The Stacks project

Lemma 10.39.14. Let $R$ be a ring. Let $M$ be an $R$-module. The following are equivalent

  1. $M$ is faithfully flat, and

  2. $M$ is flat and for all $R$-module homomorphisms $\alpha : N \to N'$ we have $\alpha = 0$ if and only if $\alpha \otimes \text{id}_ M = 0$.

Proof. If $M$ is faithfully flat, then $0 \to \mathop{\mathrm{Ker}}(\alpha ) \to N \to N'$ is exact if and only if the same holds after tensoring with $M$. This proves (1) implies (2). For the other, assume (2). Let $N_1 \to N_2 \to N_3$ be a complex, and assume the complex $N_1 \otimes _ R M \to N_2 \otimes _ R M \to N_3\otimes _ R M$ is exact. Take $x \in \mathop{\mathrm{Ker}}(N_2 \to N_3)$, and consider the map $\alpha : R \to N_2/\mathop{\mathrm{Im}}(N_1)$, $r \mapsto rx + \mathop{\mathrm{Im}}(N_1)$. By the exactness of the complex $-\otimes _ R M$ we see that $\alpha \otimes \text{id}_ M$ is zero. By assumption we get that $\alpha $ is zero. Hence $x $ is in the image of $N_1 \to N_2$. $\square$


Comments (4)

Comment #2340 by Federico Scavia on

Typo in the first line of the proof, an is missing in the exact sequence.

Comment #4941 by yogesh on

the exact sequence in the first line of the proof is always exact. maybe it should be N in place of

Comment #4942 by yogesh on

oops, nevermind. disregard my previous comment

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Your email address will not be published.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 00HO. Beware of the difference between the letter 'O' and the digit '0'.