The Stacks project

Lemma 15.70.2. Let $R$ be a ring. Let $\varphi : M \to N$ be an $R$-module map. If $\varphi $ factors through a projective module and $M$ is a finite $R$-module, then $\varphi $ factors through a finite projective module.

Proof. By Lemma 15.70.1 we can factor $\varphi = \tau \circ \sigma $ where the target of $\sigma $ is $\bigoplus _{i \in I} R$ for some set $I$. Choose generators $x_1, \ldots , x_ n$ for $M$. Write $\sigma (x_ j) = (a_{ji})_{i \in I}$. For each $j$ only a finite number of $a_{ij}$ are nonzero. Hence $\sigma $ has image contained in a finite free $R$-module and we conclude. $\square$

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Your email address will not be published.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 0G91. Beware of the difference between the letter 'O' and the digit '0'.