Manuscripta Mathematica

, Volume 144, Issue 3–4, pp 373–400 | Cite as

Cohomology of locally closed semi-algebraic subsets

  • Florent MartinEmail author


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.

Mathematics Subject Classification

14F20 14G22 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Laboratoire Paul PainlevéUniversité de Lille 1Villeneuve d’AscqFrance

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