Timeline for Colimits of schemes
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
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| Jul 14, 2020 at 17:42 | answer | added | David Benjamin Lim | timeline score: 2 | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Nov 12, 2011 at 11:20 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
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| May 8, 2010 at 23:12 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| May 8, 2010 at 23:12 | vote | accept | Martin Brandenburg | ||
| May 8, 2010 at 21:55 | answer | added | Anton Geraschenko | timeline score: 37 | |
| Apr 24, 2010 at 20:25 | comment | added | BCnrd | As another clarification, if $R \rightrightarrows U$ is an etale equivalence relation in schemes s.t. the alg space $X = U/R$ is not a scheme, for it to be an "example" one must prove there's no "initial map" from $X$ to schemes (i.e., map $\pi:X \rightarrow S$, or equivalently map $U \rightarrow S$ inducing same composites back to $R$, to a scheme $S$ which is initial among all such maps). But just as for "quotients" of schemes, one needs properties beyond "categorical" for it to be useful. So this is an exercise in pathology. Not that there's anything wrong with that... | |
| Apr 24, 2010 at 18:20 | comment | added | Martin Brandenburg | I've made bad experiences with bounties for a difficult question with many answers, which are rather long comments instead of answers. after one week, the answer with the most votes becomes the answer and this cannot be reversed. | |
| Apr 24, 2010 at 15:23 | comment | added | S. Carnahan♦ | Perhaps you should offer a bounty. | |
| Apr 24, 2010 at 9:07 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Jan 17, 2010 at 2:41 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Dec 29, 2009 at 15:49 | comment | added | Martin Brandenburg | there is a gap, li does not prove that X/W = X -> Y' is an isomorphism, which may be very hard: how do get a map into the quotient? I don't know what he means with "(locally, point by point," and can't find "Remark 4.2". | |
| Dec 29, 2009 at 13:38 | comment | added | Kevin Buzzard | If you take the naive approach and just google "no categorical quotient" then you seem to get lots of examples. I just looked through a few and perhaps the one you'll like most is the one in "push-out of schemes" by Li (p538, example 2). I should stress that I did not check anything here though. | |
| Dec 29, 2009 at 12:18 | comment | added | Martin Brandenburg | yes, exactly :-). besides, I'm interested in special cases where the colimit exists (see the comments in emertons answer). | |
| Dec 29, 2009 at 11:44 | comment | added | Kevin Buzzard | I'm going to attempt to clarify your question, because I misunderstood it twice. Your question is not about locally-ringed spaces. Your question is not about the famous free Z/2Z-action on the smooth proper 3-fold. Your question is simply to give, with full proof, an example of a diagram in the category of schemes, with no colimit in the category of schemes. Right? | |
| Dec 29, 2009 at 5:58 | answer | added | Evgeny Shinder | timeline score: 1 | |
| Dec 29, 2009 at 3:56 | answer | added | Emerton | timeline score: 24 | |
| Dec 28, 2009 at 18:53 | answer | added | David Zureick-Brown | timeline score: 6 | |
| Dec 28, 2009 at 17:54 | answer | added | Kevin Buzzard | timeline score: 14 | |
| Dec 28, 2009 at 16:13 | history | asked | Martin Brandenburg | CC BY-SA 2.5 |