Lemma 30.10.2 (01Y9)—The Stacks project

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Part 2: Schemes
Chapter 30: Cohomology of Schemes
Section 30.10: Coherent sheaves on Noetherian schemes
Lemma 30.10.2 (cite)

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Lemma 30.10.2. Let $X$ be a Noetherian scheme. 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 subscheme $Z \subset X$ . 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. This follows immediately from Algebra, Lemma 10.62.4 because $X$ has a finite covering by spectra of Noetherian rings. $\square$

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