Cohomology of locally closed semi-algebraic subsets
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 cycles for formal schemes III'', preprint, \url{http://www.wisdom.weizmann.ac.il/vova/FormIII_2013.pdf (2013)}, the author proves the following result: Suppose \(k\) is algebraic closed and \(X\) is a compact \(k\)-analytic space. For any locally closed \(G\)-semianalytic subset \(S\) of \(X\) and any finite abelian group \(\Lambda\) with cardinal prime to the characteristic of the residue field \(\tilde k\) of \(k\), the groups \(H_c^q((X,S),\Lambda)\) are finite. Let \(\mathcal A\) be a \(k\)-affinoid algebra, \(\mathcal B\) an \(\mathcal A\)-algebra of finite type, and \(\mathcal X=\mathrm{Spec}\, \mathcal B\). A subset \(S\) of \({\mathcal X}^{an}\) is called \textit{semi-algebraic} if its is a Boolean combination of subsets of the form \[ \{x\in{\mathcal X}^{an}\mid |f(x)|\leq \lambda |g(x)|\} \] for some \(f,g\in\mathcal B\) and some positive real number \(\lambda\). Let \(\mathcal X\) be a separated \(\mathcal A\)-scheme of finite type. A subset \(S\) of \({\mathcal X}^{an}\) is called \textit{semi-algebraic} if there exists a finite covering \(\mathcal X=\bigcup_i{\mathcal U}_i\) by affine open subsets \({\mathcal U}_i\) such that \(A\cap {\mathcal U}_i^{an}\) are semi-algebraic in \({\mathcal U}_i^{an}\). This implies that for any affine open subset \(\mathcal U\) of \(\mathcal X\), \(A\cap {\mathcal U}^{an}\) is semi-algebraic in \({\mathcal U}^{an}\). The author proves that if \(k\) is algebraically closed and \(S\) is a locally closed semi-algebraic subset of \(\mathcal X^{an}\), then the groups \(H_c^q((\mathcal X^{an}, S),\Lambda)\) are finite. In general, let \(\widehat{k^a}\) be the completion of the algebraic closure of \(k\), let \(\overline{\mathcal X}={\mathcal X}^{an}{\widehat\otimes}_k\widehat{k^a}\), and let \(\pi:\overline{\mathcal X}\to {\mathcal X}^{an}\) be the projection. If \(U\) is a subset of \({\mathcal X}^{an}\), we set \(\overline U=\pi^{-1}(U)\). The author proves the following results: Let \(U\) be a locally closed semi-algebraic subset of \({\mathcal X}^{an}\), and let \(\ell\) be a prime number distinct from \(\mathrm{char}\, \tilde k\). Then \(H_c^i((\overline{\mathcal X}, \overline U), {\mathbb Q}_\ell)\) are finite dimensional \({\mathbb Q}_\ell\)-vector spaces endowed with a continuous \(\mathrm{Gal}(\bar k/k)\)-action, and \(H_c^i((\overline{\mathcal X}, \overline U), {\mathbb Q}_\ell)=0\) for \(i>2\mathrm{dim}\,\mathcal X\). Let \(V\subset U\) be a semi-algebraic subset open in \(U\), and let \(F=U-V\). Then we have a \(\mathrm{Gal}(\bar k/k)\)-equivariant long exact sequence \[ \cdots\to H_c^i((\overline{\mathcal X}, \overline V), {\mathbb Q}_\ell) \to H_c^i((\overline{\mathcal X}, \overline U), {\mathbb Q}_\ell) \to H_c^i((\overline{\mathcal X}, \overline F), {\mathbb Q}_\ell)\to H_c^{i+1}((\overline{\mathcal X}, \overline V), {\mathbb Q}_\ell)\to\cdots. \] Let \(\mathcal X\) (resp. \(\mathcal Y\)) be a separated \(k\)-scheme of finite type and let \(U\) (resp. \(V\)) be a locally closed semi-algebraic subset of \(\mathcal X\) (resp. \(\mathcal Y\)). Then we have isomorphisms of \(\mathrm{Gal}(\bar k/k)\)-modules \[ \bigoplus_{i+j=n} H_c^i((\overline{\mathcal X^{an}}, \overline U),\mathbb Q_\ell) \otimes H_c^i((\overline{\mathcal Y^{an}}, \overline V),\mathbb Q_\ell)\cong H_c^n((\overline{(\mathcal X\times_k \mathcal Y)^{an}}, \overline {(U\times V)}),\mathbb Q_\ell). \]
scientific article; zbMATH DE number 6336314