The Stacks project

Lemma 15.23.10. Let $R$ be a Noetherian local ring. Let $M$, $N$ be finite $R$-modules.

  1. If $N$ has depth $\geq 1$, then $\mathop{\mathrm{Hom}}\nolimits _ R(M, N)$ has depth $\geq 1$.

  2. If $N$ has depth $\geq 2$, then $\mathop{\mathrm{Hom}}\nolimits _ R(M, N)$ has depth $\geq 2$.

Proof. Choose a presentation $R^{\oplus m} \to R^{\oplus n} \to M \to 0$. Dualizing we get an exact sequence

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _ R(M, N) \to N^{\oplus n} \to N' \to 0 \]

with $N' = \mathop{\mathrm{Im}}(N^{\oplus n} \to N^{\oplus m})$. A submodule of a module with depth $\geq 1$ has depth $\geq 1$; this follows immediately from the definition. Thus part (1) is clear. For (2) note that here the assumption and the previous remark implies $N'$ has depth $\geq 1$. The module $N^{\oplus n}$ has depth $\geq 2$. From Algebra, Lemma 10.72.6 we conclude $\mathop{\mathrm{Hom}}\nolimits _ R(M, N)$ has depth $\geq 2$. $\square$

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