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98k:14025 14F30 14F40

Faltings, Gerd ^{(}1-PRIN^{)}

Crystalline cohomology and p-adic Galois-representations. Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), 25–80, Johns Hopkins Univ. Press, Baltimore, MD, 1989. In this paper, the author proves two main conjectures in the theory of p-adic periods: the de Rham conjecture and the crystalline conjecture. These conjectures have a long history starting from Grothendieck’s work in the 1960s. In 1980 J.-M. Fontaine [Ann. of Math. (2) 115 (1982), no. 3, 529–577; MR 84d:14010] made a precise statement and, with W. Messing, proved it in some cases by using the so-called “syntomic topology” [in Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), 179–207, Contemp. Math., 67, Amer. Math. Soc., Providence, RI, 1987; MR 89g:14009]. The author’s theorem can be described as follows: If X is a smooth proper scheme over the ring of integers of a p-adic field K, we can associate, following Grothendieck, to the generic fiber X_{K} its p-adic etale cohomology H_{ét}^{∗}(X\overline{K}, Q_{p}), which is a finite vector space over Q_{p} endowed with an action of the absolute Galois group Gal(\overline{K}/K), and we associate to the special fiber X_{s} its crystalline cohomology H_{cris}^{∗}(X_{s}/K_{0}), which is a vector space over K_{0}, the field of fractions of the Witt ring with coefficients in the residue field of K; H_{cris}^{∗}(X_{s}/K_{0}) is equipped with an action of Frobenius and there is an isomorphism H_{cris}^{∗}(X_{s}/K_{0}) ⊗ K ≅ H_{dR}^{∗}(X_{K}/K) where H_{dR}^{∗}(X_{K}/K) is the de Rham cohomology. Finally, let B_{dR} be the field of p-adic periods and B_{cris} the sub-ring of B_{dR} consisting of p-adic periods “with good reduction”. B_{dR} is a K-algebra endowed with an action of Gal(\overline{K}/K) and a filtration, and B_{cris} is a K_{0}-algebra stable under Gal(\overline{K}/K) and endowed with an action of Frobenius. Then we have an isomorphism:

H_{cris}^{∗}(X_{s}/K_{0}) ⊗_{K_{0}} B_{cris} ≅ H_{ét}^{∗}(X\overline{K}, Q_{p}) ⊗_{Q_{p}} B_{cris}

compatible with actions of the Galois group Gal(\overline{K}/K) and Frobenius, and the Hodge filtrations when we extend this isomorphism to B_{dR}. Moreover each cohomology group (crystalline or etale) determines the other by a concrete recipe. For the proof, the author uses his theory of “almost etale extensions” introduced in his paper on the Hodge-Tate conjecture [J. Amer. Math. Soc. 1 (1988), no. 1, 255– 299; MR 89g:14008] and a relative version of the Fontaine-Laffaille theory. Recently T. Tsuji extended this comparison theorem to the semi-stable case.

{For the entire collection see MR 98d:00013.} Abdellah Mokrane (F-PARIS13-GL)