The Stacks project

Lemma 59.80.8. 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 $H^ q_{\acute{e}tale}(Z, \underline{M}) = 0$ for $q \geq 1$.

Proof. Write $X = \mathop{\mathrm{Spec}}(R)$ and $Z = \mathop{\mathrm{Spec}}(R')$ so that we have a surjection of rings $R \to R'$. All local rings of $R'$ are strictly henselian by Lemma 59.80.4 and Algebra, Lemma 10.156.4. Furthermore, we see that for any $f' \in R'$ there is a surjection $R_ f \to R'_{f'}$ where $f \in R$ is a lift of $f'$. Since $R_ f$ is a normal domain with separably closed fraction field we see that $H^1_{\acute{e}tale}(D(f'), \underline{M}) = 0$ by Lemma 59.80.7. Thus we may apply Lemma 59.80.2 to $Z = \mathop{\mathrm{Spec}}(R')$ to conclude. $\square$

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