Timeline for rational conjugacy classes
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 12, 2013 at 12:38 | comment | added | Rupert | Could I just ask something else, you said $H^{1}(F,H)$ is finite if $F$ has characteristic zero, does that mean that every field of characteristic zero is of type (F), in the sense defined in Ch. III of Serre's "Galois Cohomology", or am I confused? | |
| Dec 12, 2013 at 12:24 | comment | added | Rupert | Thanks very much, can you recommend any good reference for reading about torsors? | |
| Dec 10, 2013 at 15:39 | comment | added | user76758 | Let $G$ act on itself by conjugation, so you ask if $Q:=(G.g)(F)/G(F).g$ is finite. Since $G\rightarrow G.g$ is a right $Z_G(g)$-torsor (for etale topology if $Z_G(g)$ is smooth, fppf in general), by Cor. 1 to Prop. 36 in section 5.4 of Ch. I of Serre's "Galois cohomology" and its fppf variant we see that $Q=\ker({\rm{H}}^1(F,Z_G(g))\rightarrow {\rm{H}}^1(F,G))$. By 4.3 in Ch. III of that book, ${\rm{H}}^1(F,H)$ is finite if $F$ has characteristic 0; if char$(F)>0$ then finiteness holds if $H$ is connected reductive (rests on Bruhat-Tits) but fails in the smooth connected commutative case. | |
| Dec 10, 2013 at 14:36 | comment | added | Jim Humphreys | I'd add to what Yves says a probably useful reference to an old paper by R. Kottwitz, "Ratiopnal conjugacy classes in reductive groups", Duke Math. J. 49 (1982). Probably there is further work along these lines in later papers, too. Whatever "finite index" means in your question, related matters have certainly been worked out over local fields. It would help to provide more context for your question, given the substantial history. | |
| Dec 10, 2013 at 14:04 | comment | added | YCor | This holds (in much greater generality) by Corollaire 6.4 in Borel-Serre "théorèmes de finitude en cohomologie galoisienne", CMH 1964, when $F$ (locally compact) has characteristic 0. This greater generality fails for local fields of finite characteristic. But I don't know about your specific statement. | |
| Dec 10, 2013 at 14:00 | comment | added | YCor | What do you mean by finite index? Do you mean that the $F$-part of the conjugacy class of $g$ in $G(\bar{F})$ splits into finitely many $G(F)$-orbits? | |
| Dec 10, 2013 at 13:58 | history | asked | Rupert | CC BY-SA 3.0 |