Timeline for Mostow's theorem on algebraic groups
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 19, 2012 at 23:15 | answer | added | Jim Humphreys | timeline score: 5 | |
| Sep 17, 2012 at 5:59 | vote | accept | Mikhail Borovoi | ||
| Sep 16, 2012 at 19:39 | answer | added | grp | timeline score: 13 | |
| Sep 16, 2012 at 13:56 | comment | added | Mikhail Borovoi | @grp: Could you please write a detailed answer? I would be happy to accept it! | |
| Sep 15, 2012 at 19:33 | comment | added | grp | (cont'd) Of course, char. 0 is being used again at the very end, to see that the formation of $G'$-invariants is (right-)exact on the category of algebraic linear representations of the possibly disconnected reductive $G'$. Indeed, by the analogue of Hochschild-Serre (or bare hands) we reduce the cohomological vanishing (or equivalently, the exactness) to 3 cases: (i) connected semisimple, (ii) torus, and (ii) finite etale. Lie algebras over $K$ settle (i), and scalar extension to $\overline{K}$ reduces (ii) and (iii) to the easy cases of split tori and finite constant groups. | |
| Sep 15, 2012 at 19:21 | comment | added | grp | Let $U = R_u(G)$. To prove existence of a $K$-rational splitting and uniqueness up to $U(K)$-conjugacy, WLOG $U = V$ is a vector group. Conjugation of $G$ on $V$ defines an action of $G' = G/V$ on $V$ that is linear since char($K$)=0 (!). Also, $G \rightarrow G'$ is an etale $V$-torsor, so it admits a section by vanishing of degree-1 quasi-coherent etale cohomology of $G'$. Thus, the problem is vanishing of Hochschild cohomology H$^i(G',V)$ (algebraic cochains) for $i=1,2$. On the category of algebraic linear representations, Hochschild cohomology is the derived functor of "invariants". QED | |
| Sep 15, 2012 at 15:08 | comment | added | YCor | [I'm in char 0] According to modern terminology you can define $N$ as the unipotent radical of $G$ itself. Also, $G$ is reductive iff $G^0$ is reductive, and this is certainly what Mostow calls "fully reducible". I think your theorem indeed is contained in Mostow's theorem 7.1. | |
| Sep 15, 2012 at 12:09 | history | asked | Mikhail Borovoi | CC BY-SA 3.0 |