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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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