The Stacks project

Lemma 15.23.7. Let $R \to R'$ be a flat homomorphism of Noetherian domains. If $M$ is a finite reflexive $R$-module, then $M \otimes _ R R'$ is a finite reflexive $R'$-module.

Proof. Choose a short exact sequence $0 \to M \to F \to N \to 0$ with $F$ finite free and $N$ torsion free, see Lemma 15.23.6. Since $R \to R'$ is flat we obtain a short exact sequence $0 \to M \otimes _ R R' \to F \otimes _ R R' \to N \otimes _ R R' \to 0$ with $F \otimes _ R R'$ finite free and $N \otimes _ R R'$ torsion free (Lemma 15.22.4). Thus $M \otimes _ R R'$ is reflexive by Lemma 15.23.6. $\square$

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