The Stacks project

Lemma 68.4.1. Let $S$ be a scheme. Let $f : X \to Y$ be an integral (for example finite) morphism of algebraic spaces. Then $f_* : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ is an exact functor and $R^ pf_* = 0$ for $p > 0$.

Proof. By Properties of Spaces, Lemma 65.18.12 we may compute the higher direct images on an étale cover of $Y$. Hence we may assume $Y$ is a scheme. This implies that $X$ is a scheme (Morphisms of Spaces, Lemma 66.45.3). In this case we may apply Étale Cohomology, Lemma 59.43.5. For the finite case the reader may wish to consult the less technical Étale Cohomology, Proposition 59.55.2. $\square$


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