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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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