The Stacks project

Lemma 10.155.10. Let $R \to S$ be a local map of local rings. Choose separable algebraic closures $R/\mathfrak m_ R \subset \kappa _1^{sep}$ and $S/\mathfrak m_ S \subset \kappa _2^{sep}$. Let $R \to R^{sh}$ and $S \to S^{sh}$ be the corresponding strict henselizations. Given any commutative diagram

\[ \xymatrix{ \kappa _1^{sep} \ar[r]_{\phi } & \kappa _2^{sep} \\ R/\mathfrak m_ R \ar[r]^{\varphi } \ar[u] & S/\mathfrak m_ S \ar[u] } \]

There exists a unique local ring map $R^{sh} \to S^{sh}$ fitting into the commutative diagram

\[ \xymatrix{ R^{sh} \ar[r]_ f & S^{sh} \\ R \ar[u] \ar[r] & S \ar[u] } \]

and inducing $\phi $ on the residue fields of $R^{sh}$ and $S^{sh}$.

Proof. Follows immediately from Lemma 10.154.6. $\square$


Comments (0)

There are also:

  • 8 comment(s) on Section 10.155: Henselization and strict henselization

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Your email address will not be published.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 04GU. Beware of the difference between the letter 'O' and the digit '0'.