Abstract
Let k be a non-Archimedean field, let ℓ be a prime number distinct from the characteristic of the residue field of k. If χ is a separated k-scheme of finite type, Berkovich’s theory of germs allows to define étale ℓ-adic cohomology groups with compact support of locally closed semi-algebraic subsets of χ an. We prove that these vector spaces are finite dimensional continuous representations of the Galois group of k sep/k, and satisfy the usual long exact sequence and Künneth formula. This has been recently used by E. Hrushovski and F. Loeser in a paper about the monodromy of the Milnor fibration. In this statement, the main difficulty is the finiteness result, whose proof relies on a cohomological finiteness result for affinoid spaces, recently proved by V. Berkovich.
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The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 246903 NMNAG.