The adic tame site (Q2039994)

Summary: For every adic space \(\mathscr{X}\) we construct a site \(\mathscr{X}_t\), the tame site of \(\mathscr{X}\). 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 \(\mathrm{Spa}(X,S)\).