The Stacks project

Lemma 69.12.9. Let $S$ be a scheme. Let $f : X \to Y$ be a finite morphism of algebraic spaces over $S$ with $Y$ locally Noetherian. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume $f$ is finite and $Y$ locally Noetherian. Then $R^ pf_*\mathcal{F} = 0$ for $p > 0$ and $f_*\mathcal{F}$ is coherent.

Proof. Choose a scheme $V$ and a surjective ├ętale morphism $V \to Y$. Then $V \times _ Y X \to V$ is a finite morphism of locally Noetherian schemes. By (69.3.0.1) we reduce to the case of schemes which is Cohomology of Schemes, Lemma 30.9.9. $\square$


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