Lemma 75.3.1. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Given an étale morphism V \to Y, set U = V \times _ Y X and denote g : U \to V the projection morphism. Then (Rf_*E)|_ V = Rg_*(E|_ U) for E in D(\mathcal{O}_ X).
Proof. Represent E by a K-injective complex \mathcal{I}^\bullet of \mathcal{O}_ X-modules. Then Rf_*(E) = f_*\mathcal{I}^\bullet and Rg_*(E|_ U) = g_*(\mathcal{I}^\bullet |_ U) by Cohomology on Sites, Lemma 21.20.1. Hence the result follows from Properties of Spaces, Lemma 66.26.2. \square
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