The Stacks project

Lemma 15.13.1. Let $(R, I)$ be a henselian pair. Let $\overline{P}$ be a finite projective $R/I$-module. Then there exists a finite projective $R$-module $P$ such that $P/IP \cong \overline{P}$.

Proof. This follows from the fact that we can lift the finite projective $R/I$-module $\overline{P}$ to a finite projective module $P'$ over some $R'$ ├ętale over $R$ with $R/I = R'/IR'$, see Lemma 15.9.11. Then, since $(R, I)$ is a henselian pair, the ├ętale ring map $R \to R'$ has a section $\tau : R' \to R$ (Lemma 15.11.6). Setting $P = P' \otimes _{R', \tau } R$ finishes the proof. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 15.13: Lifting and henselian pairs

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Your email address will not be published.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 0D4A. Beware of the difference between the letter 'O' and the digit '0'.