Timeline for Under what hypotheses are schematic fixed points of a flat deformation themselves flat?
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| Oct 25, 2010 at 21:22 | vote | accept | Ben Webster♦ | ||
| Oct 25, 2010 at 14:25 | answer | added | Angelo | timeline score: 8 | |
| Oct 25, 2010 at 12:26 | comment | added | BCnrd | Dear Angelo: Good point. So perhaps one should "remember" a shadow of connectedness of tori in the form that the resulting finite group action preserves generic points in fibers...though maybe there's another obstruction to flatness lurking around. | |
| Oct 25, 2010 at 8:58 | comment | added | Angelo | This is false for finite groups, though. Consider two copies of the affine line glued at the origin, with an action of a cyclic group of order 2 switching the two copies, mapping to the affine line. | |
| Oct 25, 2010 at 6:06 | comment | added | Dave Anderson | There's an old theorem of Iversen that, for any field $k$, for a smooth (& separated) $k$-scheme $X$ and a $k$-torus $T$, the fixed scheme $X^T$ is smooth. I'd wager that one could replace "smooth" with "Cohen-Macaulay" in this statement -- though the proof would have to be somewhat different from Iversen's. Anyone know a counterexample? (Since in your motivating situation, I think you know that the dimension of fibers of $X^T\to S$ is constant, knowing $X^T$ is C-M gives flatness over a smooth base $S$.) | |
| Oct 25, 2010 at 0:11 | comment | added | BCnrd | Note also that by the "valuative criterion" for flatness, to handle the case when (noetherian) $S$ is reduced it suffices to handle the case when $S$ is the spectrum of a dvr. | |
| Oct 25, 2010 at 0:09 | comment | added | BCnrd | Dear David: Good thought. Consider noetherian scheme $S$ and flat $S$-scheme $X$ of f. type equipped with action by $S$-torus $T$. Then $X^T$ exists as closed subscheme of $X$, & formation commutes with base change. To check flatness, WLOG $S$ is local. Pick prime $\ell$ invertible on $S$, and observe that the collection of finite etale $S$-subgroups $G_n=T[\ell^n]$ is relatively sch. dense in $T$. Thus, $X^T = X^{G_n}$ for large $n$ by noetherianness. By finite etale base change on $S$, $G_n$ constant. So problem reduces to analogue for action by finite gp of order invertible on $S$. Hmm... | |
| Oct 25, 2010 at 0:01 | comment | added | Ben Webster♦ | The behavior of reductive groups is quite different that of unipotent ones, so the counterexample doesn't worry me so much. After all, the degree 0 part of the ring has to vary flatly, and that's the coordinate ring of the categorical quotient, so nothing too horrible can happen. | |
| Oct 24, 2010 at 23:36 | comment | added | David Treumann | Another thought: maybe one can reduce to the case of a finite group, thinking of a (char zero) torus as a limit of its finite subgroups. | |
| Oct 24, 2010 at 23:35 | comment | added | David Treumann | This is false for general group actions. Think of the additive group acting on P^1 x C with x sending (z,t) to (z + tx, t); the dimension of the fixed points at t = 0 jumps. I haven't been able come up with an example like this for torus actions, so I'm starting to believe it. | |
| Oct 24, 2010 at 22:06 | history | asked | Ben Webster♦ | CC BY-SA 2.5 |