Timeline for Does a reductive group over $K$ always have a torus that becomes maximal split over $L$?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 10, 2017 at 14:56 | vote | accept | dgulotta | ||
| Jun 9, 2017 at 19:15 | answer | added | LSpice | timeline score: 3 | |
| Jun 9, 2017 at 19:00 | comment | added | LSpice | A silly example is when $K$ is finite (or, more generally, of cohomological dimension $\le 1$), at least if $L/K$ is algebraic; for then, if $S$ is a maximal $K$-split torus in $G$, then $C_G(S)_{/L}$ is a torus, hence contains a unique maximal split sub-torus, which is therefore $\mathrm{Gal}(L/K)$-stable, hence defined over $K$. | |
| Jun 9, 2017 at 18:53 | comment | added | LSpice | I assume you want conditions on $K$ and $L$ that make this work for all $G$? For your example, $L$ need only be separably, not necessarily algebraically, closed. Another case where this can be done is when $K$ is a Henselian field, and $L$ is its strict Henselisation. This is in Bruhat–Tits 2. I'm almost positive it's not true in general, but don't have an example off the top of my head. | |
| Jun 9, 2017 at 18:39 | history | asked | dgulotta | CC BY-SA 3.0 |