Lemma 28.24.4 (07ZN)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 28: Properties of Schemes
Section 28.24: Sections with support in a closed subset
Lemma 28.24.4 (cite)

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Lemma 28.24.4. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{I} \subset \mathcal{O}_ Y$ be a quasi-coherent sheaf of ideals of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of sections annihilated by $f^{-1}\mathcal{I}\mathcal{O}_ X$ . Then $f_*\mathcal{F}' \subset f_*\mathcal{F}$ is the subsheaf of sections annihilated by $\mathcal{I}$ .

Proof. Omitted. (Hint: The assumption that $f$ is quasi-compact and quasi-separated implies that $f_*\mathcal{F}$ is quasi-coherent so that Lemma 28.24.2 applies to $\mathcal{I}$ and $f_*\mathcal{F}$ .) $\square$

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