The Stacks project

Lemma 71.2.11. 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

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

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

Then $\varphi $ is injective.

Proof. The assumptions imply that $\text{WeakAss}(\mathop{\mathrm{Ker}}(\varphi )) = \emptyset $ and hence $\mathop{\mathrm{Ker}}(\varphi ) = 0$ by Lemma 71.2.5. $\square$

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