The adic tame site

TL;DR

This paper introduces the tame site for adic spaces, explores its cohomological properties, compares it with étale cohomology, and proves a purity theorem and homotopy invariance under certain conditions.

Contribution

It constructs the tame site for adic spaces, analyzes its cohomology, and establishes a purity theorem and homotopy invariance assuming resolution of singularities.

Findings

Connection between tame site cohomology and étale cohomology.

Comparison of tame fundamental group with classical tame fundamental group.

Proof of a cohomological purity theorem for regular schemes in characteristic p.

Abstract

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