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An analog of Tate’s conjecture over local and finitely generated fields. (English) Zbl 1068.14502

Summary: 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 \(K,\bar{X} = X \otimes K^a\), where \(K^a\) is an algebraic closure of \(K,X^{\text{an}}\) the non-Archimedean \(K\)-analytic space associated with \(X\), and \(\bar{X}^{\text{an}} = (X \otimes \hat{K^a})^{\text{an}}\), where \(\hat{K^a}\) is the completion of \(K^a\). The main result of the paper states that the cohomology groups of \(\bar{X}^{\text{an}}\) with coefficients in \(\mathbb{Q}_l\) (with compact support or not) coincide with the weight zero part or the “smooth” part of the étale \(l\)-adic cohomology groups of \(\bar{X}\) if \(l \neq p\) or \(l = p\), respectively. This implies that the cohomology groups of \(X^{\text{an}}\) with coefficients in \(\mathbb{Q}_l\) (with compact support or not) coincide with the \(\text{Gal}(K^a/K)\)-invariant part of the étale \(l\)-adic cohomology groups of \(\bar{X}\).

MSC:

14G20 Local ground fields in algebraic geometry
14F30 \(p\)-adic cohomology, crystalline cohomology
14G22 Rigid analytic geometry
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