Non characteristic finiteness theorems in crystalline cohomology

Non characteristic finiteness theorems in crystalline cohomology 1 Non characteristic finiteness theorems in crystalline cohomology Pierre Berthelot Université de Rennes 1 I.H.É.S., September 23, 2015 in honor of Arthur Ogus, for his 70th birthday

Non characteristic finiteness theorems in crystalline cohomology 2 1. Introduction 1.1. Torsion coefficients for crystalline cohomology Notation 1. Introduction The purpose of this talk is to give some finiteness results on torsion coefficients for crystalline cohomology. Notation: p = prime number, fixed for the whole talk; N 1 integer. k = perfect field of characteristic p; W N = W N (k), ring of Witt vectors of length N with coefficients in k.

Non characteristic finiteness theorems in crystalline cohomology 3 1. Introduction 1.1. Torsion coefficients for crystalline cohomology Finiteness of crystalline cohomology Recall that, for a smooth and proper k-scheme X 0, crystalline cohomology with constant coefficients relative to W N is a good cohomology. Here this will mean that we have: 1 Isomorphism with the de Rham cohomology of a smooth lifting over W N, whenever there exists such a lifting; 2 RΓ crys (X 0 /W N ) is a perfect complex of W N -modules; 3 RΓ crys (X 0 /W N ) satisfies Poincaré duality. These properties still hold for cohomology with coefficients in a locally free finitely generated crystal. But such crystals do not suffice to describe direct images and to get a relative version of these finiteness theorems, even for a morphism of smooth and proper k-schemes.

Non characteristic finiteness theorems in crystalline cohomology 4 1. Introduction 1.1. Torsion coefficients for crystalline cohomology Expected generalization We want here to introduce bigger categories of coefficients relative to W N for which similar results hold, and which have a sufficiently reasonable behaviour under inverse and direct image so as to have better relative versions of the finiteness properties. Note however that, since crystalline cohomology itself is not finitely generated over W N in the non proper or non smooth case, and also because of the properties of the Frobenius pullback in arithmetic D-modules theory, we do not expect a full formalism of Grothendieck s six operations in this context.

Non characteristic finiteness theorems in crystalline cohomology 5 1. Introduction 1.2. Main results Categories of coefficients Let X 0 be a smooth k-scheme, S = Spec(W N ), and O X0 /S the structural sheaf of the crystalline site Crys(X 0 /W N ). We will construct two triangulated categories of filtered complexes of O X0 /S-modules, called respectively D-perfect and D -perfect complexes, such that: 1 Whenever there exists a smooth lifting X of X 0 over W N, these are respectively equivalent and anti-equivalent to the category of perfect complexes of filtered D X -modules, where D X is the sheaf of PD-differential operators on X (filtrations will be defined later). 2 D-perfect and D -perfect complexes are related by a local biduality theorem.

Non characteristic finiteness theorems in crystalline cohomology 6 1. Introduction 1.2. Main results Non characteristic morphisms 3 These complexes have a singular support in the cotangent space T X 0, hence one can define the notion of a non-characteristic morphism f 0 : X 0 Y 0 with respect to such a complex on X 0 or Y 0. 4 With appropriate non-characteristic assumptions, one can prove comparison theorems between cohomological operations on perfect filtered complexes of D X -modules and D-perfect (resp. D -perfect) complexes, as well as finiteness and duality theorems.

Non characteristic finiteness theorems in crystalline cohomology 7 1. Introduction 1.2. Main results Remarks Additional comments: The results given here hold for complexes on the PD-nilpotent crystalline site, which in some cases forces to assume p 2. Similar results should hold for complexes on the usual crystalline site, but one can expect increased technicalities due to the nilpotency condition needed to associate a crystal to a D X -module. If one endows S with the Hyodo-Kato log structure, the theory applies to smooth fine saturated log schemes, satisfying an additional condition on the dualizing complex (see Tsuji s article on log cystalline Poincaré duality).

Non characteristic finiteness theorems in crystalline cohomology 8 2. D-perfection Notation 2. D-perfection We now consider a more general situation X where: X 0 S 0 S, S = Z/p N Z-scheme, S 0 = V (I) S, I O S = quasi-coherent ideal, endowed with a PD-nilpotent divided power structure γ; X 0 = smooth S 0 -scheme, of relative dimension d (or d X ); X = smooth S-scheme lifting X 0, if assumed to exist; then: D X = sheaf of PD-differential operators on X /S (= D (0) X ).

Non characteristic finiteness theorems in crystalline cohomology 9 2. D-perfection 2.1. D-modules and crystals The crystal associated to a D X -module Assume a smooth lifting X of X 0 is given. Classically, one defines a functor as follows: C X0 : {Left D X -modules} {O X0 /S-modules} If (U, T, δ) is a thickening in Crys(X 0 /S), there exists locally on T an S-morphism h : T X extending U X 0 X. If E is a left D X -module and h 1, h 2 are two such extensions, the Taylor formula provides a canonical isomorphism of O T -modules h2 E h1 E, satisfying a transitivity relation for three extensions. Glueing via these isomorphisms the sheaves h E when (U, T, δ) and h vary, one gets an O X0 /S-module C X0 (E), functorially in E.

Non characteristic finiteness theorems in crystalline cohomology 10 2. D-perfection 2.1. D-modules and crystals Equivalence between D-modules and crystals The following basic result is then well known: Proposition 1 The functor C X0 factors through an equivalence {Left D X -modules} {Crystals on X 0 /S} C X0 {O X0 /S-modules}. Note that, since C X0 is defined by the inverse image functors h, it is only a right exact functor on the category of left D X -modules.

Non characteristic finiteness theorems in crystalline cohomology 11 2. D-perfection 2.2. The functor CR X0 Definition of CR X0 We now want a derived category version of Proposition??. We derive C X0 and we shift so as to insure later compatibility with Borel s conventions: we define the functor by setting, for E D (D X ), CR X0 : D (D X ) D (O X0 /S) CR X0 (E ) := LC X0 (E )[d X ], where LC X0 (E ) is obtained by applying C X0 to a resolution of E by a complex of flat D X -modules, or more generally by a complex of D X -modules which are flat over O X.

Non characteristic finiteness theorems in crystalline cohomology 12 2. D-perfection 2.2. The functor CR X0 Crystalline complexes Notation. For any thickening (U, T, δ) Crys(X 0 /S) and any complex E D(O X0 /S), we denote by E T D(O T ) the complex of Zariski sheaves on T defined by E. Definition. Let E D (O X0 /S). We say that E is crystalline if, for any morphism v : (U, T, δ ) (U, T, δ) in Crys(X 0 /S), the canonical morphism Lv (E T ) E T (obtained by choosing a flat resolution P of E and taking the transition morphism for P ) is an isomorphism of D (O T ). Example: If f 0 : X 0 Y 0 is a smooth morphism, Rf 0 crys (O X0 /S) is a crystalline complex on Crys(Y 0 /S).

Non characteristic finiteness theorems in crystalline cohomology 13 2. D-perfection 2.2. The functor CR X0 Triangulated subcategories of D(O X0 /S) Notation. We define some triangulated subcategories of D(O X0 /S): D crys(o X0 /S) = the full subcategory of D(O X0 /S) whose objects are the crystalline complexes; D b ftd (O X 0 /S) = the full subcategory of D(O X0 /S) whose obejcts are the complexes of O X0 /S-modules which are bounded of finite Tor dimension; D qc (O X0 /S) = the full subcategory of D(O X0 /S) whose obejcts are the complexes of O X0 /S-modules E such that E T has O T -quasi-coherent cohomology sheaves for any thickening (U, T, δ). A sequence of indexes will denote the intersection of the corresponding subcategories.

Non characteristic finiteness theorems in crystalline cohomology 14 2. D-perfection 2.2. The functor CR X0 Properties of CR X0 The following properties follow immediately from the definition of CR X0 : Proposition 2 Let E D (D X ). 1 CR X0 (E ) D crys(o X0 /S). 2 CR X0 (E ) D b ftd (O X 0 /S) if and only if E D b ftd (D X ). 3 CR X0 (E ) D qc(o X0 /S) if and only if E D qc(d X ). Our strategy will now be to show that these properties characterize the essential image of DfTd,qc b (D X ) inside D (O X0 /S), and that CR X0 induces an equivalence of categories with this image.

Non characteristic finiteness theorems in crystalline cohomology 15 2. D-perfection 2.3. The right adjoint to CR X0 The crystalline bimodule defined by D X We will use the existence of a right adjoint functor to C X0. Recall that the crystalline topos (X 0 /S) crys projects to the Zariski topos X Zar via a morphism of topos u X0 /S : (X 0 /S) crys X Zar. It is characterized by its inverse image functor, defined by Γ((U, T, δ), u 1 X 0 /S(F)) = Γ(U, F) for all sheaves F on X and all thickenings (U, T, δ). The crystalline transfert bimodule. One can apply the functor C X0 to D X viewed as a left D X -module over itself. Then, by functoriality, the right action of D X on itself endows C X0 (D X ) with a structure of (O X0 /S, u 1 X 0 /S (D X ))-bimodule.

Non characteristic finiteness theorems in crystalline cohomology 16 2. D-perfection 2.3. The right adjoint to CR X0 Definition of DM X We define a functor M X : {O X0 /S-modules} {Left D X -modules} by setting, for any O X0 /S-module F, M X (F ) := u XO /S Hom (C OX0 /S X 0 (D X ), F ). The functor M X is right exact, and we set, for F D + (O X0 /S), DM X (F ) := RM X (F )[ d X ] Ru XO /S RHom (C OX0 /S X 0 (D X ), F )[ d X ]. The following statements are then formal:

Non characteristic finiteness theorems in crystalline cohomology 17 2. D-perfection 2.3. The right adjoint to CR X0 The adjunction formula Proposition 3 1 For any left D X -module E, there is a functorial isomorphism of O X0 /S-modules C X0 (E) C X0 (D X ) u 1 X 0 /S D X u 1 X 0 /S E. 2 The functor M X is right adjoint to the functor C X0. 3 For any E D (D X ), F D + (O X0 /S), there is a canonical isomorphism Ru X0 /S RHom OX0 /S (CR X 0 (E ), F ) RHom DX (E, DM X (F )).

Non characteristic finiteness theorems in crystalline cohomology 18 2. D-perfection 2.3. The right adjoint to CR X0 Properties of DM X Proposition 4 Let F Dcrys(O b X0 /S). 1 DM X (F ) DfTd b (D X ) if and only if F DfTd b (O X 0 /S). 2 DM X (F ) Dqc(D b X ) if and only if F Dqc(O b X0 /S). From Propositions?? and??, we get functors CR X0 : DfTd,qc b (D X ) Dcrys,fTd,qc b (O X 0 /S), DM X : Dcrys,fTd,qc b (O X 0 /S) DfTd,qc b (D X ).

Non characteristic finiteness theorems in crystalline cohomology 19 2. D-perfection 2.3. The right adjoint to CR X0 The equivalence theorem Theorem 1 The functors CR X0 and DM X are quasi-inverse equivalences between DfTd,qc b (D X ) and Dcrys,fTd,qc b (O X 0 /S). Hint: For E DfTd,qc b (D X ), F Dcrys,fTd,qc b (O X 0 /S), the adjunction formula provides canonical morphisms E DM X (CR X0 (E )), CR X0 (DM X (F )) F ). Using the classical computation of Ru X0 /S by means of the Čech-Alexander complex, one proves that these are isomorphisms.

Non characteristic finiteness theorems in crystalline cohomology 20 2. D-perfection 2.3. The right adjoint to CR X0 Some consequences Corollary 1.1 Let E, F D b ftd,qc (D X ). There is a canonical isomorphism Ru X0 /S RHom OX0 /S (CR X 0 (E ), CR X0 (F )) RHom DX (E, F ). For E = O X [ d X ], we get back the isomorphism between crystalline and de Rham cohomologies: Corollary 1.2 Let F D b ftd,qc (D X ). There is a canonical isomorphism Ru X0 /S (CR X0 (F )) DR(F ) := RHom DX (O X, F )[d X ].

Non characteristic finiteness theorems in crystalline cohomology 21 2. D-perfection 2.4. D-perfect complexes Definition of D-perfection Definition. A complex E D b (O X0 /S) is D-perfect if there exists an open covering (V 0,α ) of X 0, and, for each α, a smooth lifting V α of V 0,α over S, a perfect complex E α D b perf (D V α ) and an isomorphism E Crys(V0,α /S) CR V0,α (E α). Special case. Bounded perfect complexes of O X0 /S-modules are D-perfect. Notation. We denote by D b D-perf (X 0/S) D b (O X0 /S) the full subcategory of D-perfect complexes.

Non characteristic finiteness theorems in crystalline cohomology 22 2. D-perfection 2.4. D-perfect complexes Characterizations of D-perfection Proposition 5 Let X be a smooth lifting of X 0 over S, and E D(O X0 /S). Then E is D-perfect if and only if there exists a complex E D b perf (D X ) and an isomorphism E CR X0 (E ). Corollary 5.1 The subcategory D b D-perf (X 0/S) D b (O X0 /S) is triangulated. Proposition 6 Assume that S is locally noetherian. Let E D b crys,ftd,qc (O X 0 /S). Then E is D-perfect relative to S if and only if E Crys(X0 /S 0 ) is D-perfect relative to S 0.

Non characteristic finiteness theorems in crystalline cohomology 23 3. Filtrations, local duality and D -perfection Direct images under a closed immersion 3. Filtrations, local duality and D -perfection Note: if f 0 : X 0 Y 0 is a closed immersion between smooth S 0 -schemes, Rf 0 crys dose not preserve D-perfection. Indeed, let (V, T, δ) Crys(Y 0 /S), U = X 0 V. Let K O T be the ideal of U in T, P U (T ) the PD-envelope of K compatible with γ and δ, and K P U (T ) the PD-ideal generated by K. Then: Rf 0 crys (O X0 /S) T which is not a quasi-coherent O T -module. lim n P U (T )/K [n]. We will see that Rf 0 crys (O X0 /S) is actually the O X0 /S-linear dual of a D-perfect complex.

Non characteristic finiteness theorems in crystalline cohomology 24 3. Filtrations, local duality and D -perfection 3.1. Derived categories of filtered modules Filtered rings and modules To deal with such complexes, we will construct a triangulated category of O X0 /S-duals of D-perfect complexes. As D X is locally free of infinite rank over O X, we need to take into account some extra structure on the dual so as to be able to recover the initial complex from its dual via a biduality theorem. To this end, we will now work systematically with filtered complexes. Filtered rings and modules are defined as in [Bourbaki, Alg. Comm., Ch. III, 2, no. 1]. The functors Hom(, ) and are endowed with their natural filtration. This provides filtered versions of the usual functors, e.g. filtered inverse images of filtered modules under a morphism of filtered ringed spaces.

Non characteristic finiteness theorems in crystalline cohomology 25 3. Filtrations, local duality and D -perfection 3.1. Derived categories of filtered modules Basic examples We will use the following filtrations: 1 O X is filtered by the I-PD-adic filtration, given by Fil i O X = I [i] O X, with I [i] = O S for i 0. 2 D X is filtered by the tensor product of the filtration by the order of PD-differential operators with the I-PD-adic filtration: Fil i D X = I [j] D X, k, j+k=i where, for all n Z, D X,n is the sheaf of PD-differential operators of order n. 3 O X0 /S is filtered by the J X0 /S-PD-adic filtration, where (J X0 /S) T = J T.

Non characteristic finiteness theorems in crystalline cohomology 26 3. Filtrations, local duality and D -perfection 3.1. Derived categories of filtered modules Filtered crystals Definition: A filtered crystal on X 0 /S (called a T-crystal in [Ogus, Astérisque 221]) is a filtered O X0 /S-module E such that, for any morphism v : (U, T, δ ) (U, T, δ) of Crys(X 0 /S), the transition morphism v (E T ) E T (where v is the filtered inverse image functor), is a filtered isomorphism. Let X be a smooth S-scheme lifting X 0. With these defiinitions, the functor C X0 extends as an equivalence of categories C X0 : {Filtered left D X -modules} {Filtered crystals on X 0 /S}.

Non characteristic finiteness theorems in crystalline cohomology 27 3. Filtrations, local duality and D -perfection 3.1. Derived categories of filtered modules Derived categories and functors for filtered modules The category of filtered modules over a filtered ring (A, A i ) is not abelian, but it has a natural notion of short exact sequence which turns it into an exact category in the sense of [Quillen, SLNM 341]. We can then apply Laumon s results in [SLNM 1016] to build the derived category DF (A) of complexes of filtered A-modules. One can also define as in [Laumon] the right and left derived functors of an additive functor between categories of filtered modules. Because we work with general filtered modules (without exhaustivity or separatedness assumption), there are enough displayed objects to ensure the derivability of the usual functors. Finally, one can extend the finiteness conditions of [SGA 6, Exposé I] to complexes of filtered modules. This provides the notions of pseudo-coherence, finite tor dimension and perfection for complexes of filtered A-modules.

Non characteristic finiteness theorems in crystalline cohomology 28 3. Filtrations, local duality and D -perfection 3.1. Derived categories of filtered modules The filtered CR X0 functor We use these constructions to define the left derived filtered functor LC X0. All the results of the previous section remain valid in the filtered context. In particular, when X 0 has a smooth lifting X over S, the filtered functor CR X0 := LC X0 [d X ] induces an equivalence of categories CR X0 : D b F ftd,qc (D X ) D b F crys,ftd,qc (O X0 /S). Without liftability assumption, we can define as above the category D b F D-perf (X 0 /S) D b F crys,ftd,qc (O X0 /S): a filtered complex E belongs to D b F D-perf (X 0 /S) if and only if there exists a covering (X α ) of X such that, for each α, the restriction of E to X α belongs to the essential image of D b F perf (D Xα ).

Non characteristic finiteness theorems in crystalline cohomology 29 3. Filtrations, local duality and D -perfection 3.2. The biduality theorem Exhaustive complexes Definition. If E is a filtered module over a filtered ring A, we denote A f := Fil i A, E f := Fil i E, i Z i Z and we endow A f and E f with the induced filtrations. This turns E f into a filtered A f -module. We say that E is exhaustive if E f = E. We say that a complex E DF (A) is exhaustive if the canonical morphism E f E is an isomorphism in DF (A f ). Example: Any D-perfect complex on X 0 /S is exhaustive.

Non characteristic finiteness theorems in crystalline cohomology 30 3. Filtrations, local duality and D -perfection 3.2. The biduality theorem The finite order biduality morphism Let A be a commutative ring, endowed with an exhaustive filtration, and let E, I be two complexes of filtered A-modules. The classical biduality morphism for E relative to I sits in a commutative diagram E Hom A (Hom A (E, I ), I ) Hom f A (Hom A (E, I ), I ) E f Hom f A (Hom A (E, I ), I ) Hom f f A (Hom A (E, I ), I ). We define the finite order biduality morphism for E relative to I as being the composition of the bottom row of the diagram. This definition extends naturally to DF (A).

Non characteristic finiteness theorems in crystalline cohomology 31 3. Filtrations, local duality and D -perfection 3.2. The biduality theorem The crystalline dual functor We define the crystalline dualizing complex on X 0 /S by K X0 /S := O X0 /S(d X )[2d X ]. For E DF (O X0 /S), we define its crystalline dual by E := D crys X 0 /S (E ) := RHom f O X0 /S (E, K X0 /S) DF (O X0 /S). If X is a smooth lifting of X 0 over S, we define the functor CR X 0 : D F (D X ) D + F (O X0 /S) by setting, for E D F (D X ), CR X 0 (E ) := (CR X0 (E )).

Non characteristic finiteness theorems in crystalline cohomology 32 3. Filtrations, local duality and D -perfection 3.2. The biduality theorem Biduality for D-perfect complexes Assume now that E is exhaustive. Taking for I an appropriate resolution of K X0 /S, the previous biduality diagram provides in DF (O X0 /S) a canonical biduality morphism E E. Theorem 2 Let E D b F D-perf (X 0 /S). 1 The complex RHom OX0 /S (E, K X0 /S) is exhaustive, hence isomorphic to E, and bounded. 2 The biduality morphisms E E and E (E ) are isomorphisms.

Non characteristic finiteness theorems in crystalline cohomology 33 3. Filtrations, local duality and D -perfection 3.3. D -perfect complexes Definition of D -perfection Definition. A complex F D b F (O X0 /S) is D -perfect if there exists an open covering (V 0,α ) of X 0, and, for each α, a smooth lifting V α of V 0,α over S, a perfect complex E α D b F perf (D Vα ) and an isomorphism F Crys(V0,α /S) CR V 0,α (E α). The category of D -perfect complexes is a full subcategory of D b F (O X0 /S), denoted by D b F D -perf(x 0 /S). The following characterization implies that the condition is independent of the covering (V 0,α ) and of the liftings (V α ).

Non characteristic finiteness theorems in crystalline cohomology 34 3. Filtrations, local duality and D -perfection 3.3. D -perfect complexes Characterization of D -perfection Proposition 7 A filtered complex F D b F (O X0 /S) is D -perfect if and only if the following conditions hold: 1 F is exhaustive. 2 F is D-perfect. 3 The biduality morphism F F is an isomorphism in DF (O X0 /S). Corollary 7.1 The category of D -perfect complexes is a triangulated subcategory of DF (O X0 /S), which is anti-equivalent to D b F D-perf (X 0 /S).

Non characteristic finiteness theorems in crystalline cohomology 35 4. Comparison and finiteness theorems 4.1. Singular support and non characteristic morphisms Singular support of D- and D -perfect complexes 4. Comparison and finiteness theorems D-perfect and D -perfect complexes have a singular support. It is a closed subset of the cotangent space T X 0, defined as follows. 1 Assume first that X 0 has a smooth lifting X over S. Then gr D X gr O X OX0 gr D X0. As I is a nilpotent ideal, it follows that the topological spaces Spec(gr D X ) and T X 0 can be identified. 2 The associated graded module functor extends as an exact functor gr : DF (D X ) D(gr D X ). If E D b F perf (D X ), then the sheaves H n (gr E ) are quasi-coherent gr D X -modules, and we denote by H n (gr E ) the corresponding quasi-coherent sheaf on the affine X 0 -scheme Spec(gr D X ).

Non characteristic finiteness theorems in crystalline cohomology 36 4. Comparison and finiteness theorems 4.1. Singular support and non characteristic morphisms Singular support of D- and D -perfect complexes 3 If E D b F D-perf (X 0 /S), let E D b F perf (D X ) be such that E CR X0 (E ). One can define the singular support SS(E ) as the closed subset SS(E ) := n Supp H n (gr E ) T X 0. One checks that this does not depend on the choice of the lifting X. 4 In the general case, one can choose local liftings of X 0 and glue the local constructions. 5 If F D b F D -perf(x 0 /S), one defines SS(F ) as being SS(F ).

Non characteristic finiteness theorems in crystalline cohomology 37 4. Comparison and finiteness theorems 4.1. Singular support and non characteristic morphisms Non characteristic morphisms Let f 0 : X 0 Y 0 be a morphism of smooth S 0 -schemes, and let T X 0 ϕ 0 X 0 Y0 T Y 0 g 0 T Y 0 π X0 π X 0 X 0 f 0 Y 0. π Y0 be the associated functoriality diagram for the cotangent bundle. Definitions. If F D b F D-perf (Y 0 /S), we say that f 0 is non characteristic for F if the restriction of ϕ 0 to g0 1 (SS(F )) is proper. NB: This condition is always satisfied when f 0 is smooth. If E D b F D-perf (X 0 /S), we say that f 0 is non characteristic for E if the restriction of f 0 to Supp(E ) is proper ( g 0 ϕ 1 0 (SS(E )) is proper).

Non characteristic finiteness theorems in crystalline cohomology 38 4. Comparison and finiteness theorems 4.2. Non characteristic finiteness theorems for D X -modules Finiteness theorems for D-modules Assume now that S is locally noetherian. Let f : X Y be a morphism of smooth S-schemes, F D b F perf (D Y ), E D b F perf (D X ). As usual, we define Theorem 3 f! (F ) := D X Y L f 1 (D Y ) f 1 (F )(d X /Y )[d X /Y ], f + (E ) := Rf (D Y X L DX E ). 1 If f is non characteristic for F, then f! (F ) D b F perf (D X ). 2 If f is non characteristic for E, then f + (E ) D b F perf (D Y ). Proof as in [Laumon, Astérique 130], but using more general finiteness properties in derived categories of filtered modules.

Non characteristic finiteness theorems in crystalline cohomology 39 4. Comparison and finiteness theorems 4.3. Finiteness results for inverse images Inverse images of D-perfect complexes Lemma 8.1 Let f : X Y be a morphism of smooth S-schemes, with reduction f 0 : X 0 Y 0 over S 0. For any F D F (D Y ), there is a canonical isomorphism Lf0 crys(cr Y0 (F )) CR X0 (f! (F ))( d X /Y )[ 2d X /Y ]. Proposition 8 Let f 0 : X 0 Y 0 be a morphism of smooth S 0 -schemes. If F D b F D-perf (Y 0 /S) and f 0 is non characteristic for F, then Lf 0 crys (F ) D b F D-perf (X 0 /S).

Non characteristic finiteness theorems in crystalline cohomology 40 4. Comparison and finiteness theorems 4.3. Finiteness results for inverse images Inverse images of D -perfect complexes For inverse images of D -perfect complexes, we work with the big crystalline topos. Lemma 9.1 Let f : X Y be a morphism of smooth S-schemes, with reduction f 0 : X 0 Y 0 over S 0. For any F D b F perf (D Y ), there is a canonical isomorphism Lf0 CRYS CR Y 0 (F ) CR X 0 (f! (F )). Proposition 9 Let f 0 : X 0 Y 0 be a morphism of smooth S 0 -schemes. If F D b F D -perf(y 0 /S) and f 0 is non characteristic for F, then Lf 0 CRYS (F ) D b F D -perf(x 0 /S).

Non characteristic finiteness theorems in crystalline cohomology 41 4. Comparison and finiteness theorems 4.4. Finiteness results for direct images Direct images of D-perfect complexes (smooth case) Lemma 10.1 Let f : X Y be a smooth morphism of smooth S-schemes, with reduction f 0 : X 0 Y 0 over S 0. For any E D b F perf (D X ), there is a canonical isomorphism CR Y0 (f + (E )) Rf 0 crys CR X0 (E ). The proof proceeds by reducing to comparison with de Rham cohomology. Proposition 10 Let f 0 : X 0 Y 0 be a smooth morphism of smooth S 0 -schemes. If E D b F D-perf (X 0 /S) and f 0 is non characteristic for E, then Rf 0 crys (E ) D b F D-perf (Y 0 /S).

Non characteristic finiteness theorems in crystalline cohomology 42 4. Comparison and finiteness theorems 4.4. Finiteness results for direct images Comparison theorem for D -perfect complexes To study the direct images of D -perfect complexes, we need to make stronger assumptions on S, in order to use Grothendieck s duality theory for coherent sheaves in the crystalline context: from now on, we assume that S is a quotient of a discrete valuation ring. The following comparison theorem can be viewed as a relative duality theorem for the CR X0 functor: Theorem 4 Let f : X Y be a proper morphism between smooth S-schemes, and let E D b F perf (D X ). There exists in D b F (O Y0 /S) a canonical isomorphism CR Y 0 (f + (E )) Rf 0 crys (CR X 0 (E )).

Non characteristic finiteness theorems in crystalline cohomology 43 4. Comparison and finiteness theorems 4.4. Finiteness results for direct images Direct images of D -perfect complexes Theorem 5 Let X 0, Y 0 be proper and smooth S 0 -schemes, and f 0 : X 0 Y 0 an S 0 -morphism. If E D b F D -perf(x 0 /S), then Rf 0 crys (E ) D b F D -perf(y 0 /S). The proof uses the graph factorization to deal separately with the cases of a closed immersion, which follows from Theorem??, and of a proper and smooth morphism, which uses the next duality theorem.

Non characteristic finiteness theorems in crystalline cohomology 44 4. Comparison and finiteness theorems 4.4. Finiteness results for direct images Relative Poincaré duality Theorem 6 Let f 0 : X 0 Y 0 be a proper and smooth morphism of smooth S 0 -schemes. 1 There exists in D b F (O YO /S) a trace morphism Tr f0 : Rf 0 crys (K X0 /S) K Y0 /S, whose value on the thickening (Y 0, Y 0 ) can be identified with the de Rham trace morphism. 2 Let E D b F D-perf (X 0 /S). The pairing Rf 0 crys (E ) L OY0 /S Rf 0 crys (E ) K Y0 /S induced by Tr f0 is a perfect pairing.

Non characteristic finiteness theorems in crystalline cohomology 45 4. Comparison and finiteness theorems 4.4. Finiteness results for direct images Crystalline cohomology of D-perfect complexes Theorem 7 Let k be a perfect field of characteristic p 2, S 0 = Spec(k), S = Spec(W N (k)). Let X 0 be a proper and smooth k-scheme, and let E D b F D-perf (X 0 /S). 1 The crystalline cohomology complexes RΓ(X 0 /S, E ) and RΓ(X 0 /S, E ) are perfect complexes of W N (k)-modules. 2 The crystalline trace morphism induces a perfect pairing RΓ(X 0 /S, E ) L WN (k) RΓ(X 0 /S, E ) W N (k).