Proposition 38.13.10 (053G)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.13: Flat finite type modules, Part II
Proposition 38.13.10 (cite)

previous lemma
next lemma

numberstags

history
statistics
2 tags refer to this tag

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.

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. $\square$

Comments (0)

There are also:

2 comment(s) on Section 38.13: Flat finite type modules, Part II

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment