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Semistable sheaves and comparison isomorphisms in the semistable case. (English) Zbl 1275.14012

The paper under review considers relations between étale and crystalline locally constant systems. Over a \(p\)-adic field \(K\) with valuation ring \(V\) this has been invented by J. M. Fontaine [Ann. Math. (2) 115, 529–577 (1982; Zbl 0544.14016); Astérisque 223, 59–111, Appendix 103–111 (1994; Zbl 0940.14012)]. Here the authors deal with schemes \(X/V\) which are proper and have a fine log-structure. They define Fontaine’s rings over \(X\) and define a functor from \(p\)-adic étale local systems to crystalline objects. This induces an equivalence of categories between certain étale local systems and certain filtered Frobenius-crystals. This relation persists if we pass to cohomology.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14F30 \(p\)-adic cohomology, crystalline cohomology
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