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Non characteristic finiteness theorems in crystalline cohomology 1

Non characteristic finiteness theoremsin crystalline cohomology

Pierre Berthelot

Universite de Rennes 1

I.H.E.S., September 23, 2015

in honor of Arthur Ogus, for his 70th birthday


1. Introduction

1.1. Torsion coefficients for crystalline cohomology

Notation

1. Introduction

The purpose of this talk is to give some finiteness results ontorsion coefficients for crystalline cohomology.

Notation:

p = prime number, fixed for the whole talk;N ≥ 1 integer.

k = perfect field of characteristic p;WN = WN(k), ring of Witt vectors of length N with coefficients ink .


1. Introduction


Finiteness of crystalline cohomology

Recall that, for a smooth and proper k-scheme X0, crystallinecohomology with constant coefficients relative to WN is a “good”cohomology. Here this will mean that we have:

1 Isomorphism with the de Rham cohomology of a smoothlifting over WN , whenever there exists such a lifting;

2 RΓcrys(X0/WN) is a perfect complex of WN -modules;

3 RΓcrys(X0/WN) satisfies Poincare duality.

These properties still hold for cohomology with coefficients in alocally free finitely generated crystal. But such crystals do notsuffice to describe direct images and to get a relative version ofthese finiteness theorems, even for a morphism of smooth andproper k-schemes.


1. Introduction


Expected generalization

We want here to introduce bigger categories of coefficients relativeto WN for which similar results hold, and which have a sufficientlyreasonable behaviour under inverse and direct image so as to havebetter relative versions of the finiteness properties.

Note however that, since crystalline cohomology itself is notfinitely generated over WN in the non proper or non smooth case,and also because of the properties of the Frobenius pullback inarithmetic D-modules theory, we do not expect a full formalism ofGrothendieck’s six operations in this context.


1. Introduction

1.2. Main results

Categories of coefficients

Let X0 be a smooth k-scheme, S = Spec(WN), and OX0/S thestructural sheaf of the crystalline site Crys(X0/WN). We willconstruct two triangulated categories of filtered complexes ofOX0/S -modules, called respectively D-perfect and D∨-perfectcomplexes, such that:

1 Whenever there exists a smooth lifting X of X0 over WN ,these are respectively equivalent and anti-equivalent to thecategory of perfect complexes of filtered DX -modules, whereDX is the sheaf of PD-differential operators on X (filtrationswill be defined later).

2 D-perfect and D∨-perfect complexes are related by a localbiduality theorem.


1. Introduction

1.2. Main results

Non characteristic morphisms

3 These complexes have a singular support in the cotangentspace T ∗X0

, hence one can define the notion of anon-characteristic morphism f0 : X0 → Y0 with respect tosuch a complex on X0 or Y0.

4 With appropriate non-characteristic assumptions, one canprove comparison theorems between cohomological operationson perfect filtered complexes of DX -modules and D-perfect(resp. D∨-perfect) complexes, as well as finiteness and dualitytheorems.


1. Introduction

1.2. Main results

Remarks

Additional comments:

The results given here hold for complexes on the PD-nilpotentcrystalline site, which in some cases forces to assume p 6= 2.Similar results should hold for complexes on the usualcrystalline site, but one can expect increased technicalities dueto the nilpotency condition needed to associate a crystal to aDX -module.

If one endows S with the Hyodo-Kato log structure, thetheory applies to smooth fine saturated log schemes, satisfyingan additional condition on the dualizing complex (see Tsuji’sarticle on log cystalline Poincare duality).


2. D-perfection

Notation

2. D-perfection

We now consider a more general situation

X0� � //

��

X

��S0� � // S ,

where:

S = Z/pNZ-scheme, S0 = V (I) ⊂ S , I ⊂ OS = quasi-coherentideal, endowed with a PD-nilpotent divided power structure γ;X0 = smooth S0-scheme, of relative dimension d (or dX );X = smooth S-scheme lifting X0, if assumed to exist; then:

DX = sheaf of PD-differential operators on X/S (= D(0)X ).


2. D-perfection

2.1. D-modules and crystals

The crystal associated to a DX -module

Assume a smooth lifting X of X0 is given. Classically, one defines afunctor

CX0 : {Left DX -modules} −→ {OX0/S -modules}

as follows:

If (U,T , δ) is a thickening in Crys(X0/S), there exists locally on Tan S-morphism h : T → X extending U ↪→ X0 ↪→ X . If E is a leftDX -module and h1, h2 are two such extensions, the Taylor formulaprovides a canonical isomorphism of OT -modules h∗2E

∼−−→ h∗1E ,satisfying a transitivity relation for three extensions. Glueing viathese isomorphisms the sheaves h∗E when (U,T , δ) and h vary,one gets an OX0/S -module CX0(E), functorially in E .


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 CX0 factors through an equivalence

{Left DX -modules} ≈ //

CX0 ))

{Crystals on X0/S}� _

��{OX0/S -modules}.

Note that, since CX0 is defined by the inverse image functors h∗, itis only a right exact functor on the category of left DX -modules.


2. D-perfection

2.2. The functor CRX0

Definition of CRX0

We now want a derived category version of Proposition ??. Wederive CX0 and we shift so as to insure later compatibility withBorel’s conventions: we define the functor

CRX0 : D−(DX ) −→ D−(OX0/S)

by setting, for E• ∈ D−(DX ),

CRX0(E•) := LCX0(E•)[dX ],

where LCX0(E•) is obtained by applying CX0 to a resolution of E•by a complex of flat DX -modules, or more generally by a complexof DX -modules which are flat over OX .


2. D-perfection


Crystalline complexes

Notation. For any thickening (U,T , δ) ∈ Crys(X0/S) and anycomplex E • ∈ D(OX0/S), we denote by E •T ∈ D(OT ) the complexof Zariski sheaves on T defined by E .

Definition. Let E • ∈ D−(OX0/S). We say that E • is crystalline if,for any morphism v : (U ′,T ′, δ′)→ (U,T , δ) in Crys(X0/S), thecanonical morphism

Lv∗(E •T )→ E •T ′

(obtained by choosing a flat resolution P• of E • and taking thetransition morphism for P•) is an isomorphism of D−(OT ′).

Example: If f0 : X0 → Y0 is a smooth morphism, Rf0 crys ∗(OX0/S)is a crystalline complex on Crys(Y0/S).


2. D-perfection


Triangulated subcategories of D(OX0/S)

Notation. We define some triangulated subcategories ofD(OX0/S):

D−crys(OX0/S) = the full subcategory of D(OX0/S) whoseobjects are the crystalline complexes;

DbfTd(OX0/S) = the full subcategory of D(OX0/S) whose

obejcts are the complexes of OX0/S -modules which arebounded of finite Tor dimension;

Dqc(OX0/S) = the full subcategory of D(OX0/S) whoseobejcts are the complexes of OX0/S -modules E • such that E •Thas OT -quasi-coherent cohomology sheaves for any thickening(U,T , δ).

A sequence of indexes will denote the intersection of thecorresponding subcategories.


2. D-perfection


Properties of CRX0

The following properties follow immediately from the definition ofCRX0 :

Proposition 2

Let E• ∈ D−(DX ).

1 CRX0(E•) ∈ D−crys(OX0/S).

2 CRX0(E•) ∈ DbfTd(OX0/S) if and only if E• ∈ Db

fTd(DX ).

3 CRX0(E•) ∈ D−qc(OX0/S) if and only if E• ∈ D−qc(DX ).

Our strategy will now be to show that these properties characterizethe essential image of Db

fTd,qc(DX ) inside D−(OX0/S), and thatCRX0 induces an equivalence of categories with this image.


2. D-perfection

2.3. The right adjoint to CRX0

The crystalline bimodule defined by DX

We will use the existence of a right adjoint functor to CX0 . Recallthat the crystalline topos (X0/S)crys projects to the Zariski toposXZar via a morphism of topos uX0/S : (X0/S)crys −→ XZar. It ischaracterized by its inverse image functor, defined by

Γ((U,T , δ), u−1X0/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 functorCX0 to DX viewed as a left DX -module over itself. Then, byfunctoriality, the right action of DX on itself endows CX0(DX ) witha structure of (OX0/S , u

−1X0/S

(DX ))-bimodule.


2. D-perfection


Definition of DMX

We define a functor

MX : {OX0/S -modules} −→ {Left DX -modules}

by setting, for any OX0/S -module F ,

MX (F ) := uXO/S ∗HomOX0/S(CX0(DX ),F ).

The functor MX is right exact, and we set, for F • ∈ D+(OX0/S),

DMX (F •) := RMX (F •)[−dX ]

' RuXO/S ∗RHomOX0/S(CX0(DX ),F )[−dX ].

The following statements are then formal:


2. D-perfection


The adjunction formula

Proposition 3

1 For any left DX -module E , there is a functorial isomorphismof OX0/S -modules

CX0(E) ' CX0(DX )⊗u−1X0/SDX

u−1X0/SE .

2 The functor MX is right adjoint to the functor CX0 .

3 For any E• ∈ D−(DX ), F • ∈ D+(OX0/S), there is a canonicalisomorphism

RuX0/S ∗RHomOX0/S(CRX0(E•),F •) ' RHomDX

(E•,DMX (F •)).


2. D-perfection


Properties of DMX

Proposition 4

Let F • ∈ Dbcrys(OX0/S).

1 DMX (F •) ∈ DbfTd(DX ) if and only if F • ∈ Db

fTd(OX0/S).

2 DMX (F •) ∈ Dbqc(DX ) if and only if F • ∈ Db

qc(OX0/S).

From Propositions ?? and ??, we get functorsCRX0 : DbfTd,qc(DX ) −→ Db

crys,fTd,qc(OX0/S),

DMX : Dbcrys,fTd,qc(OX0/S) −→ Db

fTd,qc(DX ).


2. D-perfection


The equivalence theorem

Theorem 1

The functors CRX0 and DMX are quasi-inverse equivalencesbetween Db

fTd,qc(DX ) and Dbcrys,fTd,qc(OX0/S).

Hint: For E• ∈ DbfTd,qc(DX ), F • ∈ Db

crys,fTd,qc(OX0/S), theadjunction formula provides canonical morphismsE

• → DMX (CRX0(E•)),

CRX0(DMX (F •))→ F •).

Using the classical computation of RuX0/S ∗ by means of the

Cech-Alexander complex, one proves that these are isomorphisms.


2. D-perfection


Some consequences

Corollary 1.1

Let E•,F• ∈ DbfTd,qc(DX ). There is a canonical isomorphism

RuX0/S ∗RHomOX0/S(CRX0(E•),CRX0(F•))

∼−−→ RHomDX(E•,F•).

For E• = OX [−dX ], we get back the isomorphism betweencrystalline and de Rham cohomologies:

Corollary 1.2

Let F• ∈ DbfTd,qc(DX ). There is a canonical isomorphism

RuX0/S ∗(CRX0(F•))∼−−→ DR(F•) := RHomDX

(OX ,F•)[dX ].


2. D-perfection

2.4. D-perfect complexes

Definition of D-perfection

Definition. A complex E • ∈ Db(OX0/S) is D-perfect if thereexists an open covering (V0,α) of X0, and, for each α, a smoothlifting Vα of V0,α over S , a perfect complex E•α ∈ Db

perf(DVα) andan isomorphism

E •|Crys(V0,α/S) ' CRV0,α(E•α).

Special case. Bounded perfect complexes of OX0/S -modules areD-perfect.

Notation. We denote by DbD-perf(X0/S) ⊂ Db(OX0/S) the full

subcategory of D-perfect complexes.


2. D-perfection

2.4. D-perfect complexes

Characterizations of D-perfection

Proposition 5

Let X be a smooth lifting of X0 over S, and E • ∈ D(OX0/S). ThenE • is D-perfect if and only if there exists a complexE• ∈ Db

perf(DX ) and an isomorphism E • ' CRX0(E•).

Corollary 5.1

The subcategory DbD-perf(X0/S) ⊂ Db(OX0/S) is triangulated.

Proposition 6

Assume that S is locally noetherian. Let E • ∈ Dbcrys,fTd,qc(OX0/S).

Then E • is D-perfect relative to S if and only if E •|Crys(X0/S0) isD-perfect relative to S0.


3. Filtrations, local duality and D∨-perfection

Direct images under a closed immersion


Note: if f0 : X0 ↪→ Y0 is a closed immersion between smoothS0-schemes, Rf0 crys ∗ dose not preserve D-perfection.

Indeed, let (V ,T , δ) ∈ Crys(Y0/S), U = X0 ∩ V . Let K ⊂ OT bethe ideal of U in T , PU(T ) the PD-envelope of K compatible withγ and δ, and K ⊂ PU(T ) the PD-ideal generated by K. Then:

Rf0 crys ∗(OX0/S)T∼−−→ lim←−

n

PU(T )/K[n].

which is not a quasi-coherent OT -module.

We will see that Rf0 crys ∗(OX0/S) is actually the OX0/S -linear dualof a D-perfect complex.



3.1. Derived categories of filtered modules

Filtered rings and modules

To deal with such complexes, we will construct a triangulatedcategory of “OX0/S -duals of D-perfect complexes”. As DX islocally free of infinite rank over OX , we need to take into accountsome extra structure on the dual so as to be able to recover theinitial 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 filteredversions of the usual functors, e.g. filtered inverse images offiltered modules under a morphism of filtered ringed spaces.




Basic examples

We will use the following filtrations:

1 OX is filtered by the I-PD-adic filtration, given byFiliOX = I [i ]OX , with I [i ] = OS for i ≤ 0.

2 DX is filtered by the tensor product of the filtration by theorder of PD-differential operators with the I-PD-adicfiltration:

FiliDX =∑

j+k=i

I [j]DX ,−k ,

where, for all n ∈ Z, DX ,n is the sheaf of PD-differentialoperators of order ≤ n.

3 OX0/S is filtered by the JX0/S -PD-adic filtration, where(JX0/S)T = JT .




Filtered crystals

Definition: A filtered crystal on X0/S (called a T-crystal in[Ogus, Asterisque 221]) is a filtered OX0/S -module E such that,for any morphism v : (U ′,T ′, δ′)→ (U,T , δ) of Crys(X0/S), thetransition morphism v∗(ET )→ ET ′ (where v∗ is the filteredinverse image functor), is a filtered isomorphism.

Let X be a smooth S-scheme lifting X0. With these defiinitions,the functor CX0 extends as an equivalence of categories

CX0 : {Filtered left DX -modules} ≈−−→ {Filtered crystals on X0/S}.




Derived categories and functors for filtered modules

The category of filtered modules over a filtered ring (A,Ai ) is notabelian, but it has a natural notion of short exact sequence whichturns it into an exact category in the sense of [Quillen, SLNM 341].We can then apply Laumon’s results in [SLNM 1016] to build thederived category DF (A) of complexes of filtered A-modules.

One can also define as in [Laumon] the right and left derivedfunctors of an additive functor between categories of filteredmodules. Because we work with general filtered modules (withoutexhaustivity 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, ExposeI] to complexes of filtered modules. This provides the notions ofpseudo-coherence, finite tor dimension and perfection forcomplexes of filtered A-modules.




The filtered CRX0functor

We use these constructions to define the left derived filteredfunctor LCX0 . All the results of the previous section remain valid inthe filtered context. In particular, when X0 has a smooth lifting Xover S , the filtered functor CRX0 := LCX0 [dX ] induces anequivalence of categories

CRX0 : DbFfTd,qc(DX )≈−−→ DbFcrys,fTd,qc(OX0/S).

Without liftability assumption, we can define as above the categoryDbFD-perf(X0/S) ⊂ DbFcrys,fTd,qc(OX0/S): a filtered complex E•

belongs to DbFD-perf(X0/S) if and only if there exists a covering(Xα) of X such that, for each α, the restriction of E• to Xαbelongs to the essential image of DbFperf(DXα).



3.2. The biduality theorem

Exhaustive complexes

Definition. If E is a filtered module over a filtered ring A, wedenote

A f :=⋃i∈Z

FiliA, E f :=⋃i∈Z

FiliE ,

and we endow A f and E f with the induced filtrations. This turnsE f into a filtered A f-module. We say that E is exhaustive ifE f = E . We say that a complex E• ∈ DF (A) is exhaustive if thecanonical morphism E• f → E• is an isomorphism in DF (A f).

Example: Any D-perfect complex on X0/S is exhaustive.




The finite order biduality morphism

Let A be a commutative ring, endowed with an exhaustivefiltration, and let E•, I• be two complexes of filtered A-modules.The classical “biduality morphism” for E• relative to I• sits in acommutative diagram

E• // Hom•A(Hom•A(E•, I•), I•) // Hom•A(Hom• fA (E•, I•), I•)

E• f?�

OO

// Hom• fA (Hom•A(E•, I•), I•)

?�

OO

// Hom• fA (Hom

• fA (E•, I•), I•).?�

OO

We define the finite order biduality morphism for E• relative to I•as being the composition of the bottom row of the diagram. Thisdefinition extends naturally to DF (A).




The crystalline dual functor

We define the crystalline dualizing complex on X0/S by

KX0/S := OX0/S(dX )[2dX ].

For E • ∈ DF (OX0/S), we define its crystalline dual by

E •∨ := DcrysX0/S

(E •) := RHom• fOX0/S

(E •,KX0/S) ∈ DF (OX0/S).

If X is a smooth lifting of X0 over S , we define the functorCR∨X0

: D−F (DX )→ D+F (OX0/S) by setting, for E• ∈ D−F (DX ),

CR∨X0(E•) := (CRX0(E•))∨.




Biduality for D-perfect complexes

Assume now that E • is exhaustive. Taking for I• an appropriateresolution of KX0/S , the previous biduality diagram provides inDF (OX0/S) a canonical biduality morphism

E • −→ E •∨∨.

Theorem 2

Let E • ∈ DbFD-perf(X0/S).

1 The complex RHomOX0/S(E •,KX0/S) is exhaustive, hence

isomorphic to E •∨, and bounded.

2 The biduality morphisms E • → E •∨∨ and E •∨ → (E •∨)∨∨

are isomorphisms.



3.3. D∨-perfect complexes

Definition of D∨-perfection

Definition. A complex F • ∈ DbF (OX0/S) is D∨-perfect if thereexists an open covering (V0,α) of X0, and, for each α, a smoothlifting Vα of V0,α over S , a perfect complex E•α ∈ DbFperf(DVα)and an isomorphism

F •|Crys(V0,α/S) ' CR∨V0,α(E•α).

The category of D∨-perfect complexes is a full subcategory ofDbF (OX0/S), denoted by DbFD∨-perf(X0/S).

The following characterization implies that the condition isindependent of the covering (V0,α) and of the liftings (Vα).



3.3. D∨-perfect complexes

Characterization of D∨-perfection

Proposition 7

A filtered complex F • ∈ DbF (OX0/S) is D∨-perfect if and only ifthe following conditions hold:

1 F • is exhaustive.

2 F •∨ is D-perfect.

3 The biduality morphism F • → F •∨∨ is an isomorphism inDF (OX0/S).

Corollary 7.1

The category of D∨-perfect complexes is a triangulatedsubcategory of DF (OX0/S), which is anti-equivalent to

DbFD-perf(X0/S) .


4. Comparison and finiteness theorems

4.1. Singular support and non characteristic morphisms

Singular support of D- and D∨-perfect complexes


D-perfect and D∨-perfect complexes have a singular support. It isa closed subset of the cotangent space T ∗X0

, defined as follows.

1 Assume first that X0 has a smooth lifting X over S . ThengrDX ' grOX ⊗OX0

grDX0 . As I is a nilpotent ideal, itfollows that the topological spaces |Spec(grDX )| and |T ∗X0

|can be identified.

2 The “associated graded module” functor extends as an exactfunctor gr : DF (DX )→ D(grDX ). If E• ∈ DbFperf(DX ), thenthe sheaves Hn(gr E•) are quasi-coherent grDX -modules, andwe denote by Hn(gr E•) the corresponding quasi-coherentsheaf on the affine X0-scheme Spec(grDX ).




Singular support of D- and D∨-perfect complexes

3 If E • ∈ DbFD-perf(X0/S), let E• ∈ DbFperf(DX ) be such thatE • ' CRX0(E•). One can define the singular support SS(E •)as the closed subset

SS(E •) :=⋃n

Supp Hn(gr E•) ⊂ |T ∗X0|.

One checks that this does not depend on the choice of thelifting X .

4 In the general case, one can choose local liftings of X0 andglue the local constructions.

5 If F • ∈ DbFD∨-perf(X0/S), one defines SS(F •) as beingSS(F •∨).




Non characteristic morphisms

Let f0 : X0 → Y0 be a morphism of smooth S0-schemes, and let

T ∗X0

πX0##

X0 ×Y0 T ∗Y0

ϕ0oo

π′X0

��

g0 // T ∗Y0

πY0

��

X0f0 // Y0.

be the associated functoriality diagram for the cotangent bundle.

Definitions.

If F • ∈ DbFD-perf(Y0/S), we say that f0 is non characteristicfor F • if the restriction of ϕ0 to g−1

0 (SS(F •)) is proper.NB: This condition is always satisfied when f0 is smooth.

If E • ∈ DbFD-perf(X0/S), we say that f0 is non characteristicfor E • if the restriction of f0 to Supp(E •) is proper(⇔ g0|ϕ−1

0 (SS(E•)) is proper).



4.2. Non characteristic finiteness theorems for DX -modules

Finiteness theorems for D-modules

Assume now that S is locally noetherian. Let f : X → Y be amorphism of smooth S-schemes, F• ∈ DbFperf(DY ),E• ∈ DbFperf(DX ). As usual, we define

f !(F•) := DX→Y

L⊗f −1(DY ) f −1(F•)(dX/Y )[dX/Y ],

f+(E•) := Rf∗(DY←X

L⊗DX

E•).

Theorem 3

1 If f is non characteristic for F•, then f !(F•) ∈ DbFperf(DX ).

2 If f is non characteristic for E•, then f+(E•) ∈ DbFperf(DY ).

Proof as in [Laumon, Asterique 130], but using more generalfiniteness properties in derived categories of filtered modules.



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, withreduction f0 : X0 → Y0 over S0. For any F• ∈ D−F (DY ), there isa canonical isomorphism

Lf ∗0 crys(CRY0(F•))∼−−→ CRX0(f !(F•))(−dX/Y )[−2dX/Y ].

Proposition 8

Let f0 : X0 → Y0 be a morphism of smooth S0-schemes. IfF • ∈ DbFD-perf(Y0/S) and f0 is non characteristic for F •, thenLf ∗0 crys(F •) ∈ DbFD-perf(X0/S).



4.3. Finiteness results for inverse images

Inverse images of D∨-perfect complexes

For inverse images of D∨-perfect complexes, we work with the bigcrystalline topos.

Lemma 9.1

Let f : X → Y be a morphism of smooth S-schemes, withreduction f0 : X0 → Y0 over S0. For any F• ∈ DbFperf(DY ), thereis a canonical isomorphism

Lf ∗0 CRYS CR∨Y0(F•) ∼−−→ CR∨X0

(f !(F•)).

Proposition 9

Let f0 : X0 → Y0 be a morphism of smooth S0-schemes. IfF • ∈ DbFD∨-perf(Y0/S) and f0 is non characteristic for F •, thenLf ∗0 CRYS(F •) ∈ DbFD∨-perf(X0/S).



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, withreduction f0 : X0 → Y0 over S0. For any E• ∈ DbFperf(DX ), thereis a canonical isomorphism

CRY0(f+(E•))∼−−→ Rf0 crys ∗ CRX0(E•).

The proof proceeds by reducing to comparison with de Rhamcohomology.

Proposition 10

Let f0 : X0 → Y0 be a smooth morphism of smooth S0-schemes. IfE • ∈ DbFD-perf(X0/S) and f0 is non characteristic for E •, thenRf0 crys ∗(E •) ∈ DbFD-perf(Y0/S).




Comparison theorem for D∨-perfect complexes

To study the direct images of D∨-perfect complexes, we need tomake stronger assumptions on S , in order to use Grothendieck’sduality theory for coherent sheaves in the crystalline context: fromnow on, we assume that S is a quotient of a discrete valuation ring.

The following comparison theorem can be viewed as a relativeduality theorem for the CRX0 functor:

Theorem 4

Let f : X → Y be a proper morphism between smooth S-schemes,and let E• ∈ DbFperf(DX ). There exists in DbF (OY0/S) acanonical isomorphism

CR∨Y0(f+(E•)) ' Rf0 crys ∗(CR∨X0

(E•)).




Direct images of D∨-perfect complexes

Theorem 5

Let X0, Y0 be proper and smooth S0-schemes, and f0 : X0 → Y0

an S0-morphism. If E • ∈ DbFD∨-perf(X0/S), thenRf0 crys ∗(E •) ∈ DbFD∨-perf(Y0/S).

The proof uses the graph factorization to deal separately with thecases of a closed immersion, which follows from Theorem ??, andof a proper and smooth morphism, which uses the next dualitytheorem.




Relative Poincare duality

Theorem 6

Let f0 : X0 → Y0 be a proper and smooth morphism of smoothS0-schemes.

1 There exists in DbF (OYO/S) a trace morphism

Trf0 : Rf0 crys ∗(KX0/S) −→ KY0/S ,

whose value on the thickening (Y0,Y0) can be identified withthe de Rham trace morphism.

2 Let E • ∈ DbFD-perf(X0/S). The pairing

Rf0 crys ∗(E •)L⊗OY0/S

Rf0 crys ∗(E •∨) −→ KY0/S

induced by Trf0 is a perfect pairing.




Crystalline cohomology of D-perfect complexes

Theorem 7

Let k be a perfect field of characteristic p 6= 2, S0 = Spec(k),S = Spec(WN(k)). Let X0 be a proper and smooth k-scheme, andlet E • ∈ DbFD-perf(X0/S).

1 The crystalline cohomology complexes RΓ(X0/S ,E •) andRΓ(X0/S ,E •∨) are perfect complexes of WN(k)-modules.

2 The crystalline trace morphism induces a perfect pairing

RΓ(X0/S ,E •)L⊗WN(k) RΓ(X0/S ,E •∨) −→WN(k).

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Non characteristic finiteness theorems in crystalline ...abbes/Ogus/berthelot.pdf · N) satis es Poincar e duality. These properties still hold for cohomology with coe cients in a