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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