The Stacks project

Lemma 30.16.3. Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a locally projective morphism. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then $R^ if_*\mathcal{F}$ is a coherent $\mathcal{O}_ S$-module for all $i \geq 0$.

Proof. We first remark that a locally projective morphism is proper (Morphisms, Lemma 29.43.5) and hence of finite type. In particular $X$ is locally Noetherian (Morphisms, Lemma 29.15.6) and hence the statement makes sense. Moreover, by Lemma 30.4.5 the sheaves $R^ pf_*\mathcal{F}$ are quasi-coherent.

Having said this the statement is local on $S$ (for example by Cohomology, Lemma 20.7.4). Hence we may assume $S = \mathop{\mathrm{Spec}}(R)$ is the spectrum of a Noetherian ring, and $X$ is a closed subscheme of $\mathbf{P}^ n_ R$ for some $n$, see Morphisms, Lemma 29.43.4. In this case, the sheaves $R^ pf_*\mathcal{F}$ are the quasi-coherent sheaves associated to the $R$-modules $H^ p(X, \mathcal{F})$, see Lemma 30.4.6. Hence it suffices to show that $R$-modules $H^ p(X, \mathcal{F})$ are finite $R$-modules (Lemma 30.9.1). This follows from Lemma 30.16.1 (because the restriction of $\mathcal{O}_{\mathbf{P}^ n_ R}(1)$ to $X$ is ample on $X$). $\square$

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