Timeline for "Inverse problem" for Brauer groups
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Mar 2, 2014 at 12:20 | answer | added | Idanps | timeline score: 6 | |
| May 20, 2013 at 20:51 | vote | accept | Noah Schweber | ||
| May 17, 2013 at 17:25 | answer | added | Name | timeline score: 8 | |
| May 16, 2013 at 5:55 | comment | added | Noah Schweber | Ah, I see. Within this question, I'm just interested in the isomorphism types of Brauer groups, not how they interact with each other (although the latter is definitely the "right" way of looking at things in general); I just want to know what groups are Brauer groups of some field. | |
| May 16, 2013 at 4:13 | comment | added | Chandan Singh Dalawat | When I say direct sum, I mean the direct sum of B(R) and B(Q_p) for all primes p. The sum is indexed by the places of Q, not by the natural numbers. | |
| May 16, 2013 at 4:09 | comment | added | Chandan Singh Dalawat | B(Q) might be abstractly isomorphic to this direct sum (and that might suffice for an algebraist), but the point is that B(Q_p) is naturally isomorphic to Q/Z and B(R) to Z/2Z, and that there are natural maps from B(Q) to B(Q_p) and to B(R) which embed B(Q) into the direct sum; this natural embedding is not an isomorphism. | |
| May 16, 2013 at 3:23 | comment | added | Noah Schweber | Hm, I got that value for the Brauer group from pg. 20 of "Lectures on division algebras" by Saltman (books.google.com/…). What is $Br(\mathbb{Q})$? | |
| May 16, 2013 at 3:05 | comment | added | Chandan Singh Dalawat | only those which are coherent in the sense that their invariants add up to 0. This is a manifestation of reciprocity laws in arithmetic. | |
| May 16, 2013 at 3:05 | comment | added | Chandan Singh Dalawat | B(Q) is not quite equal to the direct sum you write in the edit in response to Emerton's comment. It injects into the direct sum, and the image is equal to the kernel of the "sum of the components" from this direct sum onto Q/Z. This is a way of expressing the fact that a central simple algebra over Q gives rise to a family of central simple algebras over Q_p (for every prime p, including p=\infty), but you don't get all such local families from a global object, | |
| May 16, 2013 at 2:09 | history | edited | Noah Schweber | CC BY-SA 3.0 |
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| May 16, 2013 at 2:06 | comment | added | Noah Schweber | Thanks! I didn't know that. I've edited appropriately. | |
| May 16, 2013 at 2:04 | comment | added | Emerton | Dear Noah, The Brauer groups of all local and global fields (e.g. all number fields, including $\mathbb Q$) are known, by local/global class field theory. Regards, | |
| May 16, 2013 at 1:32 | history | edited | Noah Schweber |
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| May 16, 2013 at 0:18 | history | asked | Noah Schweber | CC BY-SA 3.0 |