The Stacks project

Lemma 15.23.2. Let $R$ be a domain. If $M \to M'$ is a map of $R$-modules whose kernel and cokernel are torsion, then the rank of $M$ equals the rank of $M'$.

Proof. Omitted. Hint: the induced map $M \otimes _ R K \to M' \otimes _ R K$ is an isomorphism if $K$ is the fraction field of $R$. $\square$

Comments (0)

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Your email address will not be published.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 0H9V. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 15.23.2 as pdf