Timeline for Reference for Pic(G) and central extensions.
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 3, 2010 at 1:36 | vote | accept | Marty | ||
| Jul 2, 2010 at 20:29 | answer | added | Mikhail Borovoi | timeline score: 3 | |
| Jul 2, 2010 at 12:18 | comment | added | BCnrd | Marty, the 2nd half of p. 110 and Remark on the top of p. 111 of LNM 119 were what I had in mind, but I now see they don't quite address your question. A better reference, over alg. closed fields, is section 4 of "Local properties of algebraic group actions" by Knop, Kraft, Luna, & Vust. In late July we can discuss why their idea works (for connected ss gps) over a general field if you don't see it. (Can ignore most of that section 4 for your purposes.) I will now email you .pdf's of this and of a scan of an awe-inspiring letter I got from Gabber on a (surprise!) vast generalization. | |
| Jul 2, 2010 at 7:27 | comment | added | Marty | Great reference - Thanks! I haven't quite found the result I was looking for, word-for-word, in Raynauds thesis (LNM 119, I presume). But I suppose it follows from something in there, by viewing $G$ as the homogeneous space $G \times G / \Delta(G)$. I'll peruse some more and try to find exactly what I'm looking for. I'm in Australia right now (!), back in CA on July 11. Perhaps we can talk in person in mid-late July, if you'll be around. | |
| Jul 2, 2010 at 5:48 | comment | added | BCnrd | Marty, Raynaud's thesis was closer to 40 years ago. :) There is a very elegant proof of the relation with central extensions by using the "big cell" structure over $F_s$ (no need to assume $F$ perfect) and Rosenlicht's result relating units on algebraic groups to homomorphisms to $\mathbf{G}_ m$. No need to refer to SGA7 (Rosenlicht's unit result is discussed in there, albeit without proof). The crux is that in the simply connected case the coordinate ring is a UFD, a fact that seems not as well-known as it should be. It is easier to discuss this in person; are you around UCSC these days? | |
| Jul 2, 2010 at 3:47 | history | asked | Marty | CC BY-SA 2.5 |