Lemma 31.2.4 (05AE)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.2: Associated points
Lemma 31.2.4 (cite)

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Lemma 31.2.4. Let $X$ be a scheme. Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of quasi-coherent sheaves on $X$. Then $\text{Ass}(\mathcal{F}_2) \subset \text{Ass}(\mathcal{F}_1) \cup \text{Ass}(\mathcal{F}_3)$ and $\text{Ass}(\mathcal{F}_1) \subset \text{Ass}(\mathcal{F}_2)$.

Proof. For every point $x \in X$ the sequence of stalks $0 \to \mathcal{F}_{1, x} \to \mathcal{F}_{2, x} \to \mathcal{F}_{3, x} \to 0$ is a short exact sequence of $\mathcal{O}_{X, x}$-modules. Hence the lemma follows from Algebra, Lemma 10.63.3. $\square$

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