Lemma 67.20.3. Let $A$ be a Noetherian ring. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism of algebraic spaces. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then $H^ i(X, \mathcal{F})$ is finite $A$-module for all $i \geq 0$.

**Proof.**
This is just the affine case of Lemma 67.20.2. Namely, by Lemma 67.3.1 we know that $R^ if_*\mathcal{F}$ is a quasi-coherent sheaf. Hence it is the quasi-coherent sheaf associated to the $A$-module $\Gamma (\mathop{\mathrm{Spec}}(A), R^ if_*\mathcal{F}) = H^ i(X, \mathcal{F})$. The equality holds by Cohomology on Sites, Lemma 21.14.6 and vanishing of higher cohomology groups of quasi-coherent modules on affine schemes (Cohomology of Schemes, Lemma 30.2.2). By Lemma 67.12.2 we see $R^ if_*\mathcal{F}$ is a coherent sheaf if and only if $H^ i(X, \mathcal{F})$ is an $A$-module of finite type. Hence Lemma 67.20.2 gives us the conclusion.
$\square$

## 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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)