## 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].

### MSC:

 14F30 $$p$$-adic cohomology, crystalline cohomology

### Citations:

Zbl 0596.14015; Zbl 0632.14016