Tame and strongly étale cohomology of curves

Abstract

For a curve C over a perfect field k of characteristic p > 0 we study the tame cohomology of $$\mathcal{C}$$ = Spa(C, k) introduced in [Hüb21]. We prove that the tame cohomology groups of $$\mathcal{C}$$ with p-torsion coefficients satisfy cohomological purity (which is not true in full generality for the étale cohomology). Using purity we show Poincaré duality for the tame cohomology of $$\mathcal{C}$$ with p-torsion coefficients.