Lemma 30.19.2. Let $S = \mathop{\mathrm{Spec}}(A)$ with $A$ a Noetherian ring. Let $f : X \to S$ be a proper morphism. 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 Proposition 30.19.1. Namely, by Lemmas 30.4.5 and 30.4.6 we know that $R^ if_*\mathcal{F}$ is the quasi-coherent sheaf associated to the $A$-module $H^ i(X, \mathcal{F})$ and by Lemma 30.9.1 this is a coherent sheaf if and only if $H^ i(X, \mathcal{F})$ is an $A$-module of finite type. $\square$

There are also:

• 2 comment(s) on Section 30.19: Higher direct images of coherent sheaves

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).