The Stacks project

Lemma 15.105.2. Let $(A, \mathfrak m)$ be a Noetherian local ring. The number of branches of $A$ is the same as the number of branches of $A^\wedge $ if and only if $\sqrt{\mathfrak qA^\wedge }$ is prime for every minimal prime $\mathfrak q \subset A^ h$ of the henselization.

Proof. Follows from Lemma 15.105.1 and the fact that there are only a finite number of branches for both $A$ and $A^\wedge $ by Algebra, Lemma 10.31.6 and the fact that $A^ h$ and $A^\wedge $ are Noetherian (Lemma 15.45.3). $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 15.105: Branches of the completion

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Your email address will not be published.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 0C29. Beware of the difference between the letter 'O' and the digit '0'.