Lemma 15.23.8 (0AV3)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.23: Reflexive modules
Lemma 15.23.8 (cite)

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Lemma 15.23.8. Let $R$ be a Noetherian domain. Let $M$ be a finite $R$ -module. Let $N$ be a finite reflexive $R$ -module. Then $\mathop{\mathrm{Hom}}\nolimits _ R(M, N)$ is reflexive.

Proof. Choose a presentation $R^{\oplus m} \to R^{\oplus n} \to M \to 0$ . Then we obtain

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

with $N' = \mathop{\mathrm{Im}}(N^{\oplus n} \to N^{\oplus m})$ torsion free. We conclude by Lemma 15.23.5. $\square$

Comments (2)

Comment #4657 by Remy on November 11, 2019 at 12:09

After the second sentence, couldn't you just conclude directly from Tag 15.23.5? (In fact, you don't need to take the image, because exactness on the right would not be needed.)

Comment #4798 by Johan on December 10, 2019 at 10:22

Much better. Thanks Remy! See changes.

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