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Torsion étale and crystalline cohomologies. (English) Zbl 1035.14005
Berthelot, Pierre (ed.) et al., $$p$$-adic cohomology and arithmetic applications (II). Paris: Société Mathématique de France (ISBN 2-85629-117-1/pbk). Astérisque 279, 81-124 (2002).
The article under review is based on two courses of the authors at the Centre Émile Borel of the Insrtitut Henri Poincaré during the “Semestre $$p$$-adique” of 1997. It contains a brief survey of results of Fontaine-Laffaille and Fontaine-Messing concerning the study of $$p$$-torsion étale cohomology of varieties with good reduction over the ring of Witt vectors with coefficients in a perfect field of characteristic $$p > 0$$ [see J.-M. Fontaine and G. Laffaille, Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 547–608 (1982; Zbl 0579.14037); J.-M. Fontaine and W. Messing, Contemp. Math. 67, 179–207 (1987; Zbl 0632.14016)].
The authors then discuss an approach to the study of $$p$$-torsion étale cohomology and crystalline cohomology making use of log-syntomic methods developed in a series of publications by B. Mazur and his followers. In fact, they describe an extension of previous results of J.-M. Fontaine and W. Messing to the semi-stable case following ideas of Ch. Breuil [Ann. Sci. Éc. Norm. Supér., IV. Sér. 31, No. 3, 281–327 (1998; Zbl 0907.14006) and Duke Math. J. 95, No. 3, 523–620 (1998; Zbl 0961.14010)]. In the conclusion, some applications are considered. Among them there are conjectures of J.-M. Fontaine [in: Périodes $$p$$-adiques,Sémin. Bures-sur-Yvette 1988, Exposé III, Astérisque 223, 113–184 (1994; Zbl 0865.14009); (5.4.4)] and Fontaine-Jannsen [see K. Kato, ibid., Exposé VI, Astérisque 223, 269–293 (1994; Zbl 0847.14009); (1.1)]. The authors also raise open questions concerning generalizations to the semi-stable case of known results by G. Faltings, V. A. Abrashkin, and others; they also present a detailed list of references containing 75 items.
For the entire collection see [Zbl 0990.00020].

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 11G25 Varieties over finite and local fields 11S20 Galois theory 14F20 Étale and other Grothendieck topologies and (co)homologies 11G10 Abelian varieties of dimension $$> 1$$ 14F40 de Rham cohomology and algebraic geometry 14G20 Local ground fields in algebraic geometry