Journal Article

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

Vladimir G. Berkovich

in International Mathematics Research Notices

Volume 2000, issue 13, pages 665-680
Published in print January 2000 | ISSN: 1073-7928
Published online January 2000 | e-ISSN: 1687-0247 | DOI: https://dx.doi.org/10.1155/S1073792800000362

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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 [math], where Ka is an algebraic closure of K,Xan the non-Archimedean K-analytic space associated with X, and [math], where [math] is the completion of Ka. The main result of the paper states that the cohomology groups of [math] 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 [math] if lp or l = p, respectively. This implies that the cohomology groups of Xan with coefficients in Ql (with compact support or not) coincide with the [math]-invariant part of the étale l-adic cohomology groups of [math].

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Subjects: Mathematics