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Exposé VI: Semi-stable reduction and \(p\)-adic étale cohomology. (English) Zbl 0847.14009
Fontaine, Jean-Marc (ed.), Périodes \(p\)-adiques. Séminaire de Bures-sur-Yvette, France, 1988. Paris: Société Mathématique de France, Astérisque. 223, 269-293 (1994).
Let \(X\) be a proper smooth scheme with semi-stable reduction over \(A\), a complete discrete valuation ring of unequal characteristic \((0,p)\) with perfect residue field \(k\) and fraction field \(K\). Let \(V\) be the \(m\)-th étale cohomology of the geometric generic fiber of \(X\) with coefficients in \(\mathbb{Q}_p\) and \(D\) a \((K_0, \varphi, N)\)-structure \((K_0\) is the field of fractions of the Witt vectors on \(k)\) constructed by O. Hyodo and the author [in: Périodes \(p\)-adiques, Sém. Bures-sur-Ivette 1988, Astérisque 223, 221-268 (1994)] on the \(m\)-th de Rham cohomology group of the generic fiber of \(X\). The paper under review contains a proof, for \(p > 2 \dim (X/A) + 1\), of a conjecture of J.-M. Fontaine and U. Jannsen: There exists a canonical isomorphism compatible with monodromy, Frobenius operators and Galois action \(B_{\text{st}} \otimes_{\mathbb{Q}_p} V = B_{\text{st}} \otimes_{K_0} D\), with \(B_{\text{st}}\) the ring of “semi-stable \(p\)-adic periods” introduced by J.-M. Fontaine [in: Algebraic geometry, Proc. Jap.-Fr. Conf., Tokyo/Kyoto 1982, Lect. Notes Math. 1016, 86-108 (1983; Zbl 0596.14015)] and endowed with a monodromy operator \(N\) and a Frobenius \(\varphi\).
After some preliminaries, the author gives a cohomological interpretation of the subring \(B_{\text{st}}^+\) of \(B_{\text{st}}\) with its operators \(N, \varphi\). The main tool for doing this is logarithmic crystalline cohomology [cf. O. Hyodo and the author (op. cit.)], which is also used to give a interpretation of the kernel of \(N\) acting on \(B_{\text{st}} \otimes_{K_0} D\) via a kind of Künneth formula. – With this interpretation, the second key point in the proof of the conjecture is the construction of a certain “syntomic complex” \(s_n^{\log} (r)\) using logarithmic (crystalline) techniques. This is a complex of differential forms with logarithmic poles which is announced (theorem 5.4) to compute, at least for \(0 \leq r \leq p - 1\), the sheaf of étale \(p\)-adic vanishing cycles with values in \(\mathbb{Z}/p^n (r)\). This generalizes a construction-proof of J.-M. Fontaine and W. Messing [in: Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata 1985, Contemp. Math. 67, 179-207 (1987; Zbl 0632.14016)].
A corollary of the proof of the above conjecture is the existence of the Hodge-Tate decomposition for \(V\).
For the entire collection see [Zbl 0802.00019].

14F30 \(p\)-adic cohomology, crystalline cohomology