Timeline for Noetherian rings of infinite Krull dimension?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Mar 28, 2011 at 15:33 | answer | added | Chris | timeline score: 16 | |
| Apr 12, 2010 at 23:33 | comment | added | Jesse Burke | @Harry: Oftentimes one wants to prove a result for all noetherian rings and the infinite dimensional case is a sticking point. For example Lemma 3.1.5 in Brian Conrad's notes on Grothendieck duality has a very nice (and fairly involved) proof due to Gabber. The lemma is almost trivial when the ring has finite Krull dimension. So having examples at hand could be very helpful. | |
| Apr 12, 2010 at 9:14 | comment | added | Harry Gindi | The entire point of coming up with pathological examples is to show that certain conditions are necessary. Presumably the only motivation to come up with more pathological examples to illustrate necessity of a specific condition is if they are simpler to construct, or if you've come up with some sort of "construction scheme" that gives you a whole class of pathological examples. | |
| Apr 12, 2010 at 9:04 | answer | added | Bugs Bunny | timeline score: 10 | |
| Apr 12, 2010 at 7:20 | comment | added | Pete L. Clark | The question could be improved by giving it more focus: one could presumably modify Nagata's construction in various small ways -- e.g. by starting with $\mathbb{Z}$ instead of a field -- but you are probably not interested in such examples. So, what kind of features are you looking for in other examples? E.g. "Does there exist a Noetherian ring of infinite Krull dimension such that...X?" By filling in X, you ask a binary question, which all of a sudden mathematicians are interested in answering. Just asking "What's out there?" doesn't have the same appeal. | |
| Apr 12, 2010 at 5:23 | comment | added | moby | Such Noetherian rings are pathological by nature, so artificial examples are probably the way to go... working first from geometry, then to the related algebraic treatment? | |
| Apr 12, 2010 at 5:05 | comment | added | Harry Gindi | The fact that Nagata had to come up with a fairly involved construction means that they probably don't come up in practice, so would you like us to look up other artificial examples? | |
| Apr 12, 2010 at 3:22 | history | asked | moby | CC BY-SA 2.5 |