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The Stacks project

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

  1. \mathcal{F}_ x \to \mathcal{G}_ x is an isomorphism, or

  2. \text{depth}(\mathcal{F}_ x) \geq 2.

Then \varphi is an isomorphism.

Proof. This is a translation of More on Algebra, Lemma 15.23.14 into the language of schemes. \square


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