Lemma 10.159.4. Let $R$ be a domain. Let $f_1, \ldots , f_ n \in R$ generate the unit ideal. If each domain $R_{f_ i}$ is N-1 then so is $R$. Same for N-2.

**Proof.**
Assume $R_{f_ i}$ is N-2 (or N-1). Let $L$ be a finite extension of the fraction field of $R$ (equal to the fraction field in the N-1 case). Let $S$ be the integral closure of $R$ in $L$. By Lemma 10.35.11 we see that $S_{f_ i}$ is the integral closure of $R_{f_ i}$ in $L$. Hence $S_{f_ i}$ is finite over $R_{f_ i}$ by assumption. Thus $S$ is finite over $R$ by Lemma 10.22.2.
$\square$

## 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Your email address will not be published.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.

## Comments (0)

There are also: