On p-adic vanishing cycles. (Application of ideas of Fontaine-Messing). (English) Zbl 0645.14009

Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 207-251 (1987).
[For the entire collection see Zbl 0628.00007.]
For a smooth scheme X over a complete discrete valuation ring of characteristic \(0\) and residue characteristic \(p,\) S. Bloch and the author [Publ. Math., Inst. Hautes Étud. Sci. 63, 107-152 (1986; Zbl 0613.14017)] have defined the sheaves of p-adic vanishing cycles on the reduction Y of X; here ground field extensions to the algebraic closure of the valuation field and of its residue field are intended to be taken.
Ideas of J.-M. Fontaine and W. Messing from p-adic étale cohomology [in Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata/Calif. 1985, Contemp. Math. 67, 179-207 (1987; Zbl 0632.14016)] are invoked to define complexes \({\mathcal S}_ n(r)\) on \(Y_{et}\), the cohomology sheaves of which are shown to be, up to an r- th Tate twist, the sheaves of the vanishing cycles. Under the assumption that Y is of Hodge-Witt, the étale cohomology groups, viewed as p-adic Galois-modules, are obtained from a Tate module [See also Fontaine and Messing’s paper cited above].
Reviewer: J.H.de Boer


14F30 \(p\)-adic cohomology, crystalline cohomology
14C99 Cycles and subschemes
14B12 Local deformation theory, Artin approximation, etc.