Lemma 31.2.6 (05AG)—The Stacks project

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Part 2: Schemes
Chapter 31: Divisors
Section 31.2: Associated points
Lemma 31.2.6 (cite)

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Lemma 31.2.6. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Then

$\mathcal{F} = 0 \Leftrightarrow \text{Ass}(\mathcal{F}) = \emptyset .$

Proof. If $\mathcal{F} = 0$ , then $\text{Ass}(\mathcal{F}) = \emptyset$ by definition. Conversely, if $\text{Ass}(\mathcal{F}) = \emptyset$ , then $\mathcal{F} = 0$ by Algebra, Lemma 10.63.7. To translate from schemes to algebra, restrict to any affine and use Lemma 31.2.2. $\square$

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