Étale cohomology of Henselian rings and cohomology of abstract Riemann surfaces of fields (Q1318035)

In this short paper, we show that for every ring \(A\) which is henselian along an ideal \(I \subseteq A\), every torsion sheaf \(F\) on the étale site \(\text{Spec} (A)_{et}\) and every \(n \in \mathbb{N}\) the restriction homomorphism \(H^ n(\text{Spec} (A)_{et},F) \to H^ n(\text{Spec} (A/I)_{et},F)\) is bijective. In order to prove this, we show that the cohomology of constant torsion sheaves on certain pro-constructible subsets of abstract Riemann surfaces of algebraically closed fields vanishes.