Complexe de de\thinspace Rham-Witt et cohomologie cristalline

Abstract

This book-length article is a major advance in the theory of p-adic cohomology. To any scheme X, smooth and proper over a perfect field k of characteristic p>0, the article associates a complex WΩ⋅X, called the de Rham-Witt complex, of sheaves whose hypercohomology (computed on the Zariski site) is isomorphic to the Grothendieck-Berthelot crystalline cohomology H∗(X/W). Furthermore, the construction of the de Rham-Witt complex is very elementary, and the present article is an excellent introduction to the study of crystalline cohomology. In fact, many of the main theorems of crystalline cohomology can be given simple proofs using WΩ⋅X (notable exception at the present date: Poincaré duality). One of the first attempts to define a Weil cohomology theory was made by J.-P. Serre [Symposium Internacional de Topología Algebraica, pp. 24–53, Univ. Nac. Autónom. México, Mexico, 1958; MR0098097 (20 #4559)], who associated to any scheme X over a perfect field k of characteristic p the pro-system of sheaves WnOX and the sheaf WOX=lim←WnOX, where WnOX is the sheaf of Witt vectors of length n. Serre suggested that the cohomology groups Hq(WOX)=limHq(WnOX), which are modules over W(k), should be the "part of type (0,q)'' of some p-adic cohomology theory. S. Lubkin [Compositio Math. 34 (1977), no. 3, 225–277; MR0453745 (56 #12005)] suggested that Serre's theory should be extended by taking the hypercohomology of a suitable completed quotient of the ordinary de Rham complex of WOX/W(k). In another direction, Grothendieck and Berthelot defined the crystalline cohomology H∗(X/W) and showed that it has all the properties needed for a (weak) Weil cohomology. The construction of crystalline cohomology (as described by P. Berthelot [Cohomologie cristalline des schémas de caractéristique p>0, Lecture Notes in Math., 407, Springer, Berlin, 1974; MR0384804 (52 #5676)]) is complicated and involves some of the finer aspects of topos theory. When X is smooth and proper over a perfect field k of characteristic p, dimX 2, S. Bloch [Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 187–268 (1978); MR0488288 (81j:14011)] associated to X a complex C⋅X (the complex of typical curves on K-theory whose hypercohomology is isomorphic to H∗(X/W). Bloch proved that the first hypercohomology spectral sequence (called the slope spectral sequence) Epq1=Hq(X,CpX)⇒H∗(X/W) degenerates at E1 after tensoring with K, the fraction field of W(k), and that the resulting filtration on H∗(X/W)⊗K is the slope filtration. Furthermore, using this theory, Bloch was able to relate crystalline cohomology to Serre's Witt vector cohomology and to other characteristic p cohomology theories, such as the de Rham cohomology and the flat cohomology. Deligne found the link between the work of Lubkin and that of Bloch, and sketched an elemeńtary construction of the de Rham-Witt complex WΩ⋅X for general X smooth and proper over k. WΩ⋅X is isomorphic to C⋅X whenever the latter is defined. Again, there is a slope spectral sequence Hq(X,WΩpX)⇒H∗(X/W) with the same properties as Bloch's. Furthermore, WΩ0X is isomorphic to Serre's WOX, so that after twenty years, the hope of Serre has been realized (in a modified form, since the slope spectral sequence, unlike the somewhat analogous Hodge-de Rham spectral sequence Epq1=Hq(X,ΩpX)⇒H∗DR(X,C) in characteristic zero, does not degenerate, and in fact, H2(WOX) is not in general finitely generated over W(k)). The article under review works out Deligne's construction of WΩ⋅X in detail and gives a number of applications. The paper starts from scratch, and can be read independently of Bloch's (although, as the author notes, many of the proofs are paraphrases of those of Bloch). No use of topos theory or of K-theory is made, and except in the sections leading up to the comparison with crystalline cohomology, no prior knowledge of crystalline cohomology is necessary. Section 0 of the paper consists of preliminaries on Witt vectors, the usual de Rham complex, and the Cartier operator. In Section 0.2.2, a thorough discussion is given of the higher cycles and boundaries ZnΩiX and BnΩiX. Recall that the Cartier operator C is defined only on the subsheaf of closed i-forms Z1ΩiX⊂ΩiX, and the kernel of C is the sheaf of boundaries B1ΩiX⊂Z1ΩiX. Roughly speaking, the higher cycles ZnΩiX are the largest subsheaf of ΩiX on which Cn is defined, and BnΩiX⊂ZnΩiX is the kernel of Cn. ZnΩiX and BnΩiX can also be characterized in terms of p-adic divisibility properties of liftings, as is shown in Section 0.2.3. Section I is devoted to the definition and the local study of the de Rham-Witt pro-complex W⋅Ω⋅X and the de Rham-Witt complex WΩ⋅X=lim←WnΩ⋅X. One wishes to extend the pro-system of rings WnOX with the usual operators F (Frobenius) and V (Verschiebung) to a pro-system of differential graded algebras WnΩiX together with operators F:WnΩiX→Wn−1ΩiX and V:WnΩiX→Wn−1ΩiX satisfying (1) FV=VF=p, FdV=d, Fdx−=x−p−1dx− (x∈OX,x−=(x,0,0,⋯)); (2) Fx⋅Fy=F(xy), xVy=V(Fx⋅y); (3) V(xdy)=Vx⋅dVy; (4) dx−⋅Vy=V(x−p−1dx−⋅y). The author first proves in Section I.1 that it is possible to construct a "universal VDR pro-complex'' W⋅Ω⋅X of differential graded algebras together with an operator V satisfying (3) and (4), which in degree zero is the pro-system W⋅OX. This is quite elementary. For each n, WnΩ⋅X is a quotient of the usual de Rham complex of WnOX, and is a quasicoherent sheaf of WnOX modules. The main difficulty is the construction of F. In Section I.2, he shows that WnΩ⋅X is isomorphic to a pro-complex obtained from a canonical filtration on an explicitly constructed "complex of integral forms'' in the case where X=SpecFp[T1,⋯,Tr]=(Gra)Fp. (The author actually does this for (Gra×Gsm)Fp, but this extension is only needed for the comparison to Bloch's theory.) F has an easy definition on the complex of integral forms, and one extends to the general case by étale localization. (This explains the use of the pro-complex W⋅Ω⋅X since this commutes with étale localization, while WΩ⋅X does not.) Section I.2 also contains the important result (2.8) that on WΩiX, F−−=piF, where F−− is the endomorphism induced by the Frobenius morphism. From now on, X is smooth over a perfect field. k. Section I.3 studies various exact sequences associated to W and its canonical filtration, which is defined by FilnWΩ⋅X=Ker(WΩ⋅X→WnΩ⋅X) for n≥1. The most important result is (3.9), which shows that the nth associated graded piece fits into two exact sequences 0→Fn+1∗ΩiX/BnΩiX→VngrnWΩiX→Fn+1∗Ωi−1X/ZnΩi−1X→00→Fn+1∗Ωi−1X/Zn+1Ωi−1X→dVngrnWΩiX→Fn+1∗ΩiX/Bn+1Ω⋅X→0. This implies that WnΩiX is of finite type over WnOX, which is important in the proof of the finiteness theorem. This section also contains the proof that WΩ⋅X is free from p-torsion (3.5), and the quasi-isomorphism WΩ⋅X/pWΩ⋅X→Ω⋅X (3.16), which gives a new proof that de Rham cohomology is "crystalline cohomology modp'', and therefore pathologies in de Rham cohomology are explained by p-torsion in crystalline cohomology. Section I.3F gives an exact sequence relating flat cohomology and WΩ⋅X which is needed to define the crystalline Chern class of a line bundle and to extend J. S. Milne's flat duality theorem [Ann. Sci. École Norm. Sup. (4) 9 (1976), 171–201; MR0460331 (57 #325)] to all characteristics. (The proof of (3.21) contains an easily corrected minor error.) Section I.4 gives the relation of WΩ⋅X to Lubkin's theory by proving that WΩ⋅X is isomorphic to a completed quotient of Ω⋅WOX/W(k). Section I.5 gives the isomorphism between WΩ⋅X and Bloch's C⋅X. These two sections are not used in the rest of the paper. Section II is devoted to the global theory, that is, to the study of the cohomology of WΩ⋅X when X is smooth and proper over a perfect field. In Section II.1, the isomorphism between H∗(WΩ⋅X) and H∗cris(X/W) is proved. (Only smoothness of X is used.) In Section II.2, it is proved that H∗(WnΩ⋅X) is a Wn(k) module of finite length, and that RΓ(WΩ⋅X) is a perfect complex of W(k)-modules of perfect amplitude [0,2n] (n=dimX). This gives a new proof of the finite generation over W(k) of Hi(X/W). (The author appeals to an appendix in the book by Berthelot and A. Ogus [Notes on crystalline cohomology, Princeton Univ. Press, Princeton, N.J., 1978; MR0491705 (58 #10908)] for part of this proof. Ogus has informed the reviewer that since we are working over W(k), this can be simplified considerably.) The finiteness modulo torsion Hj(X,WΩiX) is also proved in this section (2.13). In Section II.3, the slope spectral sequence Epq1=Hq(X,WΩpX)⇒H∗(X/W) is studied, and the de Rham-Witt version of Bloch's theorem on the degeneration modulo torsion at E1 is proved. The main idea is to use the fact that F−−=piF in degree i, and then to apply a slope argument. Some more precise degeneration theorems are also proved in this section, notably the theorem of N. O. Nygaard that d1:H2(WOX)→H2(WΩ1X) is the only nonvanishing differential in the slope spectral sequence of a surface. In Section II.4, the author uses the de Rham-Witt complex to prove Katz's conjecture that the Newton polygon lies above the Hodge polygon, under the hypothesis that Hj(X,WΩiX) is torsion-free for all i, j. Under this hypothesis, he also shows that the Newton polygon meets the Hodge polygon in each interval where the latter has constant slope. (This was observed by Bloch.) Nygaard ["A p-adic proof of the Rudakov-Šafarevič theorem'', to appear] has recently given a simple proof using WΩ⋅X of a generalized version of the Katz conjecture. Section II.5 applies the exact sequences of Section I.3 to give relationships among the étale cohomology H∗ét(X,Z/pn), the flat cohomology H∗(X,μpn), and the crystalline cohomology H∗(X,WnΩ⋅X). This is used to prove the inequality ρ≤b2−2h, where ρ=rank NS(X), b2=rankH2(X/W) and h=rankH2(WOX)⊗K. (If the formal Brauer group Br(X) exists, then h is the height of the p-divisible part of Br(X). Therefore this result is a generalization of an inequality prove