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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)