Timeline for What is the "strongest" non-local property of a ring/module that is true of all localizations at maximal ideals?
Current License: CC BY-SA 3.0
13 events
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| Oct 24, 2011 at 20:39 | answer | added | Qing Liu | timeline score: 3 | |
| Oct 24, 2011 at 13:56 | comment | added | Chuck | @Georges @Martin I have edited to clarify - did it in a rush so apologies if it still sounds muddled. Thanks for all your comments and responses. | |
| Oct 24, 2011 at 13:55 | history | edited | Chuck | CC BY-SA 3.0 |
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| Oct 24, 2011 at 8:49 | comment | added | Martin Brandenburg | @Chuch: My suggestion asks for a stronger counterexample as the answers so far. Please clarify. | |
| Oct 24, 2011 at 8:38 | comment | added | Georges Elencwajg | Dear Chuck, I think your formulation is slightly ambiguous. My interpretation is that you want a property that holds for the local rings of $A$ at maximal ideals, but does not hold for some (all?) localizations at non-maximal primes. Since the other two answers interpret your question as finding a property that holds at *all primes but not true for the ring itself, it might be useful if you merged your two requirements in grey boxes into a statement that we can all agree upon. Thanks in advance and sorry for giving you some work: your interesting question deserves a crystal-clear formulation. | |
| Oct 24, 2011 at 4:39 | answer | added | Paul Balmer | timeline score: 9 | |
| Oct 24, 2011 at 2:45 | answer | added | Sándor Kovács | timeline score: 6 | |
| Oct 23, 2011 at 23:44 | comment | added | Sándor Kovács | @Martin: this is a pretty widely used notion of a property being local. The idea is that if your property is open and local in this sense, then it is local in your sense. A lot of the properties for which this interpretation of local is used is a singularity condition, such as being regular, Cohen-Macaulay, Gorenstein, $S_n$, etc. are inherently open, so the two interpretations of local agrees for them. | |
| Oct 23, 2011 at 21:35 | comment | added | Tommaso Centeleghe | A standard example is the ideal class group of a Dedekind domain $R$, which measures the failure of freeness for projective rank one modules over $R$. Locally the group is trivial. | |
| Oct 23, 2011 at 14:50 | comment | added | Chuck | @Martin Yes, i.e. there is a prime ideal for which P doesn't hold at $A_p$ even though it holds for all maximal $m$ at $A_m$ | |
| Oct 23, 2011 at 12:55 | answer | added | Georges Elencwajg | timeline score: 10 | |
| Oct 23, 2011 at 7:47 | comment | added | Martin Brandenburg | I think the more natural and common definition is: $P$ is local if it can be tested locally with respect to the Zariski topology. This means that $A$ has $P$ iff there is a partition of unity $f_1,...,f_n$ such that each $A_{f_i}$ has $P$. Anyway, your question is interesting. So you're asking for a property such that "$A_m$ has P for all maximal ideals $m$ iff $A$ has P", but it is not true for prime ideals? | |
| Oct 22, 2011 at 22:21 | history | asked | Chuck | CC BY-SA 3.0 |