Lemma 69.20.3. 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 69.20.2. Namely, by Lemma 69.3.1 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 69.12.2 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 69.20.2 gives us the conclusion. \square
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