Timeline for Homogeneous spaces of affine algebraic groups
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 3, 2018 at 23:44 | answer | added | Wenxue Huang | timeline score: 1 | |
| Dec 10, 2015 at 19:54 | comment | added | nfdc23 | @JimHumphreys: Even "better", Gabber has an argument to show that Borel's error for disconnected $H$ only arises in characteristic 2 (i.e., he provided a detour around the error for disconnected $H$ whenever ${\rm{char}}(k) \ne 2$). | |
| Dec 9, 2015 at 22:51 | comment | added | Jim Humphreys | A few more comments: 1) Richardson was told by both Borel and Springer after circulating his draft in 1975 that Borel had already verified the difficult implication in positive characteristic; so Richardson quotes in his published note from Borel's letter to him. 2) The paper by CPS was submitted a little later than Richardson's but gives a wider perspective. 3) Borel's Theorem 1.1(i) shows that $G/H$ is affine iff the identity component $H^\circ$ is reductive. [In his collected papers IV, he also notes (following comments by Knop) that his Theorem 1.1(ii) fails in char 2.] | |
| Dec 9, 2015 at 21:39 | comment | added | Jim Humphreys | Borel's paper On affine algebraic homogeneous spaces appears in Arch. Math. (Basel) 45 (1985), no. 1, 74–78 (but may also require library use to access online). | |
| Dec 9, 2015 at 21:27 | comment | added | Jim Humphreys | Richardson's 1977 paper is found (probably by library use) in the Bulletin of the London Mathematical Society, but another relevant 1977 paper by Cline-Parshall-Scott is freely available online: gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002314673 | |
| Dec 9, 2015 at 20:04 | comment | added | nfdc23 | Borel proved the result in every characteristic earlier than Richardson (by a slick technique with etale cohomology inspired by his work with Harish-Chandra over C, not using Haboush's theorem as Richardson does), though Borel only published his argument later than Richardson. Borel's title is something like "Affine homogeneous spaces". In the Introduction he summarizes some of the history around this result. | |
| Dec 9, 2015 at 17:48 | comment | added | pro | for the lazy: Matsushima says G/H is affine if and only if H is reductive. arxiv.org/abs/math/0506430 | |
| Dec 9, 2015 at 17:33 | comment | added | Jason Starr | This is Matsushima's Criterion (proved by Richardson in positive characteristic). | |
| Dec 9, 2015 at 17:28 | history | asked | Ehud Meir | CC BY-SA 3.0 |