Timeline for On Noetherian and Japanese rings
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Feb 22, 2020 at 21:32 | answer | added | Takumi Murayama | timeline score: 10 | |
| Apr 26, 2012 at 19:25 | comment | added | David White | Wow. For 18 months this question had 0 votes and today it got 5 upvotes after a minor syntax edit. I guess syntax is pretty important 'round here. | |
| Apr 26, 2012 at 13:55 | history | edited | David White | CC BY-SA 3.0 |
Edited syntax
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| Oct 3, 2010 at 19:33 | comment | added | BCnrd | In fact, since your hypothesis is inherited by any domain that is a module-finite ring extension of $R$, to prove an affirmative answer in general it is equivalent to prove in general that your hypothesis implies that the normal locus in Spec($R$) contains a non-empty open set (which actually implies it is open, and conversely such openness is always true in the Japanese case; see 6.13.2--6.13.4 of EGA IV$_2$). Still seems like it may be a useless viewpoint on the question... | |
| Oct 3, 2010 at 17:26 | comment | added | BCnrd | EGA also only defines "Japanese" for noetherian domains. Basic result of Nagata (see EGA IV$_2$, 6.13.6) is that when localizations of a noetherian domain $R$ at all primes are Japanese then integral closure $R'$ in a finite extension $K'/K$ of fraction field is $R$-finite if and only if $R'_r$ is $R_r$-finite for some nonzero $r \in R$. Using EGA IV$_2$, 5.10.17 and 6.13.2--6.13.4, it then follows that your $R$ is Japanese iff each $R$-finite ring extension that is a domain has non-empty open normal locus in Spec. Not sure if it's of any use... | |
| Oct 3, 2010 at 15:32 | comment | added | Guntram | "In commutative algebra, an integral domain A is called an N-1 ring if its integral closure in its quotient field is a finite A module. It is called a Japanese ring (or an N-2 ring) if for every finite extension L of its quotient field K, the integral closure of A in L is a finite A module." - wikipedia | |
| Oct 3, 2010 at 14:31 | history | asked | Jaakko | CC BY-SA 2.5 |