Timeline for Colimits of schemes
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jun 1, 2022 at 1:11 | history | edited | Kevin Buzzard | CC BY-SA 4.0 |
fix typo
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| Dec 29, 2009 at 4:17 | comment | added | Tyler Lawson | I think he's asking for an example of a diagram of schemes where there is no categorical colimit, and not for an example where there is no categorical colimit that looks "correct" geometrically. E.g. if E is an elliptic curve over the complex numbers with a point x of infinite order, then translation by x induces an action of Z on E, and I believe the categorical colimit exists but it is Spec(C), because there are not many translation-invariant functions. So the idea is to construct an example where there are not too few functions out that the colimit exists in some trivial way. | |
| Dec 28, 2009 at 23:29 | comment | added | Kevin Buzzard | "that the naive approach does not work, does not prove anything." Of course it doesn't. But it does prove that I misunderstood what you were asking. Let me try again then: just go and read Mumford's "Geometric Invariant Theory" and see his careful explanation of why the quotient isn't a scheme. Will this do? If not, what are you asking? I'm trying my best ;-) | |
| Dec 28, 2009 at 20:14 | comment | added | Martin Brandenburg | again: we have a full subcategory C of D and a diagram F in C, whose colimit in D does not lie in C. this is no reason that the colimit of F in C does not exist. of course, this situation is likely to be an example, but one has to prove it. | |
| Dec 28, 2009 at 19:03 | comment | added | Martin Brandenburg | that the naive approach does not work, does not prove anything. in fact, there are categorical constructions which are in some sense unexpected. in this case, I want to understand a prove of a counterexample. | |
| Dec 28, 2009 at 17:54 | history | answered | Kevin Buzzard | CC BY-SA 2.5 |