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 $X\mathrm{/}S$ an adic space $\mathit{\text{Spa}}(X,S)$ and considering the tame site $\mathit{\text{Spa}}(X,S{)}_t$. 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 $\mathbb{Z}\mathrm{/}p\mathbb{Z}$ on $\mathit{\text{Spa}}(X,S{)}_t$. As a corollary we obtain homotopy invariance for the tame cohomology groups of $\mathit{\text{Spa}}(X,S)$.