The Stacks project

Lemma 59.80.3. Let $S$ be an affine scheme such that (1) all points are closed, and (2) all residue fields are separably algebraically closed. Then for any abelian sheaf $\mathcal{F}$ on $S_{\acute{e}tale}$ we have $H^ i(S_{\acute{e}tale}, \mathcal{F}) = 0$ for $i > 0$.

Proof. Condition (1) implies that the underlying topological space of $S$ is profinite, see Algebra, Lemma 10.26.5. Thus the higher cohomology groups of an abelian sheaf on the topological space $S$ (i.e., Zariski cohomology) is trivial, see Cohomology, Lemma 20.22.3. The local rings are strictly henselian by Algebra, Lemma 10.153.10. Thus étale cohomology of $S$ is computed by Zariski cohomology by Lemma 59.80.1 and the proof is done. $\square$

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