Lemma 10.161.11. A Noetherian domain whose fraction field has characteristic zero is N-1 if and only if it is N-2 (i.e., Japanese).

Proof. This is clear from Lemma 10.161.8 since every field extension in characteristic zero is separable. $\square$

Comment #4965 by Rankeya on

When you say a "Noetherian domain of characteristic 0" do you mean the domain contains the rationals or that its fraction field has characteristic 0? I think this result holds as long as the fraction field of the domain has characteristic 0, so may be it is good to specify this in the statement?

Comment #7448 by Haohao Liu on

From the proof we see that the condition "fraction field has characteristic zero " can be relaxed to "fraction field is perfect".

Comment #7449 by Laurent Moret-Bailly on

@ #7448: True, but if $R$ is a noetherian domain with perfect fraction field $K$ of char. $p>0$, then $R=K$. Otherwise, by Krull-Akizuki, there is a discrete valuation ring $V$ between $R$ and $K$, and a uniformizer of $V$ cannot be a $p$-th power in $K$. (There may be simpler arguments).

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).