The Stacks project

Proposition 38.13.10. Let $R$ be a domain. Let $R \to S$ be a ring map of finite type. Let $M$ be a finite $S$-module.

  1. If $S$ is flat over $R$, then $S$ is a finitely presented $R$-algebra.

  2. If $M$ is flat as an $R$-module, then $M$ is finitely presented as an $S$-module.

Proof. Part (1) is a special case of Lemma 38.13.9. For Part (2) choose a surjection $R[x_1, \ldots , x_ n] \to S$. By Lemma 38.13.7 we find that $M$ is finitely presented as an $R[x_1, \ldots , x_ n]$-module. We conclude by Algebra, Lemma 10.6.4. $\square$

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