Timeline for Does every nontrivial sheaf of rings have a maximal ideal?
Current License: CC BY-SA 2.5
7 events
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| Nov 2, 2018 at 3:06 | comment | added | Ingo Blechschmidt | I was recently shocked to learn that, if Zorn's lemma holds in the metatheory, it also does so internally to any localic topos (not only toposes of sheaves over spaces which are close to being discrete). This is in the Elephant. However, you're right in that the axiom of choice does not descend in this way, and maximal ideals are only really useful in the presence of the law of excluded middle (for instance the proof that maximal ideals are prime requires this). The axiom of choice is equivalent to Zorn's lemma + law of excluded middle. | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Aug 3, 2010 at 13:15 | comment | added | Peter Arndt | Cool, I hereby sign up to join the project when I'm done with my current duties ;-) | |
| Aug 2, 2010 at 18:38 | comment | added | Martin Brandenburg | Yes I rushed to the library and made a copy of the article; the introduction is very promising. It does not bother me if these ideas are 40 years old and no big theory has been developed out of that. Then I'll do it. ;-) | |
| Aug 2, 2010 at 15:51 | comment | added | Peter Arndt | Ok, before you devote too much time to this I should warn you that these ideas are 40 years old but haven't been pursued very far. So what you will get out of your reading will not be a bunch of concrete theorems on sheaves of rings, but a general idea for producing them. One may wonder why this hasn't been happily exploited and while one answer may be that there are just few people with appropriate background and interests, another one is probably that it's just darn difficult! Anyway, it's beautiful mathematics!! I suggest to start with Mulvey! | |
| Aug 2, 2010 at 13:43 | comment | added | Martin Brandenburg | Thank you very much for this overview! It is exactly what I was looking for. I will start to read the papers and books you cited :). | |
| Aug 2, 2010 at 12:10 | history | answered | Peter Arndt | CC BY-SA 2.5 |