Timeline for If split algebraic groups are potentially isomorphic, are they isomorphic?
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 22, 2010 at 20:16 | comment | added | BCnrd | @George: I see. So near top of p. 167, how is he getting that the map between $\alpha$-root groups over $K$ is defined over $k$? That is, what structure does he have which rules out composition with scaling by an element of $K^{\times}$ not in $k^{\times}$? Hmm, let's discuss by email, this is silly to discuss via MO. | |
| Apr 22, 2010 at 19:35 | comment | added | George McNinch | @Brian: The argument for the isomorphism theorem in 1.15 does not make that assumption -- the first thing Jantzen does there is choose an algebraic closure K of the field of fractions of k. He uses II.1.9(5) to "descend" from K to k. (He does the alg closed field case first, in 1.14). | |
| Apr 22, 2010 at 18:10 | comment | added | BCnrd | George, I have taken a look and the isomorphism theorem seems to be in 1.14--1.15, where it is assumed that $k$ is algebraically closed. I agree that some generalities are done over a more general ring, but for the big theorems linking with root data, seems to be over alg. closed field. | |
| Apr 22, 2010 at 17:08 | comment | added | George McNinch | Section II.1 of Jantzen's book Representations of Algebraic Groups contains an account of the existence and isomorphism theorems (but not the isogeny theorem) for split reductive groups G over a PID k (or rather, over Spec(k)), following arguments of Takeuchi. | |
| Feb 16, 2010 at 11:46 | comment | added | Kevin Buzzard | I'll remark that when Toby Gee and I at some point wanted such statements (at some point in our pre-preprint we wanted the L-group of a conn red group to be a group over Q) we invoked SGA3... | |
| Feb 16, 2010 at 11:21 | comment | added | Kevin Buzzard | I can't find a clean reference. How about this: Remark 2.10 gives you the existence in the semi-simple case (over any field). Some homework now gives you existence in the reductive case. G split implies that the map he calls "mu_G" is trivial. Now say G and H are both split forms of the same conn red group over k-bar. The remark at the end of 3.2 implies G,H are inner forms of one another. Now some more homework implies G and H are isomorphic. Most definitely not as good as I had hoped in terms of the "reference" stakes... | |
| Feb 15, 2010 at 23:46 | comment | added | BCnrd | Kevin, where in Corvallis does anyone state the result for split connected reductive groups over a general field? I just flipped through Springer's article; did I miss it? Apart from that, it is useful that the classification is a bit better than isomorphism classes: one can capture isogenies as well (and distinguish central from Frobenius from the weird types that only exist in small nonzero characteristics). For years it drove me up the wall that I couldn't find a non-SGA3 reference allowing a general ground field; now that problem is solved. :) | |
| Feb 15, 2010 at 23:19 | comment | added | Kevin Buzzard | In short: over any field you like there's a canonical bijection between iso classes of based root data (this is just a bunch of linear algebra) and iso classes of split reductive groups. If you just want to see statements then I've always found the articles by Springer and Borel in Corvallis to be quite precise (although for proofs you'll have to dig deeper e.g. following up Brian's references). | |
| Feb 15, 2010 at 23:04 | history | edited | BCnrd | CC BY-SA 2.5 |
added 6 characters in body
|
| Feb 15, 2010 at 22:40 | vote | accept | blt | ||
| Feb 15, 2010 at 22:34 | history | answered | BCnrd | CC BY-SA 2.5 |