Tame and strongly \'etale cohomology of curves

For a curve $C$ over a perfect field $k$ of characteristic $p>0$ we study the tame cohomology of $X=\mathit{\text{Spa}}(C,k)$ 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.