The Stacks project

Lemma 15.115.10. Let $A$ be a local ring annihilated by a prime $p$ whose maximal ideal is nilpotent. There exists a ring map $\sigma : \kappa _ A \to A$ which is a section to the residue map $A \to \kappa _ A$. If $A \to A'$ is a local homomorphism of local rings, then we can choose a similar ring map $\sigma ' : \kappa _{A'} \to A'$ compatible with $\sigma $ provided that the extension $\kappa _{A'}/\kappa _ A$ is separable.

Proof. Separable extensions are formally smooth by Algebra, Proposition 10.158.9. Thus the existence of $\sigma $ follows from the fact that $\mathbf{F}_ p \to \kappa _ A$ is separable. Similarly for the existence of $\sigma '$ compatible with $\sigma $. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 15.115: Eliminating ramification

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Your email address will not be published.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 09EZ. Beware of the difference between the letter 'O' and the digit '0'.