Cohomology of locally closed semi-algebraic subsets |
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Authors: | Florent Martin |
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Institution: | 1. Laboratoire Paul Painlevé, Université de Lille 1, Cité scientifique, 59655, Villeneuve d’Ascq, France
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Abstract: | Let k be a non-Archimedean field, let ? 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 ?-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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