Tame and strongly \'etale cohomology of curves

Abstract

For a curve $C$ over a perfect field $k$ of characteristic $p > 0$ we study the tame cohomology of $X = \textit{Spa}(C,k)$ introduced in arXiv:1801.04776. We prove that the tame cohomology groups of $X$ with $p$-torsion coefficients satisfy cohomological purity (which is not true in full generality for the \'etale cohomology). Using purity we show Poincar\'e duality for the tame cohomology of $X$ with $p$-torsion coefficients.