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p-adic étale cohomology. (English) Zbl 0584.14009
Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. I: Arithmetic, Prog. Math. 35, 13-26 (1983).
[For the entire collection see Zbl 0518.00004.]
A ”p-adic field” K is, generalizing $${\mathbb{Q}}_ p$$, defined as the quotient field of a henselian discrete valuation ring with perfect residue field k of characteristic p. Let $$\bar K$$ be the algebraic closure of K and $${\mathcal C}$$ the completion of $$\bar K.$$ Let V be a smooth complete variety over K, $$\bar V$$ be obtained by base change to $$\bar K,$$ Y be obtained from V by reduction to k and $$B^ n\subset \Omega^ n_ Y$$ the sheaf of locally exact Kähler n-forms on Y. If for all q and n, $$H^ q(Y,B^ n)=(0)$$, Y is said to be ordinary. In that case, Tate’s conjecture on the structure of the étale cohomology $$H_{et}(\bar V,{\mathcal C})$$ holds. Other results are stated towards the non-ordinary case as well as some new conjectures. A paper with full proofs by the author and Kato shall appear.
Reviewer: J.H.de Boer

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F20 Étale and other Grothendieck topologies and (co)homologies