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
Comments (0)