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.

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

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

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