The adic tame site

Abstract

For every adic space $\mathcal{X}$ we construct a site ${\mathcal{X}}_t$, the tame site of $\mathcal{X}$. For a scheme $X$ over a base scheme $S$ we obtain a tame site by associating with $X/S$ an adic space $\mathrm{Spa}(X,S)$ and considering the tame site $\mathrm{Spa}(X,S{)}_t$. 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 $\mathrm{Spa}(X,S{)}_t$. As a corollary we obtain homotopy invariance for the tame cohomology groups of $\mathrm{Spa}(X,S)$.