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The Comparison Isomorphisms Ccris The Comparison Isomorphisms Ccris Fabrizio Andreatta July 9, 2009 Contents 1 Introduction 2 2 The crystalline conjecture with coe±cients 3 3 The complex case. The strategy in the p-adic setting 4 4 The p-adic case; Faltings's topology 5 5 Sheaves of periods 7 i 6 The computation of R v¤Bcris 13 7 The proof of the comparison isomorphism 16 1 1 Introduction Let p > 0 denote a prime integer and let k be a perfect ¯eld of characteristic p. Put OK := W (k) and let K be its fraction ¯eld. Denote by K a ¯xed algebraic closure of K and set GK := Gal(K=K). We write Bcris for the crystalline period ring de¯ned by J.-M. Fontaine in [Fo]. Recall that Bcris is a topological ring, endowed with a continuous action of GK , an ² exhaustive decreasing ¯ltration Fil Bcris and a Frobenius operator '. Following the paper [AI2], these notes aim at presenting a new proof of the so called crys- talline conjecture formulated by J.-M. Fontaine in [Fo]. Let X be a smooth proper scheme, geometrically irreducible, of relative dimension d over OK : Conjecture 1.1 ([Fo]). For every i ¸ 0 there is a canonical and functorial isomorphism com- muting with all the additional structures (namely, ¯ltrations, GK {actions and Frobenii) i et » i H (XK ; Qp) ­Qp Bcris = Hcris(Xk=OK ) ­OK Bcris: i » It follows from a classical theorem for crystalline cohomology, see [BO], that Hcris(Xk=OK ) = i HdR(X=OK ). The latter denotes the hypercohomology of the de Rham complex 0 ¡! OX ¡! ­1 ¡! ­2 ¡! ¢ ¢ ¢ ­d ¡! 0. The ¯rst interpretation provides a Frobenius op- X=OK X=OK X=OK erator, the second one provides a ¯ltration, the Hodge ¯ltration. For every n 2 N we put n i i Fil HdR(X=OK ) = HdR(X=OK ) if n · 0, to be 0 if n ¸ d + 1 and to be the image of the hypercohomology of the complex 0 ¡! ­n ¡! ­2 ¡! ¢ ¢ ¢ ­d ¡! 0 if 0 · n · d. X=OK X=OK X=OK Then, in the statement of the conjecture 1) Frobenius on the left is de¯ned by the identity operator on Hi(Xet; Q ) and by Frobenius K p i on Bcris. Frobenius on the right hand side is de¯ned by Frobenius on Hcris(Xk=OK ) and by Frobenius on Bcris; 2) the ¯ltration on the left hand side is de¯ned by Hi(Xet; Q ) ­ FilnB , while on K ¡ p Qp cris ¢ n i the right hand side it is given by the composite ¯ltration Fil HdR(X=OK ) ­OK Bcris := P a i b a+b=n Fil HdR(X=OK ) ­OK Fil Bcris; 3) the Galois action on the left hand side is induced by the action of G on Hi(Xet; Q ) K K p and on Bcris. The Galois action on the right hand side is given by letting GK act trivially on i Hcris(Xk=OK ) and through its natural action on Bcris. The conjecture is now a theorem, proven by G. Faltings in [F2]. In fact it holds without assuming that K is unrami¯ed and with non-constant coe±cients. But these assumptions will simplify our considerations. There are various approaches to the proof of the conjecture. One (and the ¯rst) is based on ideas of Fontaine and Messing in [FM] using the syntomic cohomology on X; a full proof (for constant coe±cients) using these methods was given by T. Tsuji in [T]. There is also an approach (for constant coe±cients) based on a comparison isomorphism in K- theory which is due to W. Niziol [N]. We will follow the approach by Faltings, not in its original version but using a certain topology described in [F3]. Our approach, which works also for non constant coe±cients, is based on the papers [AI2] and [AB]. The strategy consists in de¯ning a new cohomology theory associated to X and proving that it computes both the left hand side (via the theory of almost ¶etaleextensions) and the right hand side of conjecture 1.1. The new inputs, compared to Faltings's original approach, are: i) we systematically study the underlying sheaf theory of Faltings' topology. Faltings uses Galois cohomology of a±nes and then patches his computations to global results using hyper- coverings; 2 ii) we introduce certain acyclic resolutions of sheaves of periods on Faltings' topology. This allows to avoid the use of Poincar¶eduality. In Faltings' original approach one needed to prove Poincar¶eduality in Faltings' theory and to show that it is compatible both with poicar¶eduality on crystalline or de Rham cohomology and with Poincar¶eduality on ¶etalecohomology. We explain how to avoid this. In order to simplify the proof, we also assume that there exists a morphism F : X ! X lifting Frobenius on OK and the Frobenius morphism on the special ¯ber Xk. This is a strong hypothesis. For example, if X is an abelian scheme over OK with ordinary reduction, it amounts to require that X is the canonical lift of Xk. In fact, in [AI2] the existence of such a lift of Frobenius is assumed only Zariski locally on X (and this is harmless). On the other hand, i i this stronger assumption allows us to work with HdR(X=OK ) instead of Hcris(Xk=OK ) with the Frobenius operator induced by F . We can then avoid the use of crystalline cohomology (and its technicalities) completely. 2 The crystalline conjecture with coe±cients We present the statement of the comparison isomorphism conjecture in the case of non constant coe±cients. As before X ! Spec(OK ) is a proper and smooth scheme, geometrically irreducible. We make no assumption on OK but we assume that X is obtained by base change from a scheme de¯ned over W (k) where k is the residue ¯eld of OK (this is needed in [AI2] not in Faltings' original approach). We consider two categories: et et The category Qp ¡ Sh(XK ) of Qp-adic ¶etalesheaves. By a p{adic ¶etalesheaf L on XK we et N mean an inverse system L := fLng 2 Sh(XK ) such that Ln is a locally constant and locally free n n of ¯nite rank sheaf of Z=p Z{modules for the ¶etaletopology of XK and Ln = Ln+1=p Ln+1 for every n 2 N. Given two such objects M = fMngn and L := fLngn a morphism of p-adic ¶etale sheaves f : M! L is a collection of morphisms fn : Mn ! Ln of sheaves such that fn ´ fn+1 n et modulo p for every n 2 N. The category Qp ¡ Sh(XK ) has the p-adic ¶etalesheaves as objects and one de¯nes the morphisms by tensoring the Zp-module of morphisms as p-adic ¶etalesheaves with Q . p ¡ ¢ The category Fil¡F ¡Isoc(XK =K) of ¯ltered F -isocrystals on XK . Let F ¡Isoc Xk=W (k) be ¤ the category of isocrystals E on Xk=W (k), endowed with a non-degenerate Frobenius ': ' (E) ¡! E. By isocrystal we mean a crystal of OXk=W (k)-modules up to isogeny. See also [B3], especially Thm. 2.4.2, for a di®erent point of view in the context of the rigid analytic spaces. To any such object one can associate a coherent OXK -module EXK with integrable, quasi-nilpotent con- nection r: E ¡! E ­ ­1 . The category Fil ¡ F ¡ Isoc(X =K) consists of an XK XK OXK XK =K K ² F -isocrystal E on Xk=W (k) and an exhaustive descending ¯ltration Fil EXK on EXK by OXK - submodules satisfying Gri±th's transversality with respect to the connection r on EXK i. e., such that rFilnE ½ Filn¡1E ­ ­1 . Then, XK XK OXK XK =K Conjecture 2.1. Assume that there exist a Qp-adic ¶etalesheaf L and a ¯ltered-F -isocrystal E which are associated. Then, for every i ¸ 0 there is a canonical and functorial isomorphism commuting with all the additional structures (namely, ¯ltrations, GK {actions and Frobenii) i et » i H (XK ; L) ­Qp Bcris = Hcris(Xk=W (k); E) ­W (k) Bcris: i i Here, the¡ ¯ltration¢ on Hcris(Xk=W (k); E) ­W (k) Bcris is obtained¡ via Hcris(¢Xk=W (k); E) ­W (k) » i i Bcris = HdR XK ; EXK ­K Bcris using the Hodge ¯ltration on HdR XK ; EXK . The clari¯cation 3 of what \being associated" means is part of the conjecture. As an example, let f : E ! X be a relative elliptic curve or an abelian scheme. Then, one can consider the Qp-adic ¶etalesheaf (Ln)n n n where Ln := EK [p ] is the groups scheme of p -torsion points of EK . On the other hand, one can consider the ¯ltered F -isocrystal E de¯ned by R1f Ocris (the ¯rst derived functor in the cris;¤ Ek=W (k) crystalline sense of the crystal Ocris with respect to f ). The module with connection E Ek=W (k) k XK on XK is the ¯rst de Rham cohomology of EK relative to XK with the Gauss-Manin connection. The ¯ltration has only one step and Fil1E ½ E is f (­1 . Then, in this case L and E XK XK ¤ EK =XK are associated and the conjecture holds. See [AI2]. 3 The complex case. The strategy in the p-adic setting Assume that X is a complex analytic projective variety. Let OX denote the sheaf of holomorphic functions on X. In this consider on the one hand a locally constant sheaf L of C-vector spaces and a locally free coherent sheaf E of OX -modules endowed with an integrable connection r.
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