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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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