The adic tame site

For every adic space \(Z\) we construct a site \(Z_t\), the tame site of \(Z\). For a scheme \(X\) over a base scheme \(S\) we obtain a tame site by associating with \(X/S\) an adic space \(\textit{Spa}(X,S)\) and considering the tame site \(\textit{Spa}(X,S)_t\). We examine the connection of the co...

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Bibliographic Details
Published inarXiv.org
Main Author Hübner, Katharina
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 31.05.2021
Subjects
Homology
Singularities
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Summary:For every adic space \(Z\) we construct a site \(Z_t\), the tame site of \(Z\). For a scheme \(X\) over a base scheme \(S\) we obtain a tame site by associating with \(X/S\) an adic space \(\textit{Spa}(X,S)\) and considering the tame site \(\textit{Spa}(X,S)_t\). We examine the connection of the cohomology of the tame site with étale cohomology and compare its fundamental group with the conventional tame fundamental group. Finally, assuming resolution of singularities, for a regular scheme \(X\) over a base scheme \(S\) of characteristic \(p > 0\) we prove a cohomological purity theorem for the constant sheaf \(\mathbb{Z}/p\mathbb{Z}\) on \(\textit{Spa}(X,S)_t\). As a corollary we obtain homotopy invariance for the tame cohomology groups of \(\textit{Spa}(X,S)\).
ISSN:2331-8422