Syntomic cohomology and $p$-adic étale cohomology
Copyright policy )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.
- University of Minnesota System United States
- University of Tokyo Japan
- University of Minnesota United States
- Tokyo University of Science Japan
14G20, 14F20, Local ground fields in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology, 14F30, Hodge-Tate decomposition, Étale and other Grothendieck topologies and (co)homologies, \(p\)-adic étale cohomology, crystalline conjecture, 11G25, \(p\)-adic vanishing cycles
14G20, 14F20, Local ground fields in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology, 14F30, Hodge-Tate decomposition, Étale and other Grothendieck topologies and (co)homologies, \(p\)-adic étale cohomology, crystalline conjecture, 11G25, \(p\)-adic vanishing cycles
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