The Stacks project

Lemma 75.7.10. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ locally Noetherian. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module with support proper over $Y$. Then $R^ pf_*\mathcal{F}$ is a coherent $\mathcal{O}_ Y$-module for all $p \geq 0$.

Proof. By Lemma 75.7.7 there exists a closed immersion $i : Z \to X$ with $g = f \circ i : Z \to Y$ proper and $\mathcal{F} = i_*\mathcal{G}$ for some coherent module $\mathcal{G}$ on $Z$. We see that $R^ pg_*\mathcal{G}$ is coherent on $S$ by Cohomology of Spaces, Lemma 69.20.2. On the other hand, $R^ qi_*\mathcal{G} = 0$ for $q > 0$ (Cohomology of Spaces, Lemma 69.12.9). By Cohomology on Sites, Lemma 21.14.7 we get $R^ pf_*\mathcal{F} = R^ pg_*\mathcal{G}$ and the lemma follows. $\square$

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