Lemma 31.11.13. Let X be an integral 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, \mathcal{G} is torsion free, 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.
Then \varphi is an isomorphism.
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