Timeline for Naive question on adelic groups
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Aug 18, 2013 at 1:30 | comment | added | David Roberts♦ | @user36938 I wouldn't have searched for 'adelic topology', since I know next to nothing about this, and didn't know that it was a thing. But thanks for pointing it out. | |
| Aug 17, 2013 at 19:41 | comment | added | user36938 | @David: As with everything else in life, Google already provides the link to a complete answer (did you try "adelic topology" and see what you get on the first page?). That's why I was surprised the question arose here in the first place. | |
| Aug 17, 2013 at 0:04 | comment | added | David Roberts♦ | @user36938 - would you mind expanding your comment into an answer? This makes it linkable, and won't get missed in a wash of comments. | |
| Aug 16, 2013 at 22:55 | comment | added | Yemon Choi | I just rolled back the latest (well-intentioned!) edit - I think that originally David's question was not asking for "intuition" but it was a "soft question" in the sense that he was deliberately not formulating a "sharply delineated, yes/no question", but something inviting more open-ended answers. | |
| Aug 16, 2013 at 22:54 | history | rollback | Yemon Choi |
Rollback to Revision 2
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| Aug 16, 2013 at 22:43 | history | edited | Ricardo Andrade |
mild nitpicking about tags: replaced 'soft-question' with 'intuition' which seems more appropriate, if not entirely adequate
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| Aug 16, 2013 at 7:05 | comment | added | David Roberts♦ | Thanks all. I guess the wikipedia article needs editing, then. | |
| Aug 16, 2013 at 6:51 | history | edited | Yemon Choi |
edited tags
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| Aug 16, 2013 at 4:37 | comment | added | Yemon Choi | Also, with tongue mostly but not entirely in cheek, I'd quibble with your last sentence: the fact multiplication is merely separately continuous is not so much annoying as fascinating. (Google for "dual Banach algebras" for a sample of the horrors and fun that can arise) | |
| Aug 16, 2013 at 4:32 | answer | added | Yemon Choi | timeline score: 8 | |
| Aug 16, 2013 at 3:45 | comment | added | user36938 | This limit construction coincides with the more familiar one in the affine case using the crutch of a closed immersion into an affine space. So there are no surprises if one is careful about the definitions. | |
| Aug 16, 2013 at 3:43 | comment | added | user36938 | Let $X$ be a separated finite type scheme over a global field $K$ with adele ring $A$. Choose a finite set $S_0$ of places of $K$ containing the archimedean ones so that $X$ extends to a separated flat $O_{K,S_0}$-scheme $X_0$ of finite type. Then $X(A)$ is the direct limit of the sets $X_0(A_K^S)$ topologized as direct products $\prod_{v\in S} X(K_v) \times \prod_{v\not\in S} X_0(O_v)$ for increasing $S$ containing $S_0$. The transition maps are open and the resulting locally compact Hausdorff topology is independent of $S_0$ and $X_0$. It is functorial in $X$ and respects fiber products. QED | |
| Aug 16, 2013 at 3:26 | comment | added | Will Sawin | If one does get a topological group the argument goes: Commutative linear group operations are continuous, so addition and multiplication are continuous, so polynomial maps between affine spaces are continuous, so algebraic maps between algebraic varieties are continuous, so algebraic group operations are continuous. I don't see any problem with this, but maybe one of the steps fails for subtle reasons? | |
| Aug 16, 2013 at 3:19 | comment | added | Pete L. Clark | It is news to me that one does not actually get a topological group in this way. Of course the article does not say this, but only suggests it. Certainly in the case of commutative linear groups one has a group topology: one does Fourier analysis on it after all. So I think the first question needs to be whether and in what circumstances the adelic topology is in fact not a group topology. | |
| Aug 16, 2013 at 1:16 | history | asked | David Roberts♦ | CC BY-SA 3.0 |