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

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

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

Then $\varphi$ is an isomorphism.

Proof. This is a special case of Lemma 15.23.13. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).