The Stacks project

Lemma 10.12.14. Let $R$ be a ring. Let $M$ and $N$ be $R$-modules.

  1. If $N$ and $M$ are finite, then so is $M \otimes _ R N$.

  2. If $N$ and $M$ are finitely presented, then so is $M \otimes _ R N$.

Proof. Suppose $M$ is finite. Then choose a presentation $0 \to K \to R^{\oplus n} \to M \to 0$. This gives an exact sequence $K \otimes _ R N \to N^{\oplus n} \to M \otimes _ R N \to 0$ by Lemma 10.12.10. We conclude that if $N$ is finite too then $M \otimes _ R N$ is a quotient of a finite module, hence finite, see Lemma 10.5.3. Similarly, if both $N$ and $M$ are finitely presented, then we see that $K$ is finite and that $M \otimes _ R N$ is a quotient of the finitely presented module $N^{\oplus n}$ by a finite module, namely $K \otimes _ R N$, and hence finitely presented, see Lemma 10.5.3. $\square$

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