Lemma 70.14.2 (0856)—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 70.14.2 (cite)

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Lemma 70.14.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ . Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Consider the sheaf of $\mathcal{O}_ X$ -modules $\mathcal{F}'$ which associates to every object $U$ of $X_{\acute{e}tale}$ the module

$\mathcal{F}'(U) = \{ s \in \mathcal{F}(U) \mid \mathcal{I}s = 0\}$

Assume $\mathcal{I}$ is of finite type. Then

$\mathcal{F}'$ is a quasi-coherent sheaf of $\mathcal{O}_ X$ -modules,
for affine $U$ in $X_{\acute{e}tale}$ we have $\mathcal{F}'(U) = \{ s \in \mathcal{F}(U) \mid \mathcal{I}(U)s = 0\}$ , and
$\mathcal{F}'_ x = \{ s \in \mathcal{F}_ x \mid \mathcal{I}_ x s = 0\}$ .

Proof. It is clear that the rule defining $\mathcal{F}'$ gives a subsheaf of $\mathcal{F}$ . Hence we may work étale locally on $X$ to verify the other statements. Thus the lemma reduces to the case of schemes which is Properties, Lemma 28.24.2. $\square$

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