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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