The adic tame site

• Main Author: Hübner, Katharina
• Format: Journal Article
• Language: English
• Published: 15.01.2018
• Subjects: Mathematics - Algebraic Geometry
• DOI: 10.48550/arxiv.1801.04776

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 \'etale 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 $\textit{Spa}(X,S)$.