Lemma 70.14.7 (085B)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 70: Limits of Algebraic Spaces
Section 70.14: Sections with support in a closed subset
Lemma 70.14.7 (cite)

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Lemma 70.14.7. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$ . Let $T \subset |Y|$ be a closed subset. Assume $|Y| \setminus T$ corresponds to an open subspace $V \subset Y$ such that $V \to Y$ is quasi-compact. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of sections supported on $|f|^{-1}T$ . Then $f_*\mathcal{F}' \subset f_*\mathcal{F}$ is the subsheaf of sections supported on $T$ .

Proof. Omitted. Hints: $|X| \setminus |f|^{-1}T$ is the support of the open subspace $U = f^{-1}V \subset X$ . Since $V \to Y$ is quasi-compact, so is $U \to X$ (by base change). The assumption that $f$ is quasi-compact and quasi-separated implies that $f_*\mathcal{F}$ is quasi-coherent. Hence Lemma 70.14.5 applies to $T$ and $f_*\mathcal{F}$ as well as to $|f|^{-1}T$ and $\mathcal{F}$ . The equality of the given quasi-coherent modules is immediate from the definitions. $\square$

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