Lemma 15.23.14. Let $R$ be a Noetherian domain. Let $\varphi : M \to N$ be a map of $R$-modules. Assume $M$ is finite, $N$ is torsion free, and that for every prime $\mathfrak p$ of $R$ one of the following happens

$M_\mathfrak p \to N_\mathfrak p$ is an isomorphism, or

$\text{depth}(M_\mathfrak p) \geq 2$.

Then $\varphi $ is an isomorphism.

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