Timeline for Structure on $X(k)$ for separated finite type alg. space $X$, for complete valued $k$.
Current License: CC BY-SA 2.5
15 events
| when toggle format | what | by | license | comment | |
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| Aug 13, 2010 at 15:59 | vote | accept | BCnrd | ||
| Aug 13, 2010 at 15:59 | history | bounty ended | BCnrd | ||
| Aug 9, 2010 at 13:45 | answer | added | Laurent Moret-Bailly | timeline score: 22 | |
| Aug 9, 2010 at 0:27 | history | edited | BCnrd | CC BY-SA 2.5 |
added 498 characters in body; edited title
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| Aug 8, 2010 at 19:04 | comment | added | BCnrd | Tom, I'm pessimistic that this finite group case (even if it works) could be adapted to handle the local compactness aspect of more general equivalence relations, but would be happy to be proved wrong. Anyway, I agree that (in char. 0) the space of interest is a union of images of finitely many closed subsets of $U(k_s^H)$. Then to control local compactness of an image, does it help to make a direct proof that the quotient topological map (hmm, which one?) is an open mapping (using openness of maps on rational points induced by \'etale maps between schemes of finite type)? | |
| Aug 8, 2010 at 16:49 | comment | added | Tom Goodwillie | I'm sticking with the case when $U$ has a finite group $G$ acting on it in a suitable way and $X$ is the quotient -- hoping that any ideas about this case can be adapted to the general case. Does this work in this case? Elements of $X(k)$ are given by pairs $(h,u)$ where $h$ is a homomorphism of (a finite quotient $\Gamma/H$ of) $\Gamma=Gal(k_s/k)$ to $G$ and $x:Spec(k_s)\to U$ is map respecting the $\Gamma$-actions. For every such $h$, this means looking at the fixed points of the finite group $\Gamma/H$ on the locally compact space $U(k_s^H)$. | |
| Aug 8, 2010 at 15:47 | comment | added | BCnrd | Tom, it doesn't seem obvious by hand that the preimage of $X(k)$ in $U(k')$ is closed, so I don't see how you infer local compactness for the quotient topology (along with independence of the choice of etale scheme chart). The Mumford example of relative Pic of $X_0^2 + t X_1^2 = X_2^2$ over $\mathbf{R}[t]$ is an algebraic space that over $\mathbf{C}[t]$ is a scheme, so free action of gp of order 2 fails to have a scheme quotient (in a good sense). Discussed in Chapter 8 of "Neron Models" (over $\mathbf{R}[[t]]$). Contractions give more examples; see Intro to D. Knutson's book on alg. spaces. | |
| Aug 8, 2010 at 15:26 | comment | added | Tom Goodwillie | I guess I was thinking that $X(k)$ is a quotient of its preimage in $U(k_s)$, which preimage lies in $U(k')$ and therefore is locally compact. By the way, I don't "know" anything about algebraic spaces except what I "learned" from Wikipedia last night. What's a key example of one which is not a scheme? Are there examples involving a finite group action on a scheme (I mean an action satisfying a suitable etaleness condition)? | |
| Aug 8, 2010 at 15:07 | comment | added | BCnrd | Tom, that's an interesting observation (even though it doesn't work in positive characteristic). But unfortunately I don't see how to use it to prove that $X(k)$ is locally compact (since going to the bigger field, say $k'/k$, will usually make $U(k')$ which hits points in $X(k_s)$ outside of $X(k)$, suggesting to increase $k'$ even more...). | |
| Aug 8, 2010 at 5:30 | comment | added | Tom Goodwillie | Pursuing that idea, it appears to me that in the characteristic zero case there is a single finite extension $k'$ of $k$ such that every point of $U(k_s)$ which maps to a point of $X(k)$ is defined over $k'$. Because $k$ has only finitely many Galois extensions with a given Galois group. Is that right? | |
| Aug 7, 2010 at 14:02 | history | bounty started | BCnrd | ||
| Aug 5, 2010 at 16:56 | comment | added | BCnrd | Dear JSE: Yes, your "vague idea" (as you call it) is what led to unpleasantly increasing towers. Maybe it can be made to work, but I don't see it. | |
| Aug 5, 2010 at 16:16 | comment | added | JSE | But as for actual content; do you have to go all the way up to k_s? Each point in U(k_s) is defined over an extension k' of bounded degree; I guess I have some vague idea that it would help to work over the union of all the U(k'), which together cover X(k)? But maybe this is what you already did that led you to the increasing towers. | |
| Aug 5, 2010 at 16:07 | comment | added | JSE | I started reading this question and said to myself, "Oh, BCnrd will answer this." Then I got to the end and said "Oh, dear." | |
| Aug 5, 2010 at 13:57 | history | asked | BCnrd | CC BY-SA 2.5 |