Timeline for Adjoint semi-simple algebraic groups over non-algebraically closed fields
Current License: CC BY-SA 3.0
5 events
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| May 17, 2016 at 2:28 | comment | added | nfdc23 | Curiously, the proof becomes a bit easier if one doesn't pass to the quasi-simple case, as one can formulate the result with a strong uniqueness aspect (pinning down $(k'/k, G')$ uniquely up to unique isomorphism, with $k'$ a nonzero finite etale $k$-algebra and $G'$ a smooth affine $k'$-group whose fibers are connected semisimple, absolutely simple, and simply connected -- or adjoint); that handles the Galois descent by pure thought. See Proposition A.5.14 in the book "Pseudo-reductive groups" for such a proof, the version over fields for the reference given by Cesnavicius over rings. | |
| May 16, 2016 at 20:36 | comment | added | zeno | The second sentence of the proof: in general $G$ will only be isogenous to such a product. | |
| May 16, 2016 at 20:34 | history | edited | zeno | CC BY-SA 3.0 |
added 2 characters in body
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| May 16, 2016 at 19:27 | comment | added | Daniel Loughran | Thanks. Though can you please emphasise at which point in the argument you use that $G$ simply connected or adjoint? | |
| May 16, 2016 at 18:51 | history | answered | zeno | CC BY-SA 3.0 |