Cohomology of locally closed semi-algebraic subsets

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• 作者：Florent Martin (1)
  
• 关键词：14F20 ; 14G22
• 刊名：manuscripta mathematica
• 出版年：2014
• 出版时间：July 2014
• 年：2014
• 卷：144
• 期：3-4
• 页码：373-400
• 全文大小：327 KB
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• 作者单位：Florent Martin (1)
  
  1. Laboratoire Paul Painlevé, Université de Lille 1, Cité scientifique, 59655, Villeneuve d’Ascq, France
  
• ISSN：1432-1785

文摘

Let k be a non-Archimedean field, let ?/em> 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 ?/em>-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.