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International Mathematics Research Notices (2000) 2000:665-680, doi:10.1155/S1073792800000362
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Copyright © 2000 Hindawi Publishing Corporation. All rights reserved.

An analog of Tate's conjecture over local and finitely generated fields

Vladimir G. Berkovich

Let K be a local non-Archimedean field, p the characteristic of the residue field of K,l a prime number different from the characteristic of K,X a separated scheme of finite type over Formula, where Ka is an algebraic closure of K,Xan the non-Archimedean K-analytic space associated with X, and Formula, where Formula is the completion of Ka. The main result of the paper states that the cohomology groups of Formula with coefficients in Ql (with compact support or not) coincide with the weight zero part or the "smooth" part of the étale l-adic cohomology groups of Formula if l != p or l = p, respectively. This implies that the cohomology groups of Xan with coefficients in Ql (with compact support or not) coincide with the Formula-invariant part of the étale l-adic cohomology groups of Formula.


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