The Stacks project

Lemma 10.162.7. Let $R$ be a ring. Let $f_1, \ldots , f_ n \in R$ generate the unit ideal.

  1. If each $R_{f_ i}$ is universally Japanese then so is $R$.

  2. If each $R_{f_ i}$ is Nagata then so is $R$.

Proof. Let $\varphi : R \to S$ be a finite type ring map so that $S$ is a domain. Then $\varphi (f_1), \ldots , \varphi (f_ n)$ generate the unit ideal in $S$. Hence if each $S_{f_ i} = S_{\varphi (f_ i)}$ is N-1 then so is $S$, see Lemma 10.161.4. This proves (1).

If each $R_{f_ i}$ is Nagata, then each $R_{f_ i}$ is Noetherian and hence $R$ is Noetherian, see Lemma 10.23.2. And if $\mathfrak p \subset R$ is a prime, then we see each $R_{f_ i}/\mathfrak pR_{f_ i} = (R/\mathfrak p)_{f_ i}$ is N-2 and hence we conclude $R/\mathfrak p$ is N-2 by Lemma 10.161.4. This proves (2). $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 10.162: Nagata 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Your email address will not be published.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 032V. Beware of the difference between the letter 'O' and the digit '0'.