Lemma 31.2.10. 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 that for every $x \in X$ at least one of the following happens

1. $\mathcal{F}_ x \to \mathcal{G}_ x$ is injective, or

2. $x \not\in \text{Ass}(\mathcal{F})$.

Then $\varphi$ is injective.

Proof. The assumptions imply that $\text{Ass}(\mathop{\mathrm{Ker}}(\varphi )) = \emptyset$ and hence $\mathop{\mathrm{Ker}}(\varphi ) = 0$ by Lemma 31.2.6. $\square$

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