The Stacks project

Definition 10.39.1. Let $R$ be a ring.

  1. An $R$-module $M$ is called flat if whenever $N_1 \to N_2 \to N_3$ is an exact sequence of $R$-modules the sequence $M \otimes _ R N_1 \to M \otimes _ R N_2 \to M \otimes _ R N_3$ is exact as well.

  2. An $R$-module $M$ is called faithfully flat if the complex of $R$-modules $N_1 \to N_2 \to N_3$ is exact if and only if the sequence $M \otimes _ R N_1 \to M \otimes _ R N_2 \to M \otimes _ R N_3$ is exact.

  3. A ring map $R \to S$ is called flat if $S$ is flat as an $R$-module.

  4. A ring map $R \to S$ is called faithfully flat if $S$ is faithfully flat as an $R$-module.

Comments (0)

There are also:

  • 2 comment(s) on Section 10.39: Flat modules and flat ring maps

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00HB. Beware of the difference between the letter 'O' and the digit '0'.



View Definition 10.39.1 as pdf