The Stacks project

Lemma 69.14.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset and let $U \subset X$ be the open subspace such that $T \amalg |U| = |X|$. 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 $\varphi : W \to X$ of $X_{\acute{e}tale}$ the module

\[ \mathcal{F}'(W) = \{ s \in \mathcal{F}(W) \mid \text{the support of }s\text{ is contained in }|\varphi |^{-1}(T)\} \]

If $U \to X$ is quasi-compact, then

  1. for $W$ affine there exist a finitely generated ideal $I \subset \mathcal{O}_ X(W)$ such that $|\varphi |^{-1}(T) = V(I)$,

  2. for $W$ and $I$ as in (1) we have $\mathcal{F}'(W) = \{ x \in \mathcal{F}(W) \mid I^ nx = 0 \text{ for some } n\} $,

  3. $\mathcal{F}'$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-modules.

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 28.24.5. $\square$

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