Syntomic cohomology and \(p\)-adic étale cohomology
This article is a complement to the paper ``\(p\)-adic periods and \(p\)-adic étale cohomology'' by \textit{J.-M. Fontaine} and \textit{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 \textit{S. Bloch} and \textit{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.
scientific article; zbMATH DE number 570298