Resolving G-torsors by abelian base extensions
| dc.creator | Chernousov, V. | |
| dc.creator | Gille, Ph. | |
| dc.creator | Reichstein, Z. | |
| dc.date | 2004-04-21 | |
| dc.date | 2006-03-15 | |
| dc.date.accessioned | 2026-07-07T06:36:43Z | |
| dc.date.available | 2026-07-07T06:36:43Z | |
| dc.description | Let G be a linear algebraic group defined over a field k. We prove that, under mild assumptions on k and G, there exists a finite k-subgroup S of G such that the natural map H^1(K, S) -> H^1(K, G) is surjective for every field extension K/k. We give several applications of this result in the case where k an algebraically closed field of characteristic zero and K/k is finitely generated. In particular, we prove that for every z in H^1(K, G) there exists an abelian field extension L/K such that z_L \in H^1(L, G) is represented by a G-torsor over a projective variety. From this we deduce that z_L has trivial point obstruction. We also show that a (strong) variant of the algebraic form of Hilbert's 13th problem implies that the maximal abelian extension of K has cohomological dimension =< 1. The last assertion, if true, would prove conjectures of Bogomolov and Koenigsmann, answer a question of Tits and establish an important case of Serre's Conjecture II for the group E_8. | |
| dc.description | New material added on the no-name lemma in Section 4 and on Hilbert's 13th problem in Section 9. A mistake in the proof of Proposition 2.3 is corrected | |
| dc.identifier | https://arxiv.org/abs/math/0404392 | |
| dc.identifier | http://arxiv.org/abs/math/0404392 | |
| dc.identifier | Journal of Algebra 296 (2006), 561-581 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/100176 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Group Theory | |
| dc.subject | 11E72, 14L30, 14E20 | |
| dc.title | Resolving G-torsors by abelian base extensions | |
| dc.type | text |