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Syntomic cohomology and $$p$$-adic étale cohomology. (English) Zbl 0792.14008
This article is a complement to the paper “$$p$$-adic periods and $$p$$-adic étale cohomology” by 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)] concerning the $$p$$-adic étale cohomology of varieties over $$p$$-adic fields. In that paper, the absolute ramification index of the base $$p$$- adic field was assumed to be one in the main results. We are interested in composing the method in the paper cited above and the study of $$p$$- adic vanishing cycles in the paper by S. Bloch and K. Kato [Publ. Math., Inst. Hautes Étud. Sci. 63, 107-152 (1986; Zbl 0613.14017)]. We show that the composition gives, for a smooth proper variety with good reduction over a $$p$$-adic field and whose dimension not too big, fairly short proofs of the Hodge-Tate decomposition and of the crystalline conjecture without the assumption on the absolute ramification index. The Hodge-Tate decomposition and the crystalline conjecture were proved by Faltings without any assumption. The aim of this paper is to show the existence of a different method.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F20 Étale and other Grothendieck topologies and (co)homologies 14G20 Local ground fields in algebraic geometry
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