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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. $\square$

Comments (4)

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 is a noetherian domain with perfect fraction field of char. , then . Otherwise, by Krull-Akizuki, there is a discrete valuation ring between and , and a uniformizer of cannot be a -th power in . (There may be simpler arguments).

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  • 7 comment(s) on Section 10.161: Japanese rings

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