The Stacks project

Lemma 109.12.2. Let $R$ be a countable ring. Let $M$ be a countable $R$-module. Then $M$ is finitely presented if and only if the canonical map $M \otimes _ R R^\mathbf {N} \to M^\mathbf {N}$ is an isomorphism.

Proof. If $M$ is a finitely presented module, then the map is an isomorphism by Algebra, Proposition 10.89.3. Conversely, assume the map is an isomorphism. By Lemma 109.12.1 the module $M$ is finite. Choose a surjection $R^{\oplus m} \to M$ with kernel $K$. Then $K$ is countable as a submodule of $R^{\oplus m}$. Arguing as in the proof of Algebra, Proposition 10.89.3 we see that $K \otimes _ R R^\mathbf {N} \to K^\mathbf {N}$ is surjective. Hence we conclude that $K$ is a finite $R$-module by Lemma 109.12.1. Thus $M$ is finitely presented. $\square$

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