The Stacks project

Lemma 15.23.11. Let $R$ be a Noetherian ring. Let $M$, $N$ be finite $R$-modules.

  1. If $N$ has property $(S_1)$, then $\mathop{\mathrm{Hom}}\nolimits _ R(M, N)$ has property $(S_1)$.

  2. If $N$ has property $(S_2)$, then $\mathop{\mathrm{Hom}}\nolimits _ R(M, N)$ has property $(S_2)$.

  3. If $R$ is a domain, $N$ is torsion free and $(S_2)$, then $\mathop{\mathrm{Hom}}\nolimits _ R(M, N)$ is torsion free and has property $(S_2)$.

Proof. Since localizing at primes commutes with taking $\mathop{\mathrm{Hom}}\nolimits _ R$ for finite $R$-modules (Algebra, Lemma 10.71.9) parts (1) and (2) follow immediately from Lemma 15.23.10. Part (3) follows from (2) and Lemma 15.22.12. $\square$

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