A L
E S
E D L ’IN IT ST T U
F O U R
ANNALES
DE
L’INSTITUT FOURIER
Sachio OHKAWA
Riemann–Hilbert correspondence for unitF-crystals on embeddable algebraic varieties
Tome 68, no3 (2018), p. 1077-1120.
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Article mis en ligne dans le cadre du
RIEMANN–HILBERT CORRESPONDENCE FOR UNIT F -CRYSTALS ON EMBEDDABLE ALGEBRAIC
VARIETIES
by Sachio OHKAWA
Abstract. — For a separated schemeX of finite type over a perfect fieldk of characteristicp >0 which admits an immersion into a proper smooth scheme over the truncated Witt ringWn, we define the bounded derived category of lo- cally finitely generated unit F-crystals with finite Tor-dimension onX overWn, independently of the choice of the immersion. Then we prove the anti-equivalence of this category with the bounded derived category of constructible étale sheaves of Z/pnZ-modules with finite Tor-dimension. We also discuss the relationship of t-structures on these derived categories whenn= 1.
Résumé. — SoitX un schéma séparé de type fini sur un corps parfaitkde caractéristiquep >0 et qui admet une immersion vers un schéma propre et lisse sur l’anneau des vecteurs de Witt tronqué Wn.Nous définissons la catégorie dé- rivée bornée des F-cristaux unités localement finiment engendrés sur Wn avec Tor-dimension finie surX, indépendemment de l’immersion. Nous provons ensuite que cette catégorie est anti-équivalente à la catégorie dérivée bornée des faisceaux constructibles étales de Z/pnZ-modules avec Tor-dimension finie. Nous étudions aussi les relations det-structures sur ces catégories dérivées lorsquen= 1.
1. Introduction
For a complex manifold X, Kashiwara [9] and Mebkhout [12] indepen- dently established an anti-equivalence, which is called the Riemann–Hilbert correspondence, between the triangulated categoryDbrh(DX) ofDX-modules with regular holonomic cohomologies and thatDbc(X,C) of sheaves of C- vector spaces onX with constructible cohomologies. There is a significant property from the point of view of relative cohomology theories that this anti-equivalence respects Grothendieck’s six operationsf!,f!,f∗,f∗,RHom and⊗L defined onDbrh(DX) andDcb(X,C).
Keywords:D-modules, Frobenius structure, étale sheaves.
2010Mathematics Subject Classification:14F30, 14F10.
In [4], Emerton and Kisin studied a positive characteristic analogue of the Riemann–Hilbert correspondence. Let k be a perfect field of charac- teristic p > 0. We denote by Wn := Wn(k) the ring of Witt vectors of length n. For a smooth scheme X over Wn, Emerton and Kisin defined the sheaf DF,X of OX-algebras by adjoining to OX the differential op- erators of all orders onX and a “local lift of Frobenius”. By using DF,X, they introduced the triangulated categoryDblfgu(DF,X) ofDX-modules with Frobenius structures with locally finitely generated unit cohomologies and proved the anti-equivalence
Dblfgu(DF,X)◦ ∼−=→Dctfb (X´et,Z/pnZ)
between the subcategoryDblfgu(DF,X)◦ of Dblfgu(DF,X) consisting of com- plexes of finite Tor dimension over OX and the triangulated category Dbctf(X´et,Z/pnZ) of étale sheaves ofZ/pnZ-modules with constructible co- homologies and of finite Tor dimension over Z/pnZ, which they call the Riemann–Hilbert correspondence for unitF-crystals. They also introduced three of Grothendieck’s six operations, which are the direct imagef+, the inverse imagef! and the tensor product ⊗L on Dlfgub (DF,X), and proved that their Riemann–Hilbert correspondence exchanges these tof!,f−1and
⊗L onDbctf(X´et,Z/pnZ).
Emerton and Kisin established the Riemann–Hilbert correspondence for unitF-crystals only for smooth schemesX overWn. Since the triangulated categoryDbctf(X´et,Z/pnZ) depends only on the modpreduction ofX, it is natural to expect that there exists a definition of the triangulated category Dblfgu(DF,X)◦ and the Riemann–Hilbert correspondence depending only on the mod p reduction of X. Also, there should be the Riemann–Hilbert correspondence for algebraic varieties overkwhich are not smoothly liftable toWn. The purpose of this article is to generalize the Emerton–Kisin theory to the case of Wn-embeddable algebraic varieties over k. Here we say a separated k-scheme X of finite type is Wn-embeddable if there exists a proper smooth Wn-scheme P and an immersion X ,→ P such that the diagram
(1.1)
X
/P
Speck //SpecWn
is commutative. A quasi projective variety overk is a typical example of Wn-embeddable variety and thus Wn-embeddable varieties form a suffi- ciently wide class of algebraic varieties in some sense.
The first problem is to define a reasonable D-module category forWn- embeddable algebraic varieties overk. Our construction is based on Kashi- wara’s theorem which roughly asserts that, for any closed immersionX ,→ Pof smooth algebraic varieties, the category ofD-modules onPsupported onX is naturally equivalent to the category ofD-modules onX. Using the characteristic p > 0 analogue of Kashiwara’s theorem due to Emerton–
Kisin [4, Prop. 15.5.3], we show that, when we are given the diagram (1.1), the full triangulated subcategory ofDlfgub (DF,P)◦ consisting of complexes supported on X does not depend on the choice of immersion X ,→ P (Corollary 4.6). We denote this full subcategory byDblfgu(X/Wn)◦. Then we show the Riemann–Hilbert correspondence
Dblfgu(X/Wn)◦ ∼−=→Dctfb (X´et,Z/pnZ)
for anyWn-embeddablek-schemeX (Theorem 4.12). As in the case of [4], we can naturally introduce three of Grothendieck’s six operations, that is, direct and inverse images and tensor products. We then prove that the Riemann–Hilbert correspondence respects these operations (Theorem 4.14).
A striking consequence of the Riemann–Hilbert correspondence over com- plex numbers is that one can introduce an exotict-structure on the topo- logical side called the perverset-structure, which corresponds to the stan- dard t-structure on the D-module side. For an algebraic variety X over k, Gabber introduced in [5] the perverse t-structure on Dbc(X´et,Z/pZ), which we call Gabber’s perverset-structure. In the case whenX is smooth over k, Emerton and Kisin showed that the standard t-structure on the D-module side corresponds to Gabber’s perverset-structure. In this paper, we generalize it to the case of k-embeddable k-schemes. In the complex situation, conversely, at-structure on theD-module side corresponding to the standardt-structure on the topological side is explicitly described by Kashiwara in [10]. In this paper, we construct the analogue of Kashiwara’s t-structure on Dlfgub (X/k) and discuss the relationship of it and the stan- dardt-structure onDbc(X´et,Z/pZ).
The content of each section is as follows: In the second section, we re- call several notions, terminologies and cohomological operations onDF,P- modules from [4] which we often use in this paper. We also recall the statement of the Riemann–Hilbert correspondence for unit F-crystals of Emerton–Kisin (Theorem 2.3). In the third section, we define the local co- homology functor RΓZ for DF,P-modules and prove compatibilities with RΓZ and other operations for DF,P-modules, which are essential tools to define and study the triangulated category Dblfgu(X/Wn)◦ for any Wn- embeddable k-scheme X. In subsection 4.1, we introduce the category
Dblfgu(X/Wn)◦ for anyWn-embeddablek-schemeX and in subsection 4.2, we construct three of Grothendieck’s six operations onDblfgu(X/Wn)◦. Our arguments in these subsections are heavily inspired by that of Caro in [2].
In subsection 4.3, we prove the Riemann–Hilbert correspondence for unit F-crystals on Wn-embeddablek-schemes, which is our main result. In the fifth section, we discuss several properties onDlfgub (X/k) (in the casen= 1) related to t-structures. In subsection 5.1, we introduce the standard t- structure onDblfgu(X/k) depending on the choice of the immersionX ,→P. We prove that the standardt-structure corresponds to Gabber’s perverse t-structure via the Riemann–Hilbert correspondence. As a consequence, we know that the definition of the standardt-structure is independent of the choice ofX ,→P (Theorem 5.5). In subsection 5.2, we define the abelian category µlfgu,X as the heart of the standard t-structure on Dblfgu(X/k), and prove that the natural functorDb(µlfgu,X)→Dblfgu(X/k) is an equiv- alence of triangulated categories (Theorem 5.6), which can be regarded as an analogue of Beilinson’s theorem. In subsection 5.3, depending on the choice of the immersionX ,→P, we introduce the constructiblet-structure on Dblfgu(X/k) by following the arguments in [10] and prove that it cor- responds to the standard t-structure on the étale side via the Riemann–
Hilbert correspondence. As a consequence, we see that the constructible t-structure does not depend on the choice ofX ,→P (Corollary 5.19).
After writing the first version of this article, the author learned that, in [13] Schedlmeier also studied on closely related subjects independently in terms of the notion of Cartier crystals. In particular, he proved that, for a separatedF-finitek-scheme X which admits a closed immersion into a smooth (but not necessarily proper)k-scheme, there exists an equivalence of triangulated categories between the bounded derived category of Cartier crystals onX andDbc(X´et,Z/pZ).
Acknowledgments. This article is the doctor thesis of the author in the university of Tokyo. The author would like to express his profound gratitude to his supervisor Atsushi Shiho for valuable suggestions, discussions and the careful reading of this paper. He would like to thank Kohei Yahiro for useful discussions and Manuel Blickle for informing me a result of Schedlmeier in [13]. He would also like to thank the referee for reading the paper in detail, pointing out mistakes, and giving valuable comments. This work was supported by the Program for Leading Graduate Schools, MEXT, Japan, the Grant-in-Aid for Scientific Research (KAKENHI No. 26-9259), and the Grant-in-Aid for JSPS fellows.
Conventions.Throughout this paper, we fix a prime number pand a perfect base fieldk of characteristicp. We denote byW the ring of Witt vectors associated tokand byWnthe quotient ringW/(p)nfor any natural numbern. For a schemeX, we denote the structure sheaf ofX byOX. For a smoothWn-schemeX, the dimension ofX is a continuous integer valued function onX defined by
dX :x∈X 7→dimension of the component ofX containingx.
For a morphismf :X →Y of smoothWn-scheme, we denote a function dX−dY◦fbydX/Y and a function−dX/Y bydY /X. For an abelian category C, we denote by D(C) the derived category of C. For a schemeX and an OX-algebraA, we denote byD(A) the derived category of leftA-modules and by Dqc(A) the full triangulated subcategory of D(A) consisting of complexes whose cohomology sheaves are quasi-coherent as OX-modules.
ForA-modulesF and G, we denote by HomA(F,G) the sheaf ofA-linear homomorphism fromF toG. We denote a complex by a single letter such as M and by Mn the n-th term of M. For an object M in D(A), we denote by Hi(M) thei-th cohomology of Mand by SuppMthe support ofM, which is defined as the closure ofS
iSupp Hi(M).
2. Preliminaries
In this section we recall the notion of locally finitely generated unitDF,X- modules introduced in [4] and the Riemann–Hilbert correspondence for unit F-crystals in [4].
2.1. Locally finitely generated unit DF,X-modules
For a smooth Wn-schemeX, we denote byDX the sheaf of differential operators ofX overWn defined in [6, §16]. For a morphism of smoothWn- schemesf :X→Y and a leftDY-moduleM,f∗M:=OX⊗f−1OY f−1M has a natural structure of left DX-modules. When there exists a lifting F : X → X of the absolute Frobenius on X ⊗Wnk, the left DX-module structure onF∗Mis known to be independent of the choice of the liftingF up to canonical isomorphism by [4, Prop. 13.2.1]. Since the liftingF above always exists Zariski locally onX, we obtain a functor
F∗: (leftDX-module)→(leftDX-module)
by glueing for any smoothWn-schemeX. We set DF,X :=M
r>0
(F∗)rDX.
Then DF,X naturally forms a sheaf of associative Wn-algebras such that the natural embeddingDX → DF,X is aWn-algebra homomorphism by [4, Cor. 13.3.5]. It is proved in [4, Prop. 13.3.7] that giving a leftDF,X-module Mis equivalent to giving aDX-moduleMtogether with a morphismψM: F∗M → M of left DX-modules, which we call the structural morphism ofM.
Next let us recall the notion of locally finitely generated unit DF,X- modules. We say that a leftDF,X-moduleMis unit if it is quasi-coherent as an OX-module and the structural morphism ψM : F∗M → M is an isomorphism. We say that aDF,X-moduleMis locally finitely generated unit if it is unit and Zariski locally on X, there exists a coherent OX- submoduleM ⊂ Msuch that the natural morphismDF,X⊗OXM → Mis surjective. Then the locally finitely generated unit leftDF,X-modules form a thick subcategory of the category of quasi-coherent leftDF,X-modules [4, Prop. 15.3.4]. We say that a locally finitely generated unit DF,X-module Mis anF-crystal ifMis locally free of finite rank as anOX-module.
Finally we introduce some notations on triangulated categories. We de- note byD(DF,X) the derived category of the abelian category of leftDF,X- modules and byDqc(DF,X) (resp. Dlfgu(DF,X)) the full triangulated sub- category of D(DF,X) consisting of those complexes whose cohomology sheaves are quasi-coherent asOX-modules (resp. are locally finitely gen- erated unit left DF,X-modules). If • is one of ∅, −, +, b, we denote by D•(DF,X) the full triangulated subcategories of D(DF,X) consisting of those complexes satisfying the appropriate boundedness condition. We use the notationsD•qc(DF,X) andDlfgu• (DF,X) in a similar manner. We denote byDlfgub (DF,X)◦the full triangulated subcategory ofDblfgu(DF,X) consisting of those complexes which are of finite Tor dimension asOX-modules.
2.2. Cohomological operations for leftDF,X-modules
For a morphismf:X→Y of smoothWn-schemes,f∗DF,Y:=OX⊗f−1OY f−1DF,Y has a natural structure of left DF,X-module by [4, Cor. 14.2.2]. It also forms a rightf−1DF,Y-module via the right multiplication onf−1DF,Y. Sof∗DF,Y has a structure of DF,X, f−1DF,Y
-bimodule, which we denote byDF,X→Y. For a DF,Y-module M, we define a left DF,X-module f∗M
byDF,X→Y ⊗f−1DF,Y f−1M. Note thatf∗M ∼=OX⊗f−1OY f−1Mas an OX-module. We then define a functor
Lf∗:D−(DF,Y)→D−(DF,X)
to be the left derived functor off∗. One hasLf∗M ∼=OX⊗Lf−1OY Mas a complex ofOX-modules. We also define a functor
f!:D−(DF,Y)→D−(DF,X)
byf!M:=Lf∗M[dX/Y]. For the definition of the shift functor (−)[dX/Y] by the functiondX/Y, we refer the reader to [4, §0]. The second inverse image functor is appropriate to formulate the compatibility with the Riemann–Hilbert correspondence (see Theorem 2.3 (2) bellow). Let ? be one of qc or lfgu and∗ one of ◦ or ∅. Then the functor f! restricts to a functor
f!:Db?(DF,Y)∗→Db?(DF,X)∗ by [4, Prop. 14.2.6 and Prop. 15.5.1].
Next let us define the direct image functor f+ forDF,X-modules for a morphism f : X → Y of smooth Wn-schemes. First of all, we recall the definition of the direct image functor
f+B:D−(DX)→D−(DY)
forDX-modules. For a smoothWn-schemeY, we denote byωY the canoni- cal bundle ofY overWn. ThenDY⊗OYωY−1has two natural leftDY-module structures. The first one is the tensor product of leftDY-modulesDY and ω−1Y (cf. [1, 1.2.7.(b)]). On the other hand, using the right DY-module structure onDY defined by the multiplication ofDY on the right, one has the second leftDY-module structure onDY ⊗OY ωY−1 by [1, 1.2.7.(b)]. So DY ⊗OY ω−1Y naturally forms a left (DY,DY)-bimodule. For a morphism f : X → Y of smooth Wn-schemes, by pulling DY ⊗OY ωY−1 back with respect to the secondDY-module structure, one has a left (f−1DY,DX)- modulefd∗ DY ⊗OY ωY−1
. Here, in order to avoid confusion we use the notationfd∗ instead of f∗. By tensoringωX on the right, one obtains an (f−1DY,DX)-bimodule
DY←X :=fd∗ DY ⊗OY ω−1Y
⊗OX ωX.
On the other hand, one has a left (DX, f−1DY)-modulefg∗ DY ⊗OY ωY−1 by pulling backDY⊗OYωY−1with respect to the firstDY-module structure.
By tensoringωX on the left, we obtain an (f−1DY,DX)-bimodule DY←X0 :=ωX⊗OXfg∗ DY ⊗OY ωY−1
.
Then there exists the natural isomorphism of (f−1DY,DX)-bimodules DY←X
∼=
−→ DY←X0.
For more details see [1, 3.4.1]. We define a functor f+B : D−(DX) → D−(DY) by
f+BM:=Rf∗ DY←X⊗LDX M .
Let us go back to the situation ofDF,X-modules. We defineDF,Y←X by DF,Y←X:=ωX⊗OX f∗ DF,Y ⊗OY ωY−1
.
ThenDF,Y←X has a natural (f−1DF,Y,DF,X)-bimodule structure (see [4,
§14.3]). We remark thatDF,Y←X has finite Tor dimension as a rightDF,X- module by [4, Prop. 14.3.5] and, sinceY is a noetherian topological space, Rf∗ has finite cohomological amplitude. We define a functor
f+:D−(DF,X)→D−(DF,Y) by
f+M:=Rf∗
DF,Y←X⊗LDF,X M .
Let?be one of qc or lfgu and ∗one of◦ or∅. The functorf+ restricts to a functor
f+:Db?(DF,X)∗ →Db?(DF,Y)∗ by [4, Prop. 14.3.9 and Prop. 15.5.1].
Remark 2.1. — Letf :X→Y be a morphism of smoothWn-schemes.
The natural inclusion of (DY,DY)-bimodules DY → DF,Y induces an in- clusion of (f−1DY,DX)-bimodulesι:DY←X → DF,Y←X. We then obtain morphisms in the derived category of (f−1DY,DF,X)-bimodules
DY←X⊗LDX DF,X → DY←X⊗DX DF,X
D1⊗D27→ι(D1)D2
−−−−−−−−−−−−→ DF,Y←X. For an objectMinD−(DF,X), applying the functor Rf∗(− ⊗LD
F,X M) to the composite of the above morphisms, we obtain aDY-linear morphism
f+BM →f+M.
It is proved in [4, §14.3.10] that the morphismf+BM →f+Mis an isomor- phism.
Let X be a smooth Wn-scheme. Let Mand N be DF,X-modules with structural morphismsψMandψN. ThenM⊗OXN has a natural structure of leftDX-modules. We define the structural morphism onM ⊗OX N to be the composite ofDX-linear morphisms
F∗(M ⊗OX N)∼=F∗M ⊗OX F∗N −−−−−−→ M ⊗ψM⊗ψN OX N.
Here the first isomorphism follows from [1, 2.3.1] and its proof. We thus obtain theDF,X-module structure onM ⊗OX N and define a bi-functor
D−(DF,X)×D−(DF,X)→D−(DF,X) by (M,N)7→ M ⊗LOXN. This functor restricts to a bi-functor
Dblfgu(DF,X)×Dblfgu(DF,X)◦→Dblfgu(DF,X) by [4, Prop. 15.5.1].
Proposition 2.2. — Let f : X → Y be a morphism of smooth Wn- schemes. If M and N are objects in D−(DF,Y), then there are natural isomorphisms
Lf∗M ⊗LOX Lf∗N −∼=→Lf∗ M ⊗LOY N and
f!M ⊗LOX f!N[dY /X]−∼=→f! M ⊗LOY N .
Proof. — The second isomorphism follows from the first one. LetP →M (resp. Q → N) be a resolution of M (resp. N) by flat DF,Y-modules.
Note that P and Q are complexes of flat OY-modules. So P ⊗OY Q → M ⊗LO
Y N gives a resolution of M ⊗LO
Y N by flat DF,Y-modules (cf. [7, Lem. 4.1]) and f∗P is a complex of flat OX-modules. By the universal mapping property of the tensor product, one has a natural DF,X-linear morphismf∗P ⊗OXf∗Q →f∗(P ⊗OY Q). Evidently it is an isomorphism as a morphism inD(OX) and hence it is the desired isomorphism.
2.3. Riemann–Hilbert correspondence for unit F-crystals
Let X be a smooth Wn-scheme. We denote by Db(X´et,Z/pnZ) the bounded derived category of complexes of Z/pnZ-modules on the étale siteX´et. We letDbctf(X´et,Z/pnZ) denote the full triangulated subcategory ofDctfb (X´et,Z/pnZ) consisting of complexes whose cohomology sheaves are constructible and which have finite Tor dimension overZ/pnZ.
For a morphism f :X →Y of smoothWn-schemes, the inverse image f−1:Dbctf(Y´et,Z/pnZ)→Dctfb (X´et,Z/pnZ)
and the direct image with proper support
f! :Dctfb (X´et,Z/pnZ)→Dctfb (Y´et,Z/pnZ)
are defined. For a review of constructions of these functors, we refer the reader to [4, §8].
LetX be a smoothWn-scheme. We denote byπX:X´et→Xthe natural morphism of sites, where X means the Zariski site ofX. ThenDF,X´et :=
πX∗DF,X naturally forms a sheaf of associative Wn-algebras on X´et. By étale descent, we have an equivalence of triangulated categories (cf. [4, §7 and 16.1.1])
πX∗ :Dbqc(DF,X)→Dbqc(DF,X´et) with quasi-inverseπX∗. ForM ∈Dlfgub (DF,X)◦, we set
SolX(M) =RHomDF,X
´
et(πX∗(M),OX´et)[dX]. Then this correspondence defines a contravariant functor
SolX :Dblfgu(DF,X)◦→Dctfb (X´et,Z/pnZ)
by [4, Prop. 16.1.7]. Conversely, forL ∈Dbctf(X´et,Z/pnZ), we set MX(L) =πX∗RHomZ/pnZ(L,OX´et)[dX].
Then this correspondence defines a contravariant functor MX:Dbctf(X´et,Z/pnZ)→D+(DF,X).
Now we may state one of the main results in [4].
Theorem 2.3. — For a smooth Wn-scheme X, the functor SolX is an anti-equivalence of triangulated categories between Dblfgu(DF,X)◦ and Dbctf(X´et,Z/pnZ)with quasi-inverseMX. FurthermoreSolX and MX sat- isfy the following properties:
(1) Iff :X →Y is a morphism of smoothWn-schemes, thenSolX and MX interchangef! andf−1.
(2) Let f be a morphism of smoothWn-schemes such that f can be factored asf =g◦h, wheregis an immersion of smoothWn-schemes andhis a proper smooth morphism of smoothWn-schemes. Then SolX andMX interchangef+ andf!.
(3) SolXandMX interchange the functors⊗LO
X and⊗L
Z/pnZup to shift.
More precisely, for objectsMandN inDblfgu(DF,X)◦, there exists a canonical isomorphism
SolX(M)⊗LZ/pnZSolX(N)−∼=→SolX M ⊗LOX N [dX].
Proof. — See [4, Prop. 16.1.10 and Cor. 16.2.6].
2.4. Remark in the case n= 1
LetX be a smoothk-scheme and assume that n= 1 in this subsection.
LetOF,X denote a sheaf of the non-commutative polynomial ringOX[F] in a formal variableF, which satisfies the relationF a=apFfora∈ OX. One can naturally regardOF,X as a subring of DF,X. Giving an OF,X-module Mis equivalent to giving an OX-module Mwith a structural morphism F∗M → M, where F denotes the absolute Frobenius on X. We say an OF,X-module M is unit if it is quasi-coherent as anOX-module and the structural morphism F∗M → M is an isomorphism. We say an OF,X- moduleMis locally finitely generated unit if it is unit and locally finitely generated as anOF,X-module. Similar to the case ofDF,X-modules, the lo- cally finitely generated unitOF,X-modules form a thick subcategory of the category of quasi-coherentOF,X-modules. So one can consider the bounded derived categoryDblfgu(OF,X) of complexes ofOF,X-modules whose coho- mology sheaves are locally finitely generated unit. Similar to the case of DF,X-modules, one can define the inverse and direct image functors for a morphism of smoothk-schemes (see [4, §2 and §3]) and the derived tensor product onDblfgu(OF,X). Then Emerton and Kisin proved that the natural functor
Dlfgub (DF,X)→Dlfgub (OF,X)
induces an equivalence of categories with quasi inverse DF,X ⊗LO
F,X (−), which is compatible with the functorsf+,f! and ⊗LOX, wheref is a mor- phism of smoothk-schemes [4, Prop. 15.4.3].
Remark 2.4. — In [4], Emerton and Kisin firstly established the theory of OF,X-modules for smooth k-schemes. They proved many properties of DF,X-modules for smoothWn-schemes including Theorem 2.3 by reducing them to the corresponding properties ofOF,X⊗Wnk-modules.
3. Local cohomology functor
LetP be a smoothWn-scheme. LetZ be a closed subset ofP andjZ the canonical open immersionP\Z ,→P. For a sheaf F of abelian groups on P, we set ΓZF := Ker(F →jZ∗jZ−1F). IfM is a leftDF,P-module, then ΓZMnaturally forms a leftDF,P-module. We have a left exact functor ΓZ from the category of leftDF,P-modules to itself. Then the local cohomology functor
RΓZ :D+(DF,P)→D+(DF,P)
is defined to be the right derived functor of ΓZ. By definition, we have a distinguished triangle
(3.1) RΓZM → M →RjZ∗jZ−1M−+→
for M ∈ D+(DF,P). Note that RjZ∗ = jZ+ and jZ−1 = jZ! . We can also define the local cohomology functor
RΓZ :D+(OP)→D+(OP)
on the level of OP-modules. Then the forgetful functor D+(DF,P) → D+(OP) commutes withRΓZ. It is proved by Grothendieck thatRΓZ has finite cohomological amplitude.
Lemma 3.1. — Let P be a smooth Wn-scheme and Z a closed subset ofP. Denote by jZ the open immersionP \Z ,→P. Then the following conditions are equivalent forM ∈D(OP).
(1) RΓZM−∼=→ M, (2) RjZ∗jZ−1M= 0,
(3) SuppMis contained inZ.
Proof. — The equivalence of (1) and (2) follows from (3.1). Assuming that SuppM ⊂ Z, one has jZ−1M = 0. This shows (3) ⇒ (2). Finally
(1)⇒(3) is evident.
Lemma 3.2. — Let P be a smooth Wn-scheme and Z a closed subset ofP. There exists a natural OP-linear isomorphism
(3.2) RΓZ(M)⊗LOP N −∼=→RΓZ M ⊗LOPN
for any M ∈ D−(OP) and N ∈ Dqc−(OP). Furthermore for any M ∈ D−(DP)(resp.M ∈D−(DF,P)) andN ∈Dqc−(DP)(resp.N ∈D−qc(DF,P)) (3.2)is a DP-linear (resp.DF,P-linear) isomorphism.
Proof. — Note first that both sides are well-defined. Let us construct a natural morphism in the Lemma. LetMbe an object ofD−(OP) andN an object ofD−qc(OP). One hasRΓZ(M)⊗LOPN → M ⊗LOPN. Then since RΓZ(M)⊗LO
P N is supported onZ, we haveRΓZ RΓZ(M)⊗LO
PN ∼=
−→ RΓZ(M)⊗LO
PN by Lemma 3.1. SoRΓZ(M)⊗LO
PN → M⊗LO
PN uniquely factors as
RΓZ(M)⊗LOP N →RΓZ M ⊗LOP N
→ M ⊗LOP N
and we get the desired morphism. Note that ifMis an object of D−(DP) (resp.D−(DF,P)) and N is an object ofDqc−(DP) (resp.D−qc(DF,P)) then
(3.2) is DP-linear (resp. DF,P-linear). Let us prove that (3.2) is an iso- morphism. It suffices to show that this is an isomorphism inD(OP). The assertion is Zariski local onP and so we may assume thatP is affine. Note that the source is a way-out left functor inN. Also, sinceRΓZ is finite co- homological amplitude, the target is also a way-out left functor inN. Using the lemma on way-out functors (cf. [7, Chap. I, Prop. 7.1]), we reduce to the case whereN is a single quasi-coherentOP-module. Furthermore since any quasi-coherentOP-module is a quotient of a freeOP-module (because P is affine), we may assume thatN is a single freeOP-module. Now since RΓZ commutes with infinite direct sums, we reduce the assertion to prove the case whenN =OP. Then both sides are equal to RΓZMand we are
done.
Proposition 3.3. — Let P be a smooth Wn-scheme and Z a closed subset ofP. The local cohomology functor induces a functor
RΓZ :Dlfgub (DF,P)◦→Dblfgu(DF,P)◦.
Proof. — Let us first show that, for anyM ∈Dlfgub (DF,P),RΓZMhas locally finitely generated cohomology sheaves. LetjZ denote the open im- mersionP\Z ,→P. Then there exists a distinguished triangle
RΓZM → M →RjZ∗jZ−1M−+→.
Since Mand RjZ∗jZ−1M are objects of Dblfgu(DF,P) by [4, Prop. 15.5.1], RΓZMis also an object ofDblfgu(DF,P) by [4, Prop. 15.3.4]. Next we show that, for anyM ∈Dlfgub (DF,P)◦,RΓZMis of finite Tor dimension overOP. According to [8, I, Prop. 5.1], it is enough to show thatRΓZ(M)⊗LO
PN is a bounded complex for anyOP-moduleN. First of all, suppose thatN is a quasi-coherentOP-module. Then, by Lemma 3.2, we haveRΓZ(M)⊗LO
P
N −∼=→ RΓZ(M ⊗LO
P N). Since M is of finite Tor dimension, M ⊗LO
P N is a bounded complex ofOP-modules. So we know thatRΓZ(M)⊗LOP N is bounded since RΓZ is finite cohomological amplitude. Now if N is an arbitrary OP-module, then the stalks of RΓZ(M)⊗LOP N are uniformly bounded, hence so isRΓZ(M)⊗LO
P N because P is quasi-compact.
Lemma 3.4. — Letf :P →Qbe a morphism of smoothWn-schemes andZQ a closed subset of Q. We denote byZP the inverse image of ZQ. There exists a natural isomorphism inD−(DF,P)
Lf∗◦RΓZQ(OQ)−∼=→RΓZP(OP).
Proof. — One has natural morphisms
(3.3) Lf∗RΓZQ(OQ)→Lf∗OQ → OP.
Let us denote byjZQ (resp.jZP) the open immersionQ\ZQ ,→Q(resp.
P\ZP ,→P) and byf0 the restriction off toP\ZP. Then one has jZ−1
PLf∗RΓZQ(OQ)∼=Lf0∗jZ−1
QRΓZQ(OQ)∼= 0.
Hence, we know that Lf∗RΓZQ(OQ) is supported on ZP and the mor- phism (3.3) uniquely factors as
Lf∗RΓZQOQ
−→a RΓZP(OP)→ OP.
It suffices to prove thatais an isomorphism inD(OP). One has a morphism of distinguished triangles
Lf∗RΓZQOQ //
a
Lf∗OQ b
//Lf∗RjZQ∗j−1Z
QOQ c
+ //
RΓZP(OP) //OP //RjZP∗jZ−1
POP
+ //. Herecis defined to be the composite of morphisms
Lf∗RjZQ∗jZ−1
QOQ→RjZP∗jZ−1
PLf∗RjZQ∗jZ−1
QOQ
∼=RjZP∗Lf0∗jZ−1
QRjZQ∗jZ−1
QOQ
∼=
−→RjZP∗Lf0∗jZ−1
QOQ
∼=RjZP∗jZ−1
PLf∗OQ
→RjZP∗jZ−1
POP,
where the first morphism and the third one are induced from the adjunction morphisms id → RjZP∗jZ−1
P and jZ−1
QRjZQ∗
∼=
−→ id respectively. Then b is evidently an isomorphism andc is an isomorphism by [14, Lem. 35.18.3].
Soais an isomorphism.
Proposition 3.5. — Let f : P → Q be a morphism of smooth Wn- schemes andZQ a closed subset ofQ. We denote byZP the inverse image ofZQ. Then, for anyM ∈D−qc(DF,Q), there exists a natural isomorphism
Lf∗◦RΓZQM−∼=→RΓZP◦Lf∗M and also a natural isomorphism
f!◦RΓZQM−∼=→RΓZP◦f!M.
Proof. — The second isomorphism follows by applying the shift operator to the first one. By using Proposition 2.2, Lemma 3.2 and Lemma 3.4, we
obtain
Lf∗RΓZQ(M)←∼=−Lf∗
RΓZQ(OQ)⊗LOQM
∼=
−→Lf∗RΓZQ(OQ)⊗LO
P Lf∗M
∼=
−→RΓZP(OP)⊗LOP Lf∗M=RΓZP(Lf∗M). Next we show the compatibility of the local cohomology functor and the direct image. We begin with the corresponding result for usualDP-modules (without Frobenius structures).
Proposition 3.6. — Let f : P → Q be a morphism of smooth Wn- schemes andZQ a closed subset ofQ. We denote byZP the inverse image ofZQ. LetM be an object in Dqcb (DP). Then there exists a natural iso- morphism of functors
RΓZQ◦f+B(M)→f+B◦RΓZP(M).
We need some lemmas.
Proposition 3.7. — Let f : P → Q be a morphism of smooth Wn- schemes. IfMis an object inDqc−(DQ)andN is an object inD−(f−1DQ), then there exists a natural isomorphism
M ⊗LO
PRf∗N −∼=→Rf∗
f−1M ⊗Lf−1OQN . inD−(DQ).
Proof. — Note first that both sides are defined. Let us take anf∗-acyclic resolutionIofN and aDQ-flat resolutionPofM. Then we have a natural DQ-linear morphism
M ⊗LOP Rf∗N :=P ⊗OP f∗I →f∗ f−1P ⊗f−1OQI
→Rf∗ f−1P ⊗f−1OY I
=Rf∗
f−1M ⊗Lf−1OY N . It is enough to prove that this is an isomorphism in D(OQ). Then this
follows from [7, II, Prop. 5.6].
Lemma 3.8. — Letf :P →Qbe a morphism of smoothWn-schemes.
For an objectE in D−(DQ) and an objectF in D−(DP), there exists an isomorphism
f−1E ⊗Lf−1OQDQ←P
⊗LDP F−∼=→ DQ←P ⊗LDP Lf∗E ⊗LOP F inDb(f−1DQ).
