Lemma 10.161.4 (032H)—The Stacks project

Processing math: 100%

Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.161: Japanese rings
Lemma 10.161.4 (cite)

previous lemma
next lemma

numberstags

history
statistics
3 tags refer to this tag

Lemma 10.161.4. 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 10.36.11 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 10.23.2. $\square$

Comments (0)

There are also:

7 comment(s) on Section 10.161: Japanese rings

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).

Comments (0)

Post a comment