Timeline for When does dimension behave nicely for quotients of affine algebraic varieties by the action of a group?
Current License: CC BY-SA 3.0
9 events
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| Jul 26, 2012 at 15:38 | history | edited | Phillip Williams | CC BY-SA 3.0 |
added 100 characters in body
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| Jul 25, 2012 at 19:36 | comment | added | Olivier Benoist | Yes Phillip, "categorical quotient" would do the job ! Being a "good quotient" is stronger (roughly, one requires the properties that hold for quotients of affine varieties by a reductive group to hold for all good quotients). If a good quotient exists, then it is the categorical quotient. To be a "geometric quotient" is even stronger : for such quotients, the geometric points of the quotient should be in one-to-one correspondence with the orbits of geometric points of $X$. | |
| Jul 25, 2012 at 16:40 | comment | added | Phillip Williams | Yes Oliver I believe you are probably right. I am not an expert on this stuff. Forgive me for the naive question regarding your terminology: is "good quotient" the same thing as "categorical quotient"? | |
| Jul 25, 2012 at 16:28 | vote | accept | Phillip Williams | ||
| Jul 24, 2012 at 20:44 | comment | added | Olivier Benoist | I believe "geometric quotient" should be replaced by "good quotient" in this question as the natural quotient, given by taking the spectrum of invariant functions, is not always geometric in GIT's sense. | |
| Jul 24, 2012 at 20:40 | answer | added | Olivier Benoist | timeline score: 7 | |
| Jul 24, 2012 at 20:27 | comment | added | Jim Humphreys |
I'm also bothered by the looseness of the formulation (as well as the reason for requiring characteristic 0). What precise notion of "geometric quotient variety" is invoked here? Must a "reductive" group be connected? (For a finite group acting on an affine variety, there is a nice theory which leads to equality of dimensions here, with $\dim G =0$ as an algebraic group.) Also, the question seems open-ended, with an indefinite number of correct answers. The literature on such group actions is wide-ranging and involves GIT too.
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| Jul 24, 2012 at 17:28 | comment | added | Qiaochu Yuan | Certainly at a minimum the action ought to be faithful. Possibly you also want it to be free (otherwise consider the action of $\text{GL}_n$ on $\mathbb{G}_a^n$). | |
| Jul 24, 2012 at 17:21 | history | asked | Phillip Williams | CC BY-SA 3.0 |