Math. Z. (2011) 267:433–452 DOI 10.1007/s00209-009-0628-2
Mathematische Zeitschrift
Differential complexes and stratified pro-modules Luisa Fiorot
Received: 4 October 2007 / Accepted: 19 October 2009 / Published online: 11 December 2009 © Springer-Verlag 2009
Abstract In this paper we introduce the category of stratified pro-modules and the notion of induced object in this category. We propose a translation of Saito equivalence results (Bull Soc Math France 117:361–387, 1989) using the dual language of pro-objects. So we prove an equivalence between the derived category of stratified pro-modules and the category of prodifferential complexes. We also supply a comparison with the notion of crystal in pro-module (introduced by P. Deligne in 1970). Keywords
Differential complexes · Stratified module
Mathematics Subject Classification (2000)
14F40 · 14F10 · 14F05
1 Introduction Let X be a smooth separated Noetherian scheme of finite type over C. We first note that a differential operator of finite order m ∈ N (between O X -modules) can be defined in two different ways: the first using induced right D X,m -modules, as done by Saito in [17], the second using the sheaf of principal parts P Xm . Thus, any differential complex L • admits two −1 (L • ) := L • ⊗OX DX in “linearized” versions. The first is given by Saito’s functor DRX the category of right D X -modules while the other is given by Grothendieck’s formalization functor Q0X := {P Xm }Z ⊗O X L • in the category of stratified pro-modules. In [17] Saito proved the equivalence between the derived category of right D X -modules (quasi-coherent as O X -ones) and a suitable localized category of differential complexes. Our aim is to prove a “dual” version of this equivalence replacing quasi-coherent right D X -modules by pro-coherent stratified ones. The main idea is that of using the Grothendieck
L. Fiorot (B) Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, Via Trieste, 63, 35121 Padua, Italy e-mail:
[email protected]
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−1 formalization functor Q0X instead of Saito’s DRX . On the other hand, a functor DRX is always defined on stratified objects simply by taking horizontal sections. Suitably localizing these functors gives an equivalence between the derived category of Stratified pro-coherent modules and that of pro-differential complexes (suitably localized). We also define a category of •X -modules in pro-object (suitable localized) and we prove an equivalence with that of stratified pro-modules (as done in the dual case in [8]). In Sect. 7 we interpret stratified pro-coherent modules as objects in the crystalline site. In particular we prove that the category of stratified pro-coherent modules is equivalent to the category of “crystals in pro-modules”. This last notion was first introduced by P. Deligne in a cycle of lectures he gave at IHES. There Deligne proposed the notion of “pro-coherent crystals” attached to algebraic constructible sheaves on an analytic space X an . Moreover he proved that the category of “regular pro-coherent crystals” is equivalent to that of “algebraic” constructible sheaves on X an (unfortunately this work was not published). By Deligne’s equivalence theorem we obtain the equivalence between the derived category of regular stratified pro-coherent modules and that of “algebraic” constructible sheaves, and thus a sort of Riemann–Hilbert correspondence. As noted above the notion of stratified pro-coherent module is dual to that of quasicoherent right D -module. In a work in progress we expect to prove an anti-equivalence of categories between the category of perfect D -complexes and that of perfect complexes of stratified pro-modules. This anti-equivalence is compatible with the duality in the category of differential complexes (see [4,11,13,17]). In particular when any object of a differential complex is coherent on O X , the notion of D X -qis (see [16,17]) is equivalent to that of Q 0X -qis.
2 Pro-coherent O X -modules We briefly recall some results on the category of pro-coherent O X -modules. Definition 1 ([3], 1) By definition the category of pro-coherent O X -modules is the category Pro(Coh(OX )) := Ind(Coh(OX )◦ )◦ . Objects are filtering projective systems of coherent O X -modules, while morphisms between two such objects F I , G J are elements of HomPro (FI , GJ ) = lim lim HomOX (Fi , Gj ). For brevity we will use the notation ν(O X ) ←J →I for Pro(Coh(OX )) and μ(O X ) for Ind(Coh(OX )). Remark 1 Given a noetherian scheme X over C, the category μ(O X ) is equivalent to the category of quasi-coherent O X -modules (denoted by QCoh(OX )) (Deligne, appendix of [12]). Moreover any object in μ(O X ) may be represented as an inductive system whose transition morphisms are injective maps. Remark 2 The functor tensor product: _ ⊗O X _ : Coh(OX ) × Coh(OX ) −→ Coh(OX ) (F , G ) −→ F ⊗O X G extends to the procategory: _ ⊗O X _ : Pro(Coh(OX )) × Pro(Coh(OX )) −→ Pro(Coh(OX )) ({Fi } I , {G j } J ) −→ {Fi ⊗O X G j } I ×J . Remark 3 Let C be an abelian category, then Pro(C ) is abelian (see appendix of [2]). This result mainly concerns the description of Ker( f ) and Coker( f ), where f is a morphism in
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Pro(C ), done in [2]. If C is an Abelian category with enough injectives and whose arbitrary products are representable, then Pro(C ) has enough injectives (see [2,14]). Let us consider X a Noetherian scheme. The category of quasi-coherent O X -modules has enough injectives; moreover it is equivalent to the category of Ind-coherent O X -modules. The category of coherent O X -modules is small, its finite inductive limits are representable so ([3, I 8.9], SGA4) its ind-category has small projectives limits representable. Then the category of proquasicoherent O X -modules has enough injectives. In particular for any {Fi } I there exists a monomorphism {Fi } I −→ {Ei } I (in the pro-category) where Ei are injective quasi-coherent O X -modules for any i ∈ I and for any i < j the morphism of pro-system E j −→ Ei is a split epimorphism (so the pro-object {Ei } I is injective). In this case the restriction of {Ei } I to any open U ⊂ X is a pro-system of the same type. So ν(O X ) is an Abelian category and the pro-category of quasi-coherent O X -modules Pro(QCoh(OX )) is an Abelian category and it has enough injectives. Moreover ν(O X ) is a full thick subcategory of Pro(QCoh(OX )). In fact Coh(OX ) is a full thick subcategory of QCoh(OX ), and it is easy to prove that the same is true for their pro-categories. + Remark 4 Let us denote by Dν( O X ) (Pro(QCoh(OX ))) the derived category of the category Pro(QCoh(OX )) with bounded below cohomology in ν(O X ). + + Then Dν( O X ) (Pro(QCoh(OX ))) is equivalent to the derived category D (ν(O X )).
3 Stratified pro-modules In this paper we consider X a smooth algebraic variety over C. We will denote by {P Xm }Z the projective system of sheaves of principal parts ([9, 16.7]), by qm : P Xm −→ O X the map induced by the diagonal embedding X −→ X × X and by qm,n : P Xm −→ P Xn (m ≥ n) the maps of the projective system {P Xm }Z . By definition D X,m = H om O X (P Xm , O X ) and D X = lim D is the sheaf of differential operators. We denote by •X the De Rham → m∈N X,m i complex of algebraic differential forms and by −i X := H om O X ( X , O X ) its dual. Mored over let d := d X be the dimension of X ; we denote by ω X := X the sheaf of differential forms of maximum degree. Definition 2 ([6, 2.10]) Let X be a smooth algebraic variety over C and let F be an O X -module. A stratification on F is a collection (one for any n ∈ N) of P Xn -linear isomorphisms εF ,n : P Xn ⊗O X F −→ F ⊗O X P Xn such that εF ,n and εF ,m are compatible via qn,m for each m ≤ n, the map εF ,0 is the identity, and the cocycle condition holds. Proposition 1 ([6, 2.11]) Let F be an O X -module, the following are equivalent: (i) there is a collection of maps sF ,n : F −→ F ⊗O X P Xn “right” O X -linear such that sF ,0 = idF , (idF ⊗O X qm,n ) ◦ sF ,m = sF ,n and (sF ,n ⊗O X idP mX ) ◦ sF ,m = (idF ⊗O X δ n,m ) ◦ sF ,m+n (see [9, 16.8.9.1] for the definition of δ m,n ); (i ) there is a collection of maps
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m “left” O X -linear such that sF ,0 = idF , (qm,n ⊗O X idF )◦sF ,m = sF ,n and (idP X ⊗O X n,m ⊗ sF ) ◦ s = (δ id ) ◦ s ; OX F ,n F ,m F ,m+n (ii) F is a stratified module; (iii) F is a left D X -module, where D X is the sheaf of rings of differential operators.
Proof (i) ⇔ (ii) See Berthelot Ogus [6, 2.11]. (i) + (ii) ⇒ (i ) and (i ) + (ii) ⇒ (i). mF (iii) ⇔ (i) Let D X ⊗O X F −→ F be the multiplication on the D X -module F then m F ∈ HomO X (D X ⊗O X F , F ) = lim HomO X (D X,m ⊗O X F , F ) ← m∈Z
= lim HomO X (F , F ⊗O X P Xm ). ← m∈Z
The associative diagram induces the diagram for co-associativity, and the identity diagram induces that of the co-identity. Stratified O X -modules form a category which we denote by O X -Strat. Morphisms are O X -linear maps which respect the stratifications. We are now interested only in coherent objects so Coh(OX )-Strat will denote the full subcategory of O X -Strat whose objects are coherent. We want to extend this category to pro-objects, in order to obtain a category dual to that of quasi-coherent (so Ind(Coh(OX ))) right D X -modules. The naive way would be that of taking simply the pro-category Pro(O X -Strat), but in this way we obtain pro-objects which have a stratification at any “level” while we need a larger category, that of stratified pro-objects defined as follows. Definition 3 Let ν(P X· ) be the category whose objects are pro-coherent O X -modules {Fh } H endowed with a stratification that is a morphism of pro-objects s{Fh } H
{Fh } H −−−−→ {Fh } H ⊗O X {P Xm }Z which make the co-identity diagram {Fh } H
s{Fh } H
/ {Fh } H ⊗O X {P Xm }Z PPP PPP PPP id{Fh } H ⊗{qm }Z id{Fh } H PPP ' {Fh } H
and the co-associative one {Fh } H
s{Fh } H
/ {Fh } H ⊗ {P Xm }Z
s{Fh } H
{Fh } H ⊗ {P Xm }Z
id{Fh } H ⊗s{P
s{Fh } H ⊗id{P m }
/ {Fh } H ⊗ {P Xm }Z m X }Z
X Z
⊗ {P Xm }Z
commutative. (This is simply the category of {P Xm }Z -co-modules in the category ν(O X )).
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By definition s{P mX }Z := {δ m,n }Z×Z is the map inducing the stratification on the “right” {P Xm }Z = p1∗ (O X ) (see [6, Remark 2.13], and [10]). A morphism of pro-objects f : {Fh } H −→ {Gk } K is a morphism in ν(P X· ) if and only if the diagram f
{Fh } H s{Fh } H
{P Xm }Z ⊗ {Fh } H
/ {Gk } K s{Gk } K
id⊗ f
/ {P Xm }Z ⊗ {Gk } K
commutes. We denote by H om P ·X ({Fh } H , {Gk } K ) (or H om Strat ({Fh } H , {Gk } K ) the sheafified version of the set of morphisms in ν(P X· ). Remark 5 ([6, 2.2, 2.3]) In our setting X is smooth, hence the sheaves {P Xm }Z are locally free of finite type. A base of P Xm (for both left and right O X -module structures) is {ξ1α1 · · · ξdαd |α1 + · · · + αd ≤ n; αi ∈ N} where ξi := 1 ⊗ xi − xi ⊗ 1 and I = 1 ⊗ 1 in local coordinates. By this description the “right” stratification on the pro-system {P Xm }Z is given by the morphisms δ m, p : P Xm −→ I −→ ξi −→ ξ1α1 · · · ξdαd −→
m− p
p
PX ⊗O X P X I⊗I I ⊗ ξi + ξi ⊗ I
δ m, p (ξ1 )α1 · · · δ m, p (ξd )αd
∀α1 . . . αd ∈ N, α1 + · · · + αd ≤ m
Moreover the sheaf 1X is simply the sub-sheaf of P X1 generated by ξi for i = 1, . . . , d. Definition 4 A stratified pro-module is induced if it is isomorphic to {P Xm }Z ⊗O X L , for some L ∈ ν(O X ), endowed with the stratification induced by the canonical one on {P Xm }Z (see [10, 6.3]). We denote by νi (P X· ) the full subcategory of ν(P X· ) whose objects are induced. Proposition 2 The category ν(P X· ) is an Abelian category, small filtering projective limits are representable and exact. The forgetful functor f or : ν(P X· ) −→ ν(O X ) has a right adjoint Q0X := {P Xm }Z ⊗O X _ : ν(O X ) −→ ν(P X· ) {Fh } H −→ {P Xm }Z ⊗O X {Fh } H which takes image into νi (P X· ). Proof Kernels and cokernels in ν(P X· ) are those of ν(O X ) endowed with the induced stratification and for any morphism f in ν(P X· ), the image of f is isomorphic to its co-image. So ν(P X· ) is an Abelian category and the forgetful functor is exact. Small filtering limits are representable and exact because they are representable in ν(O X ) and they have canonical stratifications. The map α
H om Strat ({Fh } H , {P Xm }Z ⊗O X {Gk } K ) → H om ν(O X ) ( f or ({Fh } H ), {Gk } K )
f → ({qm }Z ⊗O X id{Gk } K ) ◦ f
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(co-extension of scalars) is a bijection whose inverse is the map β
H om ν(O X ) ( f or ({Fh } H ), {Gk } K ) → H om {P mX }Z ({Fh } H , {P Xm }Z ⊗ {Gk } K ) OX
g → (id{P mX }Z ⊗ g) ◦ {sF ,m }Z .
Clearly α( f ) is a morphism in ν(O X ); on the other hand in order to prove that β(g) respects the stratifications it is sufficient to remark that Q0X is a functor so the map (id{P mX }Z ⊗ g) respects the canonical stratifications on {P Xm }Z ⊗O X _. Definition 5 Let ∗ ∈ {+, −, b} and let C ∗ (P X· ) (resp. K ∗ (P X· ), resp. D ∗ (P X· )) be the category of complexes (bounded below, bounded above, bounded) (resp. up to homotopy, resp. up to quasi-isomorphisms) in ν(P X· ). We denote by Cib (P X· ) (resp. K ib (P X· ), resp. Dib (P X· )) the category of bounded complexes (resp. up to homotopy, resp. up to quasi-isomorphisms) in νi (P X· ). We note that K ib (P X· ) is a sub-triangulated category of K ∗ (P X· ) and Dib (P X· ) is obtained by K ib (P X· ) by localizing with respect to a multiplicative systems obtained by a cohomological functor. So we have a calcule of fractions in Dib (P X· ) as for usual derived categories. Remark 6 Proposition 2 also holds true on replacing ν(O X ) by the category of proquasi-coherent O X -modules Pro(μ(O X )), and ν(P X· ) by the category of stratified proquasi-coherent O X -modules denoted Pro(μ(P X· )). Corollary 1 Any object in the category Pro(μ(P X· )) induced by an injective object in Pro(μ(O X )) is injective. Moreover Pro(μ(P X· )) has enough injectives. Proof Let E be an injective pro-quasi-coherent O X -module, then the functor H om Strat (_, {P Xm } ⊗O X E ) ∼ = H om Pro(μ(O X )) ( f or (_), E )
is exact because f or (_) : Pro(μ(P X· )) → Pro(μ(O X )) is exact and E is injective; so any object in Pro(μ(P X· )) induced by an injective object in Pro(μ(O X )) is injective. For each N ∈ Pro(μ(P X· )), there exists I ∈ Pro(μ(O X )) and an injective map i : f or (N ) → I in Pro(μ(O X )). Then the map β(i) : N −→ {P Xm }Z ⊗O X I is an injective map in Pro(μ(P X· )) (and {P Xm }Z ⊗O X I is injective in Pro(μ(P X· )). Corollary 2 Derived co-extension of scalars. Let F ∈ ν(P X· ) and G ∈ ν(O X ): ∼ =
RH om Strat (F , {P Xm }Z ⊗O X G ) −→ RH om ν(O X ) ( f or (F ), G ); is a quasi-isomorphism. Proof Let denote by E • (G ) an injective resolution of G in Pro(μ(O X )). Then RH om Strat (F , {P Xm }Z ⊗O X G ) = H om Strat (F , {P Xm }Z ⊗O X E • (G )) ∼ = H om Pro(μ(O X )) ( f or (F ), E • (G )) = RH om ν(O X ) ( f or (F ), G ).
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Let consider the De Rham functor RH om ·Strat (O X , _) : D b (ν(P X· )) −→ D b (C X ) where C X denotes the category of sheaves in C-vector spaces. Then for any M · ∈ D b (ν(P X· )) deriving it in Pro(μ(P X· )) we have: RH om ·Strat (O X , M · ) ∼ = RH om ·Strat (D X ⊗ ·X , M · ) ∼ = H om · (D X ⊗ · , M · ) Strat
∼ = ·X ⊗· M ·
X
The complex ·X ⊗· M · is a complex of pro-coherent-O X -modules but its differentials are not O X -linear. In the following we will define the category ν(O X )-Diff X wherein the functor H om ·Strat (O X , _) has its image in a fully faithful way. So the De Rham functor will have its image in a suitable localization of ν(O X )-Diff X . Theorem 1 Induced stratified pro-modules are acyclic for the horizontal section functor H om Strat (O X , _). For M , N such modules M ∇ := RH om Strat (O X , M ) = H om Strat (O X , M )
and the morphism H om Strat (M , N ) −→ H om C X (M ∇ , N ∇ )
(1)
is injective. Proof By hypothesis there exist L , L in ν(O X ) such that M ∼ = {P Xm }Z ⊗O X L and m ∼ N = {P X }Z ⊗O X L . Then RH om Strat (O X , M ) = ∼ = ∼ = ∼ = =
RH om Strat (O X , {P Xm }Z ⊗O X L ) RH om ν(O X ) (O X , L ) H om ν(O X ) (O X , L ) H om Strat (O X , {P Xm }Z ⊗O X L ) H om Strat (O X , M )
which proves the first assertion. For the second statement let consider the map H om Strat (M , N ) = H om Strat ({P Xm }Z ⊗ L , {P Xm }Z ⊗ L ) OX
OX
∼ = H om ν(O X ) ({P Xm }Z ⊗ L , L ) −→ H om C X (L , L ) OX
(2)
: L → {P m } ⊗ obtained by composition with the stratification morphism sL X Z O X L . It is m injective because the image of sL generates {P X }Z ⊗O X L as pro-coherent-O X -module. We note that this theorem is the analogue (for pro-objects) of Saito’s Lemma [17, 1.2].
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4 Differential complexes of pro-modules Definition 6 Let {Li } I and {L j } J be two pro-coherent O X -modules. The sheaf of differential operators, which we denoted by H om Diff ({Li } I , {L j } J ), is the image of the injective map (2). So H om Diff X ({Li } I , {L j } J ) := H om Strat ({P Xm }Z ⊗O X L , {P Xm }Z ⊗O X L ) ∼ = H om ν(O X ) ({P Xm }Z ⊗O X {Li } I , {L j } J ) := lim lim lim H om O X (P Xm ⊗O X Li , L j )
← → → J
I
J
I
Z
∼ = lim lim H om Diff X (Li , L j ). ← → We recall that for F ∈ Coh(OX ) and G an O X -module, the sheaf H om Diff X (F , G ) is isomorphic to lim H om O X (P Xm ⊗O X F , G ) ∼ = lim H om O X (F , G ⊗O X D X,m ). →Z →Z We denote by ν(O X )-Diff X the additive category whose objects are pro-coherentO X -modules and whose morphisms are differential operators (sometimes called differential complexes). We have a functor Q0X : ν(O X )-Diff X −→ νi (P X· ) L −→ {P Xm }Z ⊗O X L
which extends that of Proposition 2. This functor was first introduced in [10, 6.2], by Grothendieck and it is called the formalization functor or linearization. By Theorem 1 this functor is an equivalence of categories. Remark 7 If we restrict the formalization functor to differential complexes L whose objects are coherent O X -modules, then the pro-objects Q 0X (L ) are always of Artin-Rees type. Moreover any morphism of pro-objects between two such objects is necessarily of Artin-Rees type (see [10, 6.2]). Definition 7 Let consider C b (ν(O X ), Diff X ) the category of bounded complexes in ν(O X )-Diff X and D b (ν(O X ), Diff X ) the category obtained from C b (ν(O X ), Diff X ) by inverting Q0X -quasi-isomorphisms. This is a triangulated category with the usual shift functor and distinguished triangles those induced by the usual mapping cones. We remark that this localizing procedure was first introduced by Andrè-Baldassarri 2001 in [1, Appendix C], following an idea of Berthelot. We obtain a localized equivalence of categories Q0X : D b (ν(O X ), Diff X ) −→ Dib (P X· ). −1
X functor. This functor would be the “dual” of Saito DR Remark 8 The morphism (1) of Theorem 1 is induced by the following commutative diagrams which are adjoint to those of D X -modules in the smooth case ([17, 1.4.1]): L
P
d 1 ⊗idL
d 1 ⊗idL
{P Xm }Z ⊗O X L
123
/ L
Q0X (P)
/ {P m }Z ⊗O X L X
L d 1 ⊗idL
/ r8 L r r rr rrQr0 (P) r r r X
{P Xm }Z ⊗O X L
P
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0 (L ), Q0 (L )) ∼ Hom 0 for any Q0X (P) ∈ HomStrat (QX = ν(OX ) (QX (L ), L ). The map X m d1 : O X −→ P X is that induced by the second projection of X × X into X .
Definition 8 An object in D b (ν(O X ), Diff X ) is said to be perfect if it is locally isomorphic to a bounded complex whose elements are locally free O X -modules of finite rank. We denote by D bp (ν(O X ), Diff X ) the category of bounded perfect complexes in D b (ν(O X ), Diff X ). Then any object in D bp (ν(O X ), Diff X ) may be represented as an object in C b (Coh(OX ), Diff X ) (see [15] for definition of perfect objects). Definition 9 Herrera-Lieberman differential complexes. ([13, §2], or [5, II.5]). Let C1b (ν(O X ), Diff X ) denote the category of bounded complexes of differential operators of order at most one, that is: (i) the objects of C1b (ν(O X ), Diff X ) are complexes whose terms are pro-coherentO X -modules and whose differentials are differential operators of order at most one; (ii) morphisms between such complexes are morphisms of complexes which are O X -linear maps. The category C1b (Coh(OX ), Diff X ) is the full subcategory of C1b (ν(O X ), Diff X ) whose objects are complexes of coherent modules. We denote by D1b (ν(O X ), Diff X ) the category obtained form C1b (ν(O X ), Diff X ) by inverting Q0X -quasi-isomorphisms in C1b (ν(O X ), Diff X ). Thus we have the functors λ : D1b (ν(O X ), Diff X ) −→ D b (ν(O X ), Diff X )
Q0X,1 : D1b (ν(O X ), Diff X ) −→ D b (P X· ) where Q0X,1 := Q0X ◦ λ. 5 De Rham functor Definition 10 Let M ∈ ν(P X· ) and DR X (M ) := •X ⊗O X M
= [0
0 /M
1
/ 1 ⊗O M X X
/ ···
d
/ d ⊗O M X X
/ 0].
(see Proposition 1) and the proThe differentials are defined using the stratification map sM 1 1 jection P X −→ X . The complex DR X (M ) belongs to C b (ν(O X ), Diff X ) and in particular it is also an object of C1b (ν(O X ), Diff X ). We define the functors
DR1,X : C b (P X· ) −→ C1b (ν(O X ), Diff X ) M • −→ (•X ⊗O X M • )tot =: •X ⊗•O X M • .
and DR X = λ ◦ DR1,X . We want to prove that this De Rham functor sends the multiplicative system of qis in C b (P X· ) into the multiplicative system of Q0X -qis in C b (ν(O X ), Diff X ). In order to
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prove this result we need the following version of the crystalline Poincaré lemma O X −→ Q0X DR X (O X ). This lemma may be found in [10, 6.5], and in [6, 6.12], where Berthelot Ogus proved a filtered version. We give here a simple proof of the result we need. We note that this proof also works well in characteristic p using the formalism of divided powers. Let us remark: Remark 9 ([6, 2.13]) The pro-object {P Xm }Z admits two different stratifications depending on the O X -module structure we chose on it. We consider on {P Xm }Z its “left” O X -structure, (that given by p0 ), (so its “right” O X -structure may be used in the tensor product with the De Rham complex). This is the construction of Grothendieck linearization. In this case {P Xm }Z is endowed with the stratification θ : P Xm+n → P Xn ⊗O X P Xm sending ( f ⊗g) → 1⊗g⊗1⊗ f . On the other hand, if we consider the “right” structure on {P Xm }Z , the stratification is given by the map δ. Lemma 1 Poincaré crystalline lemma. The linearized De Rham complex is a resolution of O X d0
O X −→ Q0X DR X (O X )
in C b (P X· ). In fact the complex OX
d0
∇
/ {P Xm }Z
0
∇
/ {P m }Z ⊗O 1 X X X
1
/ ···
∇
d−1
/ {P m }Z ⊗O d X X X
(3)
is exact and thus locally homotopic to zero since its terms are locally free. d0
Proof First of all we remark that Q0X DR X (O X ) is a complex in C b (P X· ) and the map O X −→ {P Xm }Z respects the stratifications (see the remark given below regarding the stratification 0
on {P Xm }Z ). It is evident that ∇ ◦ d 0 = 0. The complex (3) as a complex of ν(P X· ) is represented by {O X
d0
/ P Xn
0
∇ (n)
1
∇ (n)
/ P n−1 ⊗O 1 X X X
/ ···
d−1
∇ (n)
/ P n−d ⊗O d }n∈N . X X X
(4)
We will prove by induction that (4)n is exact for each n ∈ N. First of all we render the O X -linear differentials on the complex explicit by the use of the basis given by ξi (see Remark 5). Then d 0 is the map d 0 : O X −→ P Xn 1 −→ I := 1 ⊗ 1 f −→ f ⊗ 1 = f I while ∇ :=
Q0X (∇)
is the linearization of the De Rham complex ( OX
∇0
/ 1 X
∇1
/ ··· ∇
d−1
/ d ) X
obtained as p
∇ (n)
/ P n−1 ⊗O p+1 X X 5X kkk k k n,1 δ ⊗id p kkk p X kkk ⊗∇ (1) kkk idPn−1 X p P Xn−1 ⊗O X P X1 ⊗O X X p
P Xn ⊗O X X
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p
where ∇ (1) is p
p
p+1
∇ (1) : P X1 ⊗O X X −→ X
( f ⊗ g) ⊗ ω −→ f ∇ p (gω) so that in local coordinates we have I ⊗ (ξi1 ∧ · · · ∧ ξi p ) −→ 0
ξi ⊗ (ξi1 ∧ · · · ∧ ξi p ) −→ ξi ∧ ξi1 ∧ · · · ∧ ξi p ∀i ∈ {1, . . . , d}. Now we proceed by induction on the “level” n in order to prove that (3) is exact. For n = 0 the complex (4)0 reduces to idO X
0 −→ O X −→ O X −→ 0 −→ 0 which is obviously exact. When n = 1 the complex (4)1 is d0
0 −→ O X −→ P X1 −→ 1X −→ 0 which is homotopic to zero via the O X -linear homotopism q1
−→ O X f ⊗ g −→ f g
P X1
1X −→ P X1 ξi −→ ξi
∼ O X ⊕ 1 . Let us suppose that the complex (4)n−1 is exact with n ≥ 1. Then we then P X1 = X consider the diagram 0
0
/0
0
0
0
0
0
In I n+1
/
⊗O X 1X
/ ···
/ OX
/ P Xn
/ P n−1 ⊗O 1 X X X
/ OX
/ P n−1
/ P n−2 ⊗O 1 X X X
0
/
X
0
I n−1 In
0
⊗O X dX
/0
/ ···
/ P n−d ⊗O d X X X
/0
/ ···
/ P n−d−1 ⊗O d X X X
/0
0
0
/
I n−d I n−d+1
0
whose columns are exact. By inductive hypothesis the third row is exact. Then the second row is exact if and only if the first is. So we will prove that the complex 0 −→
n−d I n D 0 I n−1 D1 D d−1 I −→ ⊗O X 1X −→ · · · −→ n−d+1 ⊗O X dX −→ 0 n+1 n I I I
is exact proving that its identity is homotopic to zero. We have to construct O X -linear maps sp :
I n− p I n− p+1 p p−1 ⊗ −→ ⊗O X X O X X I n− p+1 I n− p
123
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L. Fiorot
for any p = 1, . . . , d such that in the diagram
/
In I n+1
0
id
/
0
In I n+1
/ ··· / I n−d ⊗O d X X I n−d+1 u t t uu sd ttt s2 uuu s1 ttt t u id id id tt tt uu tt tt zuu y t t t yt 1 / I n−1 / ··· / I n−d ⊗O d ⊗ n O n−d+1 X X X X I I /
I n−1 In
⊗O X 1X
/0
/0
p
the identity of I n− p /I n− p+1 ⊗O X X would be id = D p−1 ◦ s p + s p+1 ◦ D p . First we explicitly write the action of the differentials D p on a basis: n− p−1 I n− p p Dp I p+1 ⊗ −→ ⊗O X X O X X I n− p+1 I n− p d
ξ1α1 · · · ξdαd ⊗ ξi1 ∧ · · · ∧ ξi p −→
α j −1
α j ξ1α1 · · · ξ j
· · · ξdαd ⊗ ξ j ∧ ξi1 ∧ · · · ∧ ξi p
j=1
with α1 + · · · + αd = n − p. We note that the map p times
p times
1X ⊗O X · · · ⊗O X 1X −→ 1X ⊗O X · · · ⊗O X 1X α1 ⊗ · · · ⊗ α p −→ (−1)sgn(σ ) ασ (1) ⊗ · · · ⊗ ασ ( p) σ ∈ p p
p−1
induces a map σ p : X −→ 1X ⊗O X X . We define s p (up to the factor n( p − 1)!) as the composition n( p−1)!s p
/ I n− p+1 /I n− p+2 ⊗O p−1 X X ggg3 g g g g gggg id⊗σ p ggggm⊗id ggggg p−1 I n− p /I n− p+1 ⊗O X I /I 2 ⊗O X X p
I n− p /I n− p+1 ⊗O X X
where m is the map m : I n− p /I n− p+1 ⊗O X I /I 2 −→ I n− p+1 /I n− p+2 induced by the multiplication. It is well defined because I n− p+1 I = I n− p+2 = I n− p I 2 . We now explicitly calculate s p on a local basis
α
α
n− p+1 I n− p p sp I p−1 ⊗ X → n− p+2 ⊗ X n− p+1 I I OX OX
ξ1 1 · · · ξd d ⊗ ξi1 ∧ · · · ∧ ξi p →
1 n
p
m+1 α1 1 m=1 (−1)
ξ
α · · · ξd d ξim ⊗ ξi1 ∧ · · · ξim · · · ∧ ξi p
We note that this definition also makes sense in the divided powers setting. Indeed, in characteristic p we replace ξiαi by ξi[αi ] and the local description becomes:
[α1 ]
ξ1
n− p+1 I n− p p sp I p−1 ⊗O X X → n− p+2 ⊗O X X n− p+1 I I
[αd ]
· · · ξd
123
⊗ ξi1 ∧ · · · ∧ ξi p →
p
m+1 [α1 ] 1 m=1 (−1)
ξ
[α
· · · ξim im
+1] [αd ] d
ξ
⊗ ξi1 ∧ · · · ξim · · · ∧ ξi p
Differential complexes and stratified pro-modules
445
Now let us compute the composition D p−1 ◦ s p + s p+1 ◦ D p on an element of the basis. We have (D p−1 ◦ s p + s p+1 ◦ D p )(ξ1α1 · · · ξdαd ⊗ ξi1 ∧ · · · ∧ ξi p )
p αd p−1 1 m+1 α1 =D (−1) ξ1 · · · ξd ξim ⊗ ξi1 ∧ · · · ∧ ξim ∧ · · · ∧ ξi p n m=1 ⎛ ⎞ d α −1 α j ξ1α1 · · · ξ j j · · · ξdαd ⊗ ξ j ∧ ξi1 ∧ · · · ∧ ξi p ⎠ + s p+1 ⎝ j=1
p d 1 α j −1 α = (−1)m+1 α j ξ1 1 · · · ξ j · · · ξdαd ξim ⊗ ξ j ∧ ξi1 ∧ · · · ξim · · · ∧ ξi p n m=1 1= j=im p + (αim + 1)ξ1α1 · · · ξdαd ⊗ ξi1 ∧ · · · ∧ ξi p
m=1 p
+
d
α j −1
(−1)m α j ξ1α1 · · · ξ j
· · · ξdαd ξim ⊗ ξ j ∧ ξi1 ∧ · · · ∧ ξim ∧ · · · ∧ ξi p
m=1 1= j =i m d
+
α j ξ1α1 · · · ξdαd ⊗ ξi1 ∧ · · · ∧ ξi p
j=1=i 1 ,...,i p
⎞ ⎛ d 1⎝ p+ α j ⎠ ξ1α1 · · · ξdαd ⊗ ξi1 ∧ · · · ∧ ξi p = n =
ξ1α1
j=1 αd · · · ξd ⊗ ξi1
∧ · · · ∧ ξi p
as desired.
The complex {P Xm }Z ⊗ •X ∼ = DR X ({P Xm }Z ) where we take {P Xm }Z = p1∗ (O X ). Thus OX
we obtain a “Poincaré Lemma”: O X ∼ = DR X ({P Xm }Z ). This result is the “dual” of the quasi• ∼ isomorphism X ⊗ D X = ω X [d] see [7, VI.3.5]. OX
The same result holds true on taking {P Xm }Z = p0∗ (O X ). In this case we obtain the following corollary. Corollary 3 The complex d1
OX
/ {P Xm }Z
/ 1 ⊗O {P m }Z X X X
/ ···
/ d ⊗O {P m }Z X X X
is exact. Hence, O X −→ DR( p0∗ (O X ))
is a quasi-isomorphism in C b (ν(P X· )). Corollary 4 Let M ∈ ν(P X· ). The complexes M
d1
/ {P Xm }Z ⊗ M OX
/ 1X ⊗ {P Xm }Z ⊗ M OX
OX
/ ···
/ dX ⊗ {P Xm }Z ⊗ M OX
OX
123
446
L. Fiorot
and M
d0
/ M ⊗ {P Xm }Z OX
/ M ⊗ {P Xm }Z ⊗ 1X OX
OX
/ ···
/ M ⊗ {P Xm }Z ⊗ dX OX
OX
(5)
are exact. Proof Let us consider the sequence (5). Its analogue for M = O X is the sequence (3), which is exact and so locally homotopic to zero. Any additive functor respects homotopies, so the sequence (3) tensorized over O X with M gives (5) which is exact.
6 Equivalences of categories i : Remark 10 Let F • be an object in C1 (ν(O X ), Diff X ). By definition the differential dF i i+1 F −→ F is a morphism of pro-object represented by differential operators of order i at most one. It defines in a unique way a morphism d F : P X1 ⊗O X F i −→ F i+1 . This i ) : morphism extends in a unique way to a morphism of stratified pro-modules Q0X (dF {P Xm }Z ⊗O X F i −→ {P Xm }Z ⊗O X F i+1 . Using the O X -base of P X,1 given in local coordinates by I, ξ1 , · · · , ξd we have for any section s of F i i
i dF (s) = d F (I ⊗ s)
(6)
i
dxi j (s) : = d F (ξ j ⊗ s)
(where the second is taken as definition of dxi j ). The maps dxi j : F i −→ F i+1 are maps of O X -modules depending on the choice of the coordinates. Definition 11 Let F • ∈ C1b (ν(O X ), Diff X ). Then F • is a •X -module in pro-objects. Let i, j j by definition σF : X ⊗O X F j−i −→ F i be the •X -structural maps. They are: i,0 = idF i : F i −→ F i σF
i−1
O
dF•
/ Fi 9 s s ss s s ss i,1 sss σF
P X1 ⊗O X F i−1
1X ⊗O X F i−1 and in general
i−2
X
X
O
j
X ⊗ F i− j OX
123
idP 1 ⊗ d F
i−1
/ P X1 ⊗ F i−1 d Fff/3 F i OX fff fffff f f f f f fffff fffffi, j f f f f σF fff fffff
P X1 ⊗ · · · ⊗ P X1 ⊗O X F i− j O O
/ ···
X
Differential complexes and stratified pro-modules
447
where the last vertical map j X
⊗O X F
i− j
−→
j times
P X1
⊗O X
P X1
⊗O X · · · ⊗O X P X1 ⊗O X F i− j
is induced by the shuffle on a basis. Now we prove a technical lemma which will be used in the proof of our main theorem. i , d i of (6) for i ∈ Z and Lemma 2 Given F • ∈ C1 (ν(O X ), Diff X ); the morphisms dF xj j ∈ {0, . . . , d} satisfy the following conditions:
(i) (ii) (iii) (iv)
i+1 i =0 dF ◦ dF i + d i+1 ◦ d i = 0 ◦ dF dxi+1 xj F j i + d i+1 ◦ d i = 0 ◦ d dxi+1 xk xk xj j dxi+1 ◦ dxi j = 0. j
Proof The first condition is given by the hypothesis F • ∈ C1 (ν(O X ), Diff X ), so the comi+1 i = 0. In order to prove ii) to iv) we remark that: position dF ◦ dF i+1 i+1 i i ◦ dxi j (s) + dxi+1 ◦ dF (s) = dF dF (ξ j ⊗ s) = 0 dF j i+1 i ◦ dxi k (s) + dxi+1 ◦ dxi j (s) = dF dF (ξ j ξk ⊗ s) = 0 dxi+1 j k i+1 i i dxi+1 ◦ dxi j (s) + dxi+1 ◦ dF (s) = dF dF (ξ 2j ⊗ s) = 0 j j i+1 i where ξ j = 1 ⊗ x j − x j ⊗ 1 ∈ P X1 . We recall that dF dF is obtained by the composition
P X1
⊗O X
P X1 O
δ 2,1 ⊗idF i
P X2 ⊗O X F i
i idP 1 ⊗dF
d i+1
/ P 1 ⊗O F i+1 F g/3 F i+2 X X ggg ggggg g g g g ggggg ggggg d i+1 d i g g g g g F F
⊗O X
Fi
X
and δ 2,1 was defined in Proposition 1. Definition 12 Let d x1 , . . . , d xn be a local basis for 1X (where n is the dimension of X ). We define the maps i, j
j
ηF : X ⊗O X F i− j −→ F i i− j
f d xi1 ∧ · · · ∧ d xi j ⊗ s −→ f d xii−1 ◦ · · · ◦ d xi j (s) 1 where f is a section of O X and s is a section of F i− j . This definition does not depend on i, j i, j local coordinates and moreover σF = j!ηF . Definition 13 Let q = {qm }Z : {P Xm }Z −→ O X be the usual projection which is linear for both the O X -module structures of {P Xm }Z . Given F • ∈ C1 (ν(O X ), Diff X ) we define the morphisms iF :
d
j
X ⊗O X {P Xm }Z ⊗O X F i− j −→ F i
j=0
123
448
L. Fiorot
for each i ∈ Z in the following way: we consider the composition id
j
X ⊗ {P Xm }Z ⊗ F i− j OX
OX
j ⊗q⊗id i− j X FX
/ Xj ⊗ F i− j O
X VVVV VVVV i, j VVVV ηF VVVV i, j V VVVV F V+ i
F
and by definition iF :=
d
i, j j=0 F .
Theorem 2 We have two morphisms of functors 0 : idC b (P · ) −→ Q0X ◦ DR X = Q1,X ◦ DR1,X X
(functors between C b (P X· ) and itself) : DR1,X ◦ Q0X −→ idC b (ν(O X ),Diff X ) 1
(functors between C1b (ν(O X ), Diff X ) and itself ). They induce quasi-isomorphisms of complexes. So the functors DR1,X and DR X localize with respect to Q0X -quasi-isomorphisms inducing the functor DR1,X : D b (P X· ) −→ D1b (ν(O X ), Diff X ). DR X : D b (P X· ) −→ D b (ν(O X ), Diff X ). Moreover DR X (resp. DR1,X ) is an equivalence of categories whose quasi-inverse is the 0 ). functor Q0X (resp. Q1,X Proof The morphism d 0 : O X → P Xm induces a morphism of bicomplexes (see sequence (5) of Corollary 4) M • : M • −→ M • ⊗O X {P Xm }Z ⊗O X •X . Then we obtain a morphism of complexes M • −→ M • ⊗•O X {P Xm }Z ⊗O X •X
∼ = {P Xm }Z ⊗O X M • ⊗•O X •X ∼ = Q0 (DR X (M • )) X
0 ∼ (DR1,X (M • )). = Q1,X
The isomorphism between the first and the second complex is induced by the stratification on M • which is the isomorphism {P Xm }Z ⊗O X M • ∼ = M • ⊗O X {P Xm }Z . By Corollary 4 we obtain that it is a quasi-isomorphism. So the functors DR1,X and DR X send qis in D b (P X• ) into Q0X -qis which permits us to localize them.
123
Differential complexes and stratified pro-modules
449
We have to prove that the diagram di
j
OX
/ Xj ⊗ {P Xm }Z ⊗ F i+1− j
DRQ0
X ⊗ {P Xm }Z ⊗ F i− j OX
OX
OX
i+1 F
i
F
Fi
/ F i+1
di
F
is commutative. The complex DR X Q X 0 (F • ) = (G •• )tot where p
G p,q = X ⊗O X {P Xm }Z ⊗O X F q
and p,q
p
p+1
: X ⊗O X {P Xm }Z ⊗O X F q −→ X
dG
⊗O X {P Xm }Z ⊗O X F q
is p,q
dG
p
= dDR
∗ X ( p0 (O X ))
⊗ idF q
while p,q
dG
p
p
: X ⊗O X {P Xm }Z ⊗O X F q −→ X ⊗O X {P Xm }Z ⊗O X F q+1
is p,q
dG
q
= id p ⊗ Q0X (dF ); X
where DR X ( p0∗ (O X )) was considered in Corollary 3. p,q Given I •,• a bounded bicomplex with commuting differentials d I : I p,q → I p+1,q and p,q • r := p,q p,q+1 dI :I → I , the total complex associated to it is denoted by Itot with Itot p,q p,q p,q p p,q and d Itot (x) = d I (x) + (−1) d I (x) for any x ∈ I . p+q=r I • The set Hom(Atot , B• ) of morphisms of complexes between a total complex of a bicomplex p p p and a complex is the set families of maps {φ p : Atot → B p } p such that d B ◦φ p = φ p+1 ◦d Atot . • • p,q q, p−q p Then Hom(Atot , B ) is isomorphic to the set of families of maps {φ :A → B } p,q satisfying the following conditions q, p−q
p
d B ◦ φ p,q = φ p+1,q+1 d A
q, p−q
+ (−1)q φ p+1,q ◦ d A
for any p, q. So we have only to prove that p
p,q
p+1,q+1 q, p−q dG
d F ◦ F = F
p+1,q
+ (−1)q F
q, p−q
◦ dG
is true. It is enough to check these relations locally, choosing local coordinates x 1 , . . . , xn . The q sections f d xi1 ∧ · · · ∧ d xiq ⊗ I ⊗ s generate X ⊗O X {P Xm }Z ⊗O X F p−q where s is a section of F p−q . Then p
p,q
p
p−1
p−q
dF ◦ F ( f d xi1 ∧ · · · ∧ d xiq ⊗ I ⊗ s) = dF ( f dxi1 · · · dxiq (s)) p
p−1
p−q
= f dF dxi1 · · · dxiq (s) +
n ∂ f p p−1 p−q dx dx · · · dxiq (s) ∂ xi i i1 i=1
123
450
L. Fiorot
while p+1,q+1 q, p−q dG ( f d xi1 ∧ · · · ∧ d xiq ⊗ I ⊗ s)
n ∂f p+1,q+1 d xi ∧ d xi1 ∧ · · · ∧ d xiq = F ∂ xi i=1 n ∂ f p p−1 p−q = dx dx · · · dxiq (s) ∂ xi i i1 i=1
F
⊗I⊗s
For the last term we have p+1,q q, p−q dG ( f d xi1 ∧ · · · ∧ d xiq ⊗ I ⊗ s) p+1,q p−q = F ( f d xi1 ∧ · · · ∧ d xiq ⊗ I ⊗ dF (s))
F
p
p−q+1 p−q dF (s)
= f dxi1 · · · dxiq
Thus, using Lemma 2, we prove our assertion. Moreover the composition 1
d / • ⊗• {P Xm }Z ⊗O X F • F • QQ QQQ X O X QQQ QQ •F idF • QQQQ Q( F•
is the identity so F • is a Q0X -quasi-isomorphism. In particular the functor DR X localizes to DR X : D b (P X· ) −→ D b (ν(O X ), Diff X ) and it is an equivalence of categories with quasi-inverse the localized Q0X functor. Corollary 5 The functor λ : D1b (ν(O X ), Diff X ) −→ D b (ν(O X ), Diff X ) is an equivalence of categories whose quasi-inverse is the functor DR X ◦ Q0X . 7 Crystals in pro-modules We refer to Grothendieck exposé [10] for the definition of crystalline site Cris(X/C) in characteristic zero (and also to Berthelot’s thesis [5]). We denote by O X/C the sheaf on Cris(X/C) such that for any object (nilpotent closed immersion U → T with U ⊂ X open subset) its value is O X/C (U → T ) := OT . It is a ringed sheaf on Cris(X/C). Definition 14 A crystal in pro-coherent-modules {Fi }i∈I is a sheaf in the crystalline site Cris(X/C) with values in Pro(Coh(OX/C ))-modules such that for any morphism p : {U → T } → {U → T } given by the diagram U
/ T p
U
/T
X we have { p ∗ (Fi (U → T ))}i∈I ∼ = {Fi (U → T )}i∈I .
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Differential complexes and stratified pro-modules
451
Remark 11 The previous definition was introduced by P. Deligne in a conference he gave at IHES in 1970. It is consistent because the category of pro-coherent O X -modules is a stack. Theorem 3 The category ν(P X· ) is equivalent to the category of crystals in pro-coherent O X/C -modules. Proof The proof is equivalent to the classical proof of the equivalence between stratified O X/C -modules and crystals (see [5]). If {Fi }i∈I is a crystal in pro-coherent O X/C -modules id X
then {Fi (X → X )}i∈I ∈ ν(O X ). Taking the diagram defined by the diagonal thickenings X n −→ X × X with n ∈ N
/ Xn
X id
we obtain the stratification
X
{ p0∗ (Fi (X
p0
id
/X
p1
→ X n ))}i∈I ∼ = { p1∗ (Fi (X → X n ))}i∈I in the
id X
pro-object {Fi (X → X )}i∈I . Conversely if {Fi } I ∈ ν({P Xm }Z ) we define a sheaf on Cris(X/C) in the following way. For any object U → T ∈ Cris(X/C) there exists locally a section h : T → X (X is smooth). We define C R({Fi } I )(U → T ) := {h ∗ (Fi )} I . These local definitions patch together to define a sheaf in pro-coherent O X/C -Modules which is a crystal. Acknowledgments I would like to thank Pierre Berthelot for his suggestions and for his notes on Deligne lectures on “Cristaux discontinues” and Pierre Deligne for some useful comments and remarks. I would also like to thank Maurizio Cailotto and Anne Virrion for the improvements they suggested to me during the preparation of this work.
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