The Stacks project

Lemma 69.13.2. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals corresponding to a closed subspace $Z \subset X$. Then there is some $n \geq 0$ such that $\mathcal{I}^ n\mathcal{F} = 0$ if and only if $\text{Supp}(\mathcal{F}) \subset Z$ (set theoretically).

Proof. Choose an affine scheme $U$ and a surjective ├ętale morphism $U \to X$ (see Properties of Spaces, Lemma 66.6.3). Then $U$ is a Noetherian scheme (by Morphisms of Spaces, Lemma 67.23.5). Note that $\mathcal{I}^ n\mathcal{F}|_ U = 0$ if and only if $\mathcal{I}^ n\mathcal{F} = 0$ and similarly for the condition on the support. Hence the result follows from the case of schemes, see Cohomology of Schemes, Lemma 30.10.2. $\square$

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