Lemma 59.80.7. Let $X$ be a normal integral affine scheme with separably closed function field. Let $Z \subset X$ be a closed subscheme. For any finite abelian group $M$ we have $H^1_{\acute{e}tale}(Z, \underline{M}) = 0$.
Proof. By Cohomology on Sites, Lemma 21.4.3 an element of $H^1_{\acute{e}tale}(Z, \underline{M})$ corresponds to a $\underline{M}$-torsor $\mathcal{F}$ on $Z_{\acute{e}tale}$. Such a torsor is clearly a finite locally constant sheaf. Hence $\mathcal{F}$ is representable by a scheme $V$ finite étale over $Z$, Lemma 59.64.4. Of course $V \to Z$ is surjective as a torsor is locally trivial. Since $V \to Z$ has a section by Lemma 59.80.6 we are done. $\square$
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