The Stacks project

Lemma 10.153.4. Let $(R, \mathfrak m, \kappa )$ be a henselian local ring.

  1. If $R \to S$ is a finite ring map then $S$ is a finite product of henselian local rings each finite over $R$.

  2. If $R \to S$ is a finite ring map and $S$ is local, then $S$ is a henselian local ring and $R \to S$ is a (finite) local ring map.

  3. If $R \to S$ is a finite type ring map, and $\mathfrak q$ is a prime of $S$ lying over $\mathfrak m$ at which $R \to S$ is quasi-finite, then $S_{\mathfrak q}$ is henselian and finite over $R$.

  4. If $R \to S$ is quasi-finite then $S_{\mathfrak q}$ is henselian and finite over $R$ for every prime $\mathfrak q$ lying over $\mathfrak m$.

Proof. Part (2) implies part (1) since $S$ as in part (1) is a finite product of its localizations at the primes lying over $\mathfrak m$ by Lemma 10.153.3 part (10). Part (2) also follows from Lemma 10.153.3 part (10) since any finite $S$-algebra is also a finite $R$-algebra (of course any finite ring map between local rings is local).

Let $R \to S$ and $\mathfrak q$ be as in (3). Write $S = A \times B$ with $A$ finite over $R$ and $B$ not quasi-finite over $R$ at any prime lying over $\mathfrak m$, see Lemma 10.153.3 part (11). Hence $S_\mathfrak q$ is a localization of $A$ at a maximal ideal and we deduce (3) from (1). Part (4) follows from part (3). $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 10.153: Henselian local rings

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Your email address will not be published.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



View Lemma 10.153.4 as pdf