Timeline for Finite group scheme acting on a scheme such that there is an orbit NOT contained in an open affine.
Current License: CC BY-SA 2.5
7 events
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| Aug 20, 2010 at 14:42 | comment | added | BCnrd | Dear anon: for purpose of quotient questions like you are asking, it isn't necessary to read GIT thoroughly at all (not that there's anything wrong with that...). There is an awe-inspiring theorem of M. Artin that, under mild finiteness hypotheses, the category of algebraic spaces (defined relative to the etale topology) is stable under the formation of quotients by fppf equivalence relations. This is a reason why alg. spaces are a natural setting for such problems (but always nice when the qt is a scheme...). | |
| Aug 20, 2010 at 14:37 | comment | added | BCnrd | To handle qts by free actions of finite flat gp schemes (which I don't believe are handled by Mumford's book) over any base scheme, under the same "orbit in an affine" hypothesis see SGA3, Expose V. To weaken hypothesis on orbits but retaining the freeness condition (appropriately defined), you need to work with algebraic spaces; in fact, the freeness condition can be dropped as well with more input (Keel-Mori theorem on coarse moduli spaces, applied to quotient stacks). The Mumford example to which Kevin alludes is one for which the quotient does exist as an algebraic space. | |
| Aug 20, 2010 at 14:34 | answer | added | anon | timeline score: 4 | |
| Aug 20, 2010 at 14:31 | comment | added | anon | Cool, thanks. I actually had a quick flick through GIT but it didn't seem to give a quick easy criteria for when quotients by finite groups exist. Is this because a such a criteria doesn't exist or just because I haven't given the book the time it deserves? I definitely will read GIT thoroughly one day but at the moment I just want to move on with the problem at hand... | |
| Aug 20, 2010 at 14:30 | answer | added | David E Speyer | timeline score: 5 | |
| Aug 20, 2010 at 14:25 | comment | added | Kevin Buzzard | Mumford's book on geometric invariant theory has an example, the group in question being cyclic of order 2. The same book of Mumford is also the answer to your other question. | |
| Aug 20, 2010 at 14:09 | history | asked | anon | CC BY-SA 2.5 |