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A note on p-adic étale cohomology in the semi-stable reduction case. (English) Zbl 0619.14013
This paper studies p-adic étale cohomology of varieties over a p-adic field (i.e. a complete discrete valuation field of mixed characteristics (0,p)) under the assumption that the varieties have semi-stable reduction. This assumption is rather general because every variety is conjectured to admit a semi-stable model after enlarging the base field. Results of S. Bloch and K. Kato [Publ. Math., Inst. Hautes Étud. Sci. 63, 107-152 (1986; Zbl 0613.14017)] are generalized to the semi-stable reduction case (they considered the good reduction case): The structure of the sheaf of p-adic vanishing cycles are described in terms of the ”modified differential modules”. (As the special fiber may be singular, the usual differential modules do not work well.) And a global result is obtained under the ”ordinary” assumption. Namely the p-adic étale cohomology groups of varieties with ”ordinary semi-stable reduction” has a filtration whose subquotients are unramified representations of the absolute Galois group of the base field with some Tate twist.

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
14C99 Cycles and subschemes
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