Lemma 70.2.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. 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

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

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

Then $\varphi $ is injective.

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