indnilsch;all indnilsch;all uniquely extends to a functor Crys Corr(PreStklaft ) : Corr(PreStk nil-open laft ) nil-open indnilsch;all ( DGCat 2 -Cat ) 2 -op cont. indnilsch;all As in the case of [Chapter III.3, Theorem 5.5.3], the content of Theorem is the existence of the functor f dr, : Crys(Z 1 ) Crys(Z 2 ) for ind-nil-schematic morphisms of prestacks f : Z 1 Z 2, and of the base change isomorphisms compatible with proper and nil-open adjunctions. Namely, for a Cartesian diagram of prestacks Z 1 f g 1 Z1 f Z 2 g 2 Z2,

14 14 AN APPLICATION: CRYSTALS with f ind-nil-schematic, we have a canonical isomorphism (2.1) f dr, g! 1,dR g! 2,dR f dr,. Moreover, if f is ind-proper, then f dr, is the left adjoint of f dr!. Furthermore, the isomorphism (2.1) is the one arising by adjunction if either f X or g 2 is ind-proper. If f is a nil-open embedding (i.e., the map of the corresponding reduced prestacks is an open embedding), then f dr, is the right adjoint of f dr!. Furthermore, the isomorphism (2.1) is the one arising by adjunction if either f X or g 2 is a nil-open embedding Now, let us restrict the functor Crys indnilsch & ind-proper Corr(PreStklaft ) to indnilsch;all Corr(indnilSch laft ) ind-proper all;all indnilsch & ind-proper Corr(PreStk laft ) indnilsch;all. We denote the resulting functor by Crys Corr(indnilSchlaft ). From [Chapter III.3, Theorems and 5.4.3] we ind-proper all;all obtain: Corollary The restriction of Crys Corr(indnilSchlaft ) ind-proper all;all to indnilsch laft Corr(indnilSch laft ) ind-proper all;all identifies canonically with the functor Crys indnilschlaft of (1.1) Further restricting along we obtain a functor denoted by Crys (nilschaft ) proper all;all. Corr(nilSch aft ) proper all;all In particular, we obtain a functor which is also isomorphic to Crys (nilschaft ) proper all;all Crys nilschaft Corr(indnilSch laft ) ind-proper all;all, (DGCat cont ) 2 -Cat := Crys (Schaft ) proper all;all Crys indnilschlaft nilschaft. nilschaft, 2.2. The multiplicative structure of the functor of crystals. In this subsection we show how the formalism of crystals as a functor out of the category of correspondences gives rise to Verdier duality Duality. From [Chapter III.3, Theorem 6.2.2], we obtain: Theorem We have a commutative diagram of functors (Corr(indnilSch laft ) all;all ) op ϖ ( Crys Corr(indnilSchlaft ) all;all ) op ( ) DGCat dualizable op cont dualization Corr(indnilSch laft ) all;all Crys Corr(indnilSchlaft ) all;all DGCat dualizable cont.

15 AN APPLICATION: CRYSTALS Concretely, this theorem says that for Z indnilsch laft there is a canonical involutive equivalence (2.2) D Verdier Z : Crys(Z) Crys(Z), and for a map f : Z 1 Z 2 in indnilsch laft there is a canonical identification f! dr (f dr, ) As in [Chapter III.3, Sect ], we can write the unit and counit maps explicitly. µ ZdR : Vect Crys(Z) Crys(Z) and ɛ ZdR : Crys(Z) Crys(Z) Vect Namely, ɛ ZdR is the composition where and µ ZdR Crys(Z) Crys(Z) Crys(Z Z)! Z.dR Crys(Z) Γ dr(z, ) Vect, is the composition Γ dr (Z, ) := (p Z ) dr,, Vect ω Z dr ( Z ) dr, Crys(Z) Crys(Z Z) Crys(Z) Crys(Z) Verdier duality. For Z indnilsch laft, let D Verdier Z denote the canonical equivalence (Crys(Z) c ) op Crys(Z) c, corresponding to the isomorphism (2.2). In other words, D Verdier Z = D Serre Z dr As a particular case of [Chapter III.3, Corollary 6.2.9], we obtain: Corollary Let f : Z 1 Z 2 be an ind-proper map in indnilsch laft. Then we have a commutative diagram: (Crys(Z 1 ) c ) op D Verdier Z 1 Crys(Z1 ) c (f dr, ) op f dr, (Crys(Z 2 ) c ) op D Verdier Z 2 Crys(Z2 ) c. In view of Corollary 1.6.5, the above corollary gives an expression of the Verdier duality functor on Z indnilsch laft in terms of that on schemes Convolution for crystals. and the corre- indnilsch & ind-proper Returning to the entire (, 2)-category Corr(PreStk laft ) indnilsch;all sponding functor indnilsch & ind-proper, IndCoh Corr(PreStklaft ) indnilsch;all we obtain, from [Chapter III.3, Sect. 6.3], that the functor Crys indnilsch & ind-proper Corr(PreStklaft ) indnilsch;all carries a canonical right-lax symmetric monoidal structure. As in [Chapter III.3, Sect ], we have: indnilsch & ind-proper : Corr(PreStk laft ) indnilsch;all DGCat 2 cont -Cat

16 16 AN APPLICATION: CRYSTALS (i) Given a Segal object R of PreStk laft, with the target and composition maps ind-nilschematic, the category Crys(R) acquires a monoidal structure given by convolution, and as such it acts on Crys(X) (here, as in [Chapter II.2, Sect ], X = R 0 and R = R 1 ). (ii) If the composition map is ind-proper, then ω R Crys(R) acquires the structure of an algebra in Crys(R). The action of this algebra on IndCoh(X), viewed as a plain endo-functor, is given by (p t ) dr, (p s )! dr. 3. Inducing crystals In this section we study the interaction between the functors IndCoh and Crys The functor of induction. In this subsection we show that the forgetful functor Crys(Z) IndCoh(Z) admits a left adjoint, provided that Z is a prestack that admits deformation theory For an object Z PreStk laft consider the canonical map We claim: p dr,z : Z Z dr. Proposition Suppose that Z admits deformation theory. Then the map p dr,z is an inf-schematic nil-isomorphism. Proof. We need to show that for S (Sch aff aft) /ZdR, the Cartesian product (3.1) S Z dr Z is an inf-scheme. Clearly, the above Cartesian product belongs to PreStk laft, and its underlying reduced prestack identifies with red S. Hence, it remains to show that (3.1) admits deformation theory. This holds because the category PreStk def-laft is closed under finite limits From Proposition and [Chapter III.3, Proposition 3.1.2(a)] we obtain: Corollary Let Z be an object of PreStk def-laft. Then the functor admits a left adjoint. oblv dr,z : Crys(Z) IndCoh(Z) We denote the left adjoint to oblv dr,z, whose existence is given by the above corollary, by ind dr,z Thus, for Z PreStk def-laft, we obtain an adjoint pair (3.2) ind dr,z : IndCoh(Z) Crys(Z) : oblv dr,z. We claim: Lemma The pair (3.2) is monadic. Proof. Since oblv dr,z is continuous, we only need to check that it is conservative. However, this follows from [Chapter III.3, Proposition 3.1.2(b)].

17 AN APPLICATION: CRYSTALS The next corollary of Proposition expresses the functoriality of the operation of induction: Corollary There is a canonically defined natural transformation ind dr : IndCoh (PreStklaft ) indinfsch (PreStkdef-laft ) indinfsch Crys (PreStklaft ) indinfsch (PreStkdef-laft ) indinfsch, as functors (PreStk def-laft ) indinf-sch DGCat cont. In particular, the above corollary says that for an ind-inf-schematic morphism f : Z 1 Z 2 of objects of PreStk def-laft, the following diagram of functors commutes: IndCoh(Z 1 ) f IndCoh ind dr,z1 Crys(Z1 ) f dr, IndCoh(Z 2 ) ind dr,z2 Crys(Z2 ) Induction on ind-inf-schemes. In this subsection, let Z be an object of indinfsch laft. We study the interaction of the induction functor with that of Serre and Verdier dualities We have: Lemma The functor ind dr,z sends IndCoh(Z) c to Crys(Z) c. Proof. Follows from the fact that the functor oblv dr,z is continuous and conservative Induction and duality. Let us apply isomorphism [Chapter III.3, Equation (6.2)] to the map We obtain: Corollary Under the isomorphisms D Serre Z we have a canonical identification p dr,z : Z Z dr. : IndCoh(Z) IndCoh(Z) and D Verdier Z (oblv dr,z ) ind dr,z. In addition, by [Chapter III.3, Corollary 6.2.9] Corollary The following diagram of functors commutes: : Crys(Z) Crys(Z), (IndCoh(Z) c ) op (ind dr,z ) op (Crys(Z) c ) op D Serre Z IndCoh(Z) c ind dr,z D Verdier Z Crys(Z) c.

18 18 AN APPLICATION: CRYSTALS Induction and t-structure. Recall that by the definition of the t-structure on Crys(Z), the functor oblv dr,z is left t-exact. We claim: Corollary Assume that Z is an ind-scheme. Then the functor ind dr,z is t-exact. Proof. The fact that ind dr,z is right t-exact follows by adjunction. To show that it is left t-exact we use [Chapter III.3, Lemma 3.4.6]: we have to show that the p dr,z is ind-schematic. Indeed, for S Sch aff aft and S Z dr, the Cartesian product S Z dr Z identifies with the formal completion of S Z along the graph of the map red S Z Relative crystals. In this subsection we describe how the discussion of crystals generalizes to the relative situation Let Y be a fixed object of PreStk laft. Consider the -category and the corresponding (, 2)-category (PreStk laft ) /Y indinfsch & ind-proper Corr((PreStk laft ) /Y ) indinfsch;all. Restricting the functor IndCoh indinfsch & ind-proper (PreStklaft ) along the forgetful functor indinfsch;all we obtain the functor indinfsch & ind-proper Corr((PreStk laft ) /Y ) indinfsch;all Corr(PreStk laft ) inf-proper inf-sch;all, IndCoh indinfsch & ind-proper Corr((PreStklaft ) /Y ) indinfsch;all with properties specified by [Chapter III.3, Theorem 5.4.3]. In particular, let be the resulting functor. indinfsch & ind-proper : Corr((PreStk laft ) /Y ) indinfsch;all DGCat 2 cont -Cat, IndCoh! (PreStk laft ) /Y : ((PreStk laft ) /Y ) op DGCat cont The category (PreStk laft ) /Y has an endo-functor, denoted by /Y dr: Z Z/Y dr := Z dr Y dr Y. Corollary implies that the functor /Y dr gives rise to a functor Hence, /Y dr induces a functor ((PreStk laft ) /Y ) indnilsch ((PreStk laft ) /Y ) indinfsch. (3.3) Corr((PreStk laft ) /Y ) ind-proper indnilsch;all Corr((PreStk laft) /Y ) indinfsch & ind-proper indinfsch;all. Thus, precomposing IndCoh indinfsch & ind-proper Corr((PreStklaft ) /Y ) with /Y dr, we obtain the functor indinfsch;all /Y Crys Corr((PreStklaft ) /Y ) : Corr((PreStk ind-proper laft ) /Y ) ind-proper -Cat indnilsch;all DGCat2cont. indnilsch;all

19 AN APPLICATION: CRYSTALS Let /Y Crys! (PreStk laft ) /Y : ((PreStk laft ) /Y ) op DGCat cont and /Y Crys ((PreStklaft ) /Y ) indnilsch : ((PreStk laft ) /Y ) indnilsch DGCat cont denote the corresponding functors obtained by restriction. For a map Z 1 Z 2 in (PreStk laft ) indnilsch, we shall denote by f!y dr and fy dr, the corresponding functors /Y Crys(Z 1 ) /Y Crys(Z 2 ). These functors are adjoint if f is ind-proper/nil-open Let /Y indnilsch laft denote the full subcategory of (PreStk laft ) /Y given by the preimage of (indinfsch laft ) /Y under the functor /Y dr. Restricting the functor /Y Crys Corr((PreStklaft ) /Y ) to ind-proper indnilsch;all we obtain the functor Corr( /Y indnilsch laft ) ind-proper all;all /Y Crys Corr( /Y indnilsch laft ) ind-proper all;all Corr((PreStk laft ) /Y ) ind-proper indnilsch;all, : Corr( /Y indnilsch laft ) ind-proper all;all DGCat 2 -Cat cont. Furthermore, for an object Z of /Y indnilsch laft, the category /Y Crys(Z) satisfies the following properties: (1) The category /Y Crys(Z) is compactly generated, and is self-dual in the sense of Theorem (2) The category /Y Crys(Z) carries a t-structure in which an object F is coconnective if and only if its image under the forgetful functor oblv/y dr,z : /Y Crys(Z) IndCoh(Z) is coconnective, for oblv/y dr,z := (p/y dr,z)!, where p/y dr,z denotes the canonical morphism Z Z/Y dr. (3) If Z admits deformation theory over Y (see [Chapter III.1, Sect ] for what this means), then the morphism p/y dr,z is an inf-schematic nil-isomorphism, and hence the functor oblv/y dr,z admits a left adjoint, denoted ind/y dr,z. Remark The essential difference between Crys and /Y Crys is that for Z (PreStk laft ) /Y, the category /Y Crys(Z) depends not just on the underlying reduced prestack. E.g., for Z = Y, /Y Crys(Z) = IndCoh(Y) Assume for a moment that Y = Y is a smooth classical scheme, and Z = Z is also a classical scheme smooth over Y. Then, as in [GaRo2, Sect. 4.7], one shows that the category /Y Crys(Z) identifies with the DG category associated with the abelian category of quasi-coherent sheaves of modules on Y with respect to the algebra of vertical differential operators, i.e., the subalgebra of Diff(Z) that consists of differential operators that commute with O Y.

20 20 AN APPLICATION: CRYSTALS 4. Comparison with the classical theory of D-modules In this section we will identify the theory of crystals as developed in the previous sections with the theory of D-modules. This section can be regarded as a companion to [GaRo2, Sects. 6 and 7], and we shall assume the reader s familiarity with the contents of loc.cit Left D-modules and left crystals. In this subsection we will recollect (and rephrase) the contents of [GaRo2, Sect. 5]. Specifically, we will discuss the equivalence between the category of left D-modules on a smooth scheme X and the category of left crystals on X Let X be a classical scheme of finite type. Consider the category QCoh(X X) and its full subcategory (QCoh(X X) X ), consisting of objects that are set-theoretically supported on the diagonal. Let (QCoh(X X) X ) rel.flat QCoh(X X) X ) be the full subcategory, consisting of objects that are X-flat with respect to both projections p s, p t : X X X. The category (QCoh(X X) X ) rel.flat has a naturally defined monoidal structure, given by convolution. Moreover, we have a canonically defined fully faithful monoidal (!) functor (4.1) (QCoh(X X) X ) rel.flat QCoh(X X), where QCoh(X X) is a monoidal category as in [Chapter II.2, Sects and 5.3.3] Now suppose that X is smooth. In this case, we have a canonically defined object ) Diff X AssocAlg ((QCoh(X X) X ) rel.flat ), namely, the Grothendieck algebra of differential operators. Composing (4.1) with the monoidal equivalence QCoh(X X) Funct cont (QCoh(X), QCoh(X)), we obtain that Diff X gives rise to a monad on QCoh(X). We consider the category Diff X -mod(qcoh(x)). It is equipped with a t-structure, characterized by the property that the tautological forgetful functor Diff X -mod(qcoh(x)) QCoh(X) is t-exact. The category D-mod l (X) of left D-modules on X is defined as the canonical DG model of the derived category of the abelian category (Diff X -mod(qcoh(x))). As in [GaRo2, Proposition 4.7.3] one shows that the canonical functor (4.2) D-mod l (X) Diff X -mod(qcoh(x)) is an equivalence.

21 AN APPLICATION: CRYSTALS Let X be any scheme almost of finite type. Recall the category Crys l (X) := QCoh(X dr ), see [GaRo2, Sect. 2.1]. It is equipped with a forgetful functor oblv l dr,x : Crys l (X) QCoh(X). Assume now that X is eventually coconnective. According to [GaRo2, Proposition ], in this case the functor oblv l dr,x admits a left adjoint, denoted ind l dr,x, and the resulting adjoint pair of functors is monadic. ind l dr,x : QCoh(X) Crys l (X) : oblv l dr,x, The corresponding monad is given by an object I.e., we have an equivalence D l X AssocAlg(QCoh(X X) X ). Crys l (X) D l X-mod(QCoh(X)) Again, assume that X is smooth. In this case one easily shows (see, e.g., [GaRo2, Proposition 5.3.6]) that D l X (QCoh(X X) X ) rel.flat. Moreover, it is a classical fact (reproved for ( completeness in [GaRo2, ) Lemma 5.4.3]) that there is a canonical isomorphism in AssocAlg (QCoh(X X) X ) rel.flat : (4.3) D l X Diff X. In particular, we obtain a canonical equivalence of categories D-mod l (X) Diff X -mod(qcoh(x)) D l X-mod(QCoh(X)) Crys l (X), compatible with the forgetful functors to QCoh(X). We denote the resulting equivalence D-mod l (X) Crys l (X) by F l X Let f : X Y be a morphism between smooth classical schemes. In the classical theory of D-modules, one defines a functor that makes the diagram (4.4) commute. f,l : D-mod l (Y ) D-mod l (X) f,l D-mod l (Y ) D-mod l (X) oblv l dr,x QCoh(Y ) f QCoh(X) Also recall (see [GaRo2, Sect ]) that for a map f : X Y between arbitrary schemes almost of finite type, we have a functor f,l : Crys l (Y ) Crys l (X),

22 22 AN APPLICATION: CRYSTALS that makes the diagram (4.5) commute. f,l Crys l (Y ) Crys l (X) QCoh(Y ) f QCoh(X) The following can be established by a direct calculation: Lemma The following diagram of functors naturally commutes f,l D-mod l (Y ) D-mod l (X) F l X F l Y Crys l (X) f,l Crys l (X), in a way compatible with the forgeftul functors to QCoh( ) Right D-modules and right crystals. In this subsection we will discuss the equivalence between the category of right D-modules on a smooth scheme X, and the category Crys(X), considered in the earlier sections of this Chapter Note that for any scheme X, the monoidal category QCoh(X X) carries a canonical anti-involution, denoted σ, corresponding to the transposition acting on X X. In terms of the identification QCoh(X X) Funct cont (QCoh(X), QCoh(X)), we have σ(f ) F, F Funct cont (QCoh(X), QCoh(X)), where we use the canonical identification of [Chapter II.3, Equation (4.2)]. D naive X : QCoh(X) QCoh(X), In particular for an algebra object A in QCoh(X X), we can regard σ(a op ) again as an algebra object in QCoh(X X), and we have (4.6) (M A ) M σ(a op ), where M B denotes the monad on QCoh(X), corresponding to an algebra object B QCoh(X) Let X be smooth. Consider the object ( ) σ(diff op X ) AssocAlg (QCoh(X X) X ) rel.flat ), and the corresponding category σ(diff op X )-mod(qcoh(x)). The category D-mod r (X) of right D-modules on X is defined as the canonical DG model of the derived category of the abelian category (σ(diff op X )-mod(qcoh(x))). As in [GaRo2, Proposition 4.7.3] one shows that the canonical functor (4.7) D-mod r (X) σ(diff op X )-mod(qcoh(x))

23 AN APPLICATION: CRYSTALS 23 is an equivalence Let X be a scheme almost of finite type. For the duration of this section, we will denote by Crys r (X) := Crys(X) := IndCoh(X dr ), where the latter is defined as in Sect , and by ind r dr,x : IndCoh(X) Crys r (X) : oblv r dr,x the corresponding pair of adjoint functors from (3.2). I.e., we are adding the superscript r (for right ) to the notation from Sect to emphasize the comparison with right D-modules. Let D r X be the object of AssocAlg(IndCoh(X X) X ), corresponding to the monad via the equivalence of monoidal categories By Lemma 3.1.6, we have: oblv r dr,x ind r dr,x IndCoh(X X) Funct cont (IndCoh(X), IndCoh(X)). D r X-mod(IndCoh(X)) Crys r (X) Suppose that X is smooth. Recall that in this case the adjoint pairs and are both equivalences. Ξ X : QCoh(X) IndCoh(X) : Ψ X Ξ X : QCoh(X) IndCoh(X) : Ψ X = Υ X In this case one shows as in [GaRo2, Sect. categories (4.8) F r X : D-mod r (X) Crys r (X), that makes the diagram commute. D-mod r (X) QCoh(X) F r X 5.5] that there is a canonical equivalence of Crys r (X) oblvr dr,x Ξ X IndCoh(X) Remark One can obtain the equivalence of (4.8) formally from the corresponding computation for left D-modules. Namely, taking into account the equivalences (oblv l dr,x ind l dr,x) -mod(qcoh(x)) σ(diff op X )-mod(qcoh(x)) D-modr (X) (where the first equivalence comes from (4.3) and (4.6)), and (oblv r dr,x ind r dr,x)-mod(indcoh(x)) Crys r (X), it suffices to construct an isomorphism of the monads (4.9) Ψ X (oblv r dr,x ind r dr,x) Ξ X and (oblv l dr,x ind l dr,x), acting on QCoh(X).

24 24 AN APPLICATION: CRYSTALS The latter isomorphism holds for any eventually coconnective X, and follows from the isomorphism of monads see [GaRo2, Lemma 3.4.9]. oblv l dr,x ind l dr,x Ξ X (oblv r dr,x ind r dr,x) Ψ X, Remark The monads in (4.9) correspond to the pair of adjoint functors ind r dr,x : QCoh(X) Crys(X) : oblv r dr,x of [GaRo2, Sect. 4.6]. Furthermore, in terms of the D naive X we have the isomorphisms : QCoh(X) QCoh(X) and D Verdier X : Crys(X) Crys(X), ( ind r dr,x) oblv l dr,x and ( oblv r dr,x) ind l dr,x Passage between left and right D-modules/crystals. In this subsection we will compare the abstractly defined functor Υ XdR from [Chapter II.3] and the hands-on functor : Crys l (X) Crys r (X) D-mod l (X) D-mod r (X), given by tensoring a given left D-module with the right D-module det(t (X))[dim(X)] According to [GaRo2, Proposition 2.2.4], for any scheme X almost of finite type we have a canonically defined equivalence that makes the diagram commute. Υ XdR Crys l (X) oblv l dr,x QCoh(X) : Crys l (X) Crys r (X), Υ XdR Crys r (X) oblvr dr,x Υ X IndCoh(X) Concretely, the functor Υ XdR is the functor from [Chapter II.3, Sect ] applied to X dr, and it is given by M M ω XdR, where is the action of QCoh(X dr ) (= Crys l (X)) on IndCoh(X dr ) (= Crys r (X)) Now, suppose that X is a smooth classical scheme. Recall that in this case there is a canonical equivalence (4.10) D-mod l (X) D-mod r (X), given by tensoring a given left D-module with the right D-module We denote the above functor by Υ D-mod,X. ω D-mod,X := det(t (X))[dim(X)].

25 AN APPLICATION: CRYSTALS We will prove: Theorem The following diagram of functors canonically commutes: (4.11) D-mod l (X) Υ D-mod,X F l X Crys l (X) Υ XdR. D-mod r (X) F r X Crys r (X) By applying Theorem to O X D-mod l (X), we obtain: Corollary There exists a canonical isomorphism in Crys r (X): (4.12) F r X(ω D-mod,X ) ω XdR Applying the forgetful functor to the isomorphism of (4.12), we obtain: oblv r dr,x : Crys r (X) IndCoh(X), Corollary There exists a canonical isomorphism in IndCoh(X): (4.13) Ξ X (det(t (X))[dim(X)] ω X. Note that latter corollary is the well-known identification of the abstractly defined dualizing sheaf with the shifted line bundle of top forms. Remark The isomorphism (4.13) can be proved without involving D-modules or crystals, by an argument along the same lines as that proving the isomorphism (4.12) in Sect. 4.4 below. This argument is given in a more general context in [Chapter IV.4, Proposition 7.3.4] Proof of Theorem First, we make the following observation, which follows from the constructions: Lemma For M D-mod l (X) and N D-mod r (X) we have a canonical isomorphism F r X(M N) F l X(M) F r X(N). This lemma reduces the assertion of Theorem to establishing the isomorphism (4.12) Recall that for a map f : X Y between smooth schemes, one defines the functor by requiring that the diagram (4.14) commute. f,r : D-mod r (Y ) D-mod r (X) f,l D-mod l (Y ) D-mod l (X) Υ D-mod,Y Υ D-mod,X D-mod r (Y ) f,r D-mod r (X) Assume for the moment that f is a closed embedding of smooth schemes. Let D-mod r (Y ) X D-mod r (Y ) be the full subcategory consisting of objects with set-theoretic support on X. Recall that in this case we have Kashiwara s lemma which says that the functor f,r induces an equivalence D-mod r (Y ) X D-mod r (X).

26 26 AN APPLICATION: CRYSTALS For a morphism f : X Y between schemes almost of finite type, let f,r : Crys r (Y ) Crys r (X) be the corresponding pullback functor, see [GaRo2, Sect ], i.e., f,r = f! dr. Assume for the moment that f is a closed embedding. Let Crys r (Y ) X Crys r (Y ) be the full subcategory consisting of objects with set-theoretic support on X. Recall (see [GaRo2, Proposition 2.5.6]) that in this case the functor f,r induces an equivalence Crys r (Y ) X Crys r (X) Let f : X Y be a closed embedding of smooth classical schemes. The next assertion also follows from the constructions: Lemma Under the equivalences the diagram commutes. f,r : D-mod r (Y ) X D-mod r (X) and f,r : Crys r (Y ) X Crys(X), As a corollary, we obtain: F r Y D-mod r (Y ) X Crys r (Y ) X f f,r,r D-mod r (X) F r X Crys r (X) Corollary For a closed embedding of smooth schemes f : X Y, the diagram D-mod r (Y ) f,r F r Y Crys r (Y ) f,r commutes. D-mod r (X) F r X Crys r (X) Proof. Follows from the fact that the functor f,r (resp., f,r ) factors through the co-localization D-mod r (Y ) D-mod r (Y ) X (resp., Crys r (Y ) Crys r (Y ) X ) We are finally ready to construct the isomorphism (4.12) and thereby prove Theorem Consider the object Consider the isomorphism ω D-mod,X ω D-mod,X = ω D-mod,X X D-mod r (X X). (4.15) F r X,r X (ω D-mod,X X),r F r X X (ω D-mod,X ω D-mod,X ) of Corollary On the one hand, F r X,r X (ω D-mod,X X) F r X,r X Υ D-mod,X X(O X X ) F r X Υ D-mod,X,l X (O X X) F r X Υ D-mod,X (O X ) F r X(ω D-mod,X ).

27 AN APPLICATION: CRYSTALS 27 On the other hand,,r F r X X (ω D-mod,X ω D-mod,X ),r (F r X(ω D-mod,X ) F r X(ω D-mod,X )) F r X(ω D-mod,X )! F r X(ω D-mod,X ), where! denotes the symmetric monoidal operation on Crys r (X), i.e., the! tensor product on IndCoh(X dr ). Thus, from (4.15) we obtain an isomorphism in Crys r (X). F r X(ω D-mod,X ) F r X(ω D-mod,X )! F r X(ω D-mod,X ) Now, it is easy to see that F r X (ω D-mod,X) is invertible as an object of the symmetric monoidal category Crys r (X). This implies that F r X (ω D-mod,X) is canonically isomorphic to the unit object, i.e., ω XdR, as required Identification of functors. In this subsection we will show that the pullback and pushforward functors on crystals correspond to the pullback and push-forward functors defined classically for D-modules We now show: Proposition Let f : X Y be a morphism between smooth schemes. Then the diagram of functors canonically commutes. f,r D-mod r (Y ) D-mod r (X) F r Y F r X Crys r (Y ) f,r Crys r (X) Remark It follows from the construction given below that when f is a closed embedding, the isomorphism of functors of Proposition identifies canonically with one in Corollary Proof. Follows by combining the following five commutative diagrams: D-mod l (Y ) F l Y f,l D-mod l (X) F l X (of Lemma 4.1.6); Crys l (Y ) f,l f,l Crys l (X) D-mod l (Y ) D-mod l (X) Υ D-mod,Y Υ D-mod,X D-mod r (Y ) f,r D-mod r (X)