The Stacks project

Lemma 109.12.3. Let $R$ be a countable ring. Then $R$ is coherent if and only if $R^\mathbf {N}$ is a flat $R$-module.

Proof. If $R$ is coherent, then $R^\mathbf {N}$ is a flat module by Algebra, Proposition 10.90.6. Assume $R^\mathbf {N}$ is flat. Let $I \subset R$ be a finitely generated ideal. To prove the lemma we show that $I$ is finitely presented as an $R$-module. Namely, the map $I \otimes _ R R^\mathbf {N} \to R^\mathbf {N}$ is injective as $R^\mathbf {N}$ is flat and its image is $I^\mathbf {N}$ by Lemma 109.12.1. Thus we conclude by Lemma 109.12.2. $\square$

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