The Stacks project

Lemma 30.20.2. Given a morphism of schemes $f : X \to Y$, a quasi-coherent sheaf $\mathcal{F}$ on $X$, and a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ Y$. Assume $Y$ locally Noetherian, $f$ proper, and $\mathcal{F}$ coherent. Then

\[ \mathcal{M} = \bigoplus \nolimits _{n \geq 0} R^ pf_*(\mathcal{I}^ n\mathcal{F}) \]

is a graded $\mathcal{A} = \bigoplus _{n \geq 0} \mathcal{I}^ n$-module which is quasi-coherent and of finite type.

Proof. The statement is local on $Y$, hence this reduces to the case where $Y$ is affine. In the affine case the result follows from Lemma 30.20.1. Details omitted. $\square$

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