The Stacks project

Lemma 30.26.10. Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module with support proper over $S$. Then $R^ pf_*\mathcal{F}$ is a coherent $\mathcal{O}_ S$-module for all $p \geq 0$.

Proof. By Lemma 30.26.7 there exists a closed immersion $i : Z \to X$ and a finite type, quasi-coherent $\mathcal{O}_ Z$-module $\mathcal{G}$ such that (a) $g = f \circ i : Z \to S$ is proper, and (b) $i_*\mathcal{G} = \mathcal{F}$. We see that $R^ pg_*\mathcal{G}$ is coherent on $S$ by Proposition 30.19.1. On the other hand, $R^ qi_*\mathcal{G} = 0$ for $q > 0$ (Lemma 30.9.9). By Cohomology, Lemma 20.13.8 we get $R^ pf_*\mathcal{F} = R^ pg_*\mathcal{G}$ which concludes the proof. $\square$

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