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Duality in the cohomology of crystalline local systems. (English) Zbl 0917.14010

The paper under review studies the “finite” cohomology of smooth proper schemes over a \(p\)-adic discrete valuation ring. The coefficients are elements in (the appropriate version of) Fontaine’s category \({\mathcal M}{\mathcal F}\), and the cohomology is defined by extensions. For example for constant coefficients one obtains the syntomic topology of Fontaine’s sheaves. Associated to the coefficients are \(p\)-adic étale sheaves on the general fibre. There is a spectral sequence starting with the finite cohomology on the base of the étale cohomology of these sheaves on the geometric generic fibre, and this spectral sequence degenerates by degree reasons. It follows that also the usual Leray spectral sequence in étale cohomology must degenerate. Finally there are cup-products and Poincaré-duality.
Reviewer: G.Faltings (Bonn)

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
13F30 Valuation rings
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