The Stacks project

Lemma 59.80.5. Let $f : X \to Y$ be a morphism of schemes where $X$ is an integral normal scheme with separably closed function field. Then $R^ qf_*\underline{M} = 0$ for $q > 0$ and any abelian group $M$.

Proof. Recall that $R^ qf_*\underline{M}$ is the sheaf associated to the presheaf $V \mapsto H^ q_{\acute{e}tale}(V \times _ Y X, M)$ on $Y_{\acute{e}tale}$, see Lemma 59.51.6. If $V$ is affine, then $V \times _ Y X \to X$ is separated and étale. Hence $V \times _ Y X = \coprod U_ i$ is a disjoint union of open subschemes $U_ i$ of $X$, see Lemma 59.80.4. By Lemma 59.80.1 we see that $H^ q_{\acute{e}tale}(U_ i, M)$ is equal to $H^ q_{Zar}(U_ i, M)$. This vanishes by Cohomology, Lemma 20.20.2. $\square$

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