Lemma 31.2.11 (0AVM)—The Stacks project

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Part 2: Schemes
Chapter 31: Divisors
Section 31.2: Associated points
Lemma 31.2.11 (cite)

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Lemma 31.2.11. Let $X$ be a locally Noetherian scheme. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of quasi-coherent $\mathcal{O}_ X$-modules. Assume $\mathcal{F}$ is coherent and that for every $x \in X$ one of the following happens

$\mathcal{F}_ x \to \mathcal{G}_ x$ is an isomorphism, or
$\text{depth}(\mathcal{F}_ x) \geq 2$ and $x \not\in \text{Ass}(\mathcal{G})$.

Then $\varphi $ is an isomorphism.

Proof. This is a translation of More on Algebra, Lemma 15.23.13 into the language of schemes. $\square$

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