The Stacks project

Lemma 10.160.2. Let $R$ be a Noetherian complete local ring. Any quotient of $R$ is also a Noetherian complete local ring. Given a finite ring map $R \to S$, then $S$ is a product of Noetherian complete local rings.

Proof. The ring $S$ is Noetherian by Lemma 10.31.1. As an $R$-module $S$ is complete by Lemma 10.97.1. Hence $S$ is the product of the completions at its maximal ideals by Lemma 10.97.8. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 10.160: The Cohen structure theorem

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Your email address will not be published.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 0325. Beware of the difference between the letter 'O' and the digit '0'.