Timeline for divisorial ideals
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 8, 2019 at 4:08 | answer | added | mzafrullah | timeline score: 4 | |
| S Feb 24, 2016 at 17:22 | history | suggested | user26857 |
edited tags
|
|
| Feb 24, 2016 at 17:14 | review | Suggested edits | |||
| S Feb 24, 2016 at 17:22 | |||||
| Sep 16, 2014 at 9:07 | answer | added | deleted | timeline score: 1 | |
| Feb 2, 2013 at 13:25 | answer | added | user9072 | timeline score: 4 | |
| Dec 22, 2012 at 14:57 | comment | added | user9072 | If you want to ask whether such an ideal exists for every domain then you would need to exclude those domains where all ideals are divisorial. Leaving the question: For every domain possessing a nondivisorial ideal, does there exist an ideal I such that there is an ideal J strictly between I and I^v? Do you want to ask this? | |
| Dec 22, 2012 at 14:48 | comment | added | user9072 | Following the clarficantion in the answer: it is still not completely clear what you want to ask. There certainly are domains where this is the case. For example consider the example I gave with the ideal (x^2, y) the div ideal generated by it is still the full domain and you have (x,y) in between. Or a more conceptual argument: there is the notion of domains that fulfil ACC on div ideals (called Mori domains). Not all Mori domains are noethrian. So in such a domain an infinite ascending chain of usual ideals when taking the resp divisorial ideal will collapse to a finite one. | |
| Dec 22, 2012 at 2:45 | comment | added | Nazer Halimi | Thanks to every one. I meant here; Is there a non-divisorial ideal $I$ such that there exists and ideal $J$ properly located between $I$ and $I^{\nu}$? | |
| Dec 12, 2012 at 18:55 | comment | added | user9072 | Since now I hear somebody already saying I should not be too nit-picky since 'of course' it was meant that I is not divisorial (as opposed some general ideal, so only not necessarily divisorial), let me add that the answer is still essentially trivially no, since in general not all maximal ideals are divisorial and the divsiorial ideal generate by them is thus the full domain and by the very definition there cannot be some ideal between them. For a concrete example I think (X,Y) in K[X,Y] should work. | |
| Dec 12, 2012 at 18:26 | comment | added | user9072 | And since the defintion appears to be little known let me add: every divisorial ideal is also a usual ideal. Some ideal are also divisorial, and for certain dowains all are divisorial. | |
| Dec 12, 2012 at 18:23 | comment | added | user9072 | @David White: no, but neither did I. What I said in somewhat more detail: If $I$ is a divisorial ideal, then $I=I^v$. So clearly there cannot be an ideal between them. "Let ... Then... " Is something I read as "Is this true for all ..." How do you read it? In addition if the domain is Dedekind every ideal is divisisorial so it would be never true for this domain. So, could you please explain me what the question here is? Since for the only interpretation I can see the answer is trivially no. | |
| Dec 12, 2012 at 15:12 | comment | added | David White | Did he ever say $I$ was divisorial? Anyway, here is a link with the necessary definitions, which I would have provided if I was the OP to remind people of the appropriate terminology: encyclopediaofmath.org/index.php/Divisorial_ideal | |
| Dec 12, 2012 at 13:27 | comment | added | user9072 | If I itself is divisorial they are equal so how should this be possible. Voting to close. | |
| Dec 12, 2012 at 12:27 | history | asked | Nazer Halimi | CC BY-SA 3.0 |