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.


14F30 \(p\)-adic cohomology, crystalline cohomology
14F20 Étale and other Grothendieck topologies and (co)homologies
14G20 Local ground fields in algebraic geometry
Full Text: DOI


[1] P. BERTHELOT, Cohomologie cristalline des schemas de caracteristique p 0, Lecture Notes in Math. 407, Springer-Verlag, Berlin, Heidelberg, New York 1974. · Zbl 0298.14012
[2] S. BLOCH AND K. KATO, /?-adic etale cohomology, Publ. Math. IHES. 63 (1986), 107-152 · Zbl 0613.14017 · doi:10.1007/BF02831624
[3] P. BERTHELOT AND A. OGUS, Notes on crystalline cohomology, Princeton University Press, Princeton, 1978. · Zbl 0383.14010
[4] P. BERTHELOT AND A. OGUS, F-isocrystals and de Rham cohomology, I, Invent. Math. 72 (1983), 159-199. · Zbl 0516.14017 · doi:10.1007/BF01389319
[5] G. FALTINGS, p-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), 255-299 JSTOR: · Zbl 0764.14012 · doi:10.2307/1990970
[6] G. FALTINGS, Crystalline cohomology and /?-adic etale cohomology, Algebraic Analysis, Geometr and Number Theory (J. -I. Igusa, ed.), The Johns Hopkins Univ. Press (1989), 25-80. · Zbl 0805.14008
[7] J. -M. FONTAINE AND W. MESSING, p-adic periods and p-adic etale cohomology, Contemporar Math. 67 (1987), 179-209. · Zbl 0632.14016
[8] J. -M. FONTAINE, Sur certains types de representations /?-adiques du groupe de Galois d’un corp local, construction d’un anneau de Barsotti-Tate, Ann. of Math. 115 (1982), 529-577. JSTOR: · Zbl 0544.14016 · doi:10.2307/2007012
[9] J. -M. FONTAINE, Cohomologie de de Rham, cohomologie cristalline et representations/?-adiques, i Algebraic Geometry, Lecture Notes in Math. 1016, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo (1983), 86-108. · Zbl 0596.14015
[10] K. kATO, On /?-adic vanishing cycles (Applications of ideas of Fontaine-Messing), in Algebrai Geometry, Sendai, 1985, Advanced Studies in Pure Math. 10, Kinokuniya Tokyo and North. Holland (1987), 207-251. · Zbl 0645.14009
[11] M. KURIHARA, A note on/7-adic etale cohomology, Proc. Japan Acad. 63 (1987), 275-278 · Zbl 0647.14006 · doi:10.3792/pjaa.63.275
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.