Cohomology of locally closed semi-algebraic subsets
type: Item
instance of: scholarly article
title: Cohomology of locally closed semi-algebraic subsets
zbMATH Open document ID: 1300.14028
author: Florent Martin
published in: Manuscripta Mathematica
publication date: 2014-08-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1210.4521
review text: Let \(k\) be a non-Archemedean field, let \(X\) be a Hausdorff \(k\)-analytic space in the sense of Berkovich, and let \(S\) be a locally closed subset of \(X\). We call the pair \((X,S)\) a \(k\)-germ. A morphism of \(k\)-germs \(f:(Y,T)\to (X,S)\) is a morphism of \(k\)-analytic spaces \(f:Y\to X\) such that \(f(T)\subset S\). We thus get a category of \(k\)-germs. Define the category \(k\)-\(\mathscr G\)erms as the localization of this category by morphisms of \(k\)-germs \((Y,T)\to (X,S)\) which induce an isomorphism of \(Y\) with some open neighborhood of \(S\) in \(X\). A morphism in \(k\)-\(\mathscr G\)erms \(f\) is called étale if it has a representative \(f:(Y,T)\to (X,S)\) such that \(f:Y\to X\) is étale and \(T=f^{-1}(S)\). A family \(f_i:(Y_i,T_i)\to (Y,T)\) \((i\in I)\) of étale morphisms is called an étale covering if \(\bigcup_i f_i(T_i)=T\). We thus get the étale site \((X,S)_{\mathrm{\'et}}\) of the \(k\)-germ \((X,S)\). Denote by \({\mathbf S}(X,S)_{\mathrm{\'et}}\) the category of sheaves of abelian groups on this site. The right derived functors of the functor \[ \Gamma_c((X,S), -):{\mathbf S}(X,S)_{\mathrm{\'et}}\to \mathrm{Ab}, \quad F\mapsto \{s\in F(X,S)\mid \mathrm{supp}(s) \text{ is compact}\} \] are denoted by \(H_c^q((X,S),-)\). Let \(\mathcal A\) be a \(k\)-affinoid algebra, and let \(X=\mathcal M(\mathcal A)\) be the associated \(k\)-affinoid space. A subset \(S\) of \(X\) is called \textit{semianalytic} if it is a finite Boolean combination of sets of the form \[ \{x\in X\mid |f(x)|\leq \lambda |g(x)|\} \] for some \(f, g\in\mathcal A\) and a positive real number \(\lambda\). Let \(X\) be a compact \(k\)-analytic space. A subset \(S\) of \(X\) is called \textit{\(G\)-semianalytic} if there exists a finite covering \(X=\bigcup_{i=1}^n U_i\) by affinoid domains such that \(S\cap U_i\) is semianalytic in \(U_i\) for each \(i\). Using the cohomological finiteness of affinoid spaces recently proved by \textit{V. G. Berkovich}, ``Vanishing c…
reviewed by: Lei Fu
Mathematics Subject Classification ID: 14G22
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14P10
zbMATH DE Number: 6336314
zbMATH Keywords: analytic space
zbMATH Keywords: \(k\)-germ
zbMATH Keywords: cohomology with compact support
zbMATH Keywords: \(G\)-semianalytic subset
zbMATH Keywords: semi-algebraic subset
MaRDI profile type: Publication
OpenAlex ID: W2122790062
arXiv ID: 1210.4521
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DOI: 10.1007/S00229-013-0650-X
description: scientific article; zbMATH DE number 6336314