Timeline for Naive question on adelic groups
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 16, 2013 at 22:20 | comment | added | Yemon Choi | @user36938 As a non-algebraist, I am inclined to agree :) | |
| Aug 16, 2013 at 22:19 | history | edited | Yemon Choi | CC BY-SA 3.0 |
added extra detail/link for continuity of inversion
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| Aug 16, 2013 at 22:18 | comment | added | user36938 | Ah, OK, since the MathReview didn't mention inversion I wasn't sure where it was lurking. I agree that once one has understood the continuity for the action it is not unreasonable that much of the work to get at inversion is also done (as in the case of Lie groups). Very good. But I bet that even for a non-algebraist it is easier to go the route of understanding the definition of the adelic topology than to make that the oracle and use Ellis' paper. :) | |
| Aug 16, 2013 at 22:14 | comment | added | Yemon Choi | @user36938 Automatic continuity of inversion is the bit that surprised me most when I learned of Ellis's theorem, although I don't claim it's the hardest part of the result. I've added some more information. | |
| Aug 16, 2013 at 22:08 | comment | added | user36938 | @Yemon: How does the "Ellis oracle" prove that inversion is continuous in the topology (so it is a topological group)? That is the basic reason for the topological complications when working with adelic constructions, namely that in the topological ring of adeles the inversion on units is not continuous for the subspace topology. | |
| Aug 16, 2013 at 20:56 | history | edited | Yemon Choi | CC BY-SA 3.0 |
added reference to Ellis's paper and a quote from the MathReview
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| Aug 16, 2013 at 18:20 | comment | added | Yemon Choi | @user36938 Fair enough, but the point of my answer was that one doesn't even need to know any functoriality, or indeed anything about adeles. In other words, even a lowly analysis-type who is uncultured in number theory can give an answer:) I agree that if you do know the precise definition of the topology, then it is better to show joint continuity by looking at the topology rather than appealing to an oracle | |
| Aug 16, 2013 at 17:57 | comment | added | user36938 | Fair enough, though after one has actually given a definition of the topology being considered in the first place (which we need for anything to make sense), the desired answer just drops right out by functoriality considerations, so to invoke Ellis' theorem (treating the adelic topology as a "black box") seems like invoking the inverse function theorem to prove the smoothness of matrix inversion. | |
| Aug 16, 2013 at 14:40 | comment | added | Benjamin Steinberg | probably it is in Hindman-Strauss. | |
| Aug 16, 2013 at 4:32 | history | answered | Yemon Choi | CC BY-SA 3.0 |