The Stacks project

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. $\square$


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