Resolving G-torsors by abelian base extensions

Details Resolving G-torsors by abelian base extensions Resolving G-torsors by abelian base extensions

2004-04-21

arxiv.org/abs/math/0404392v2

Journal of Algebra 296 (2006), 561-581

Mathematics

Algebraic Geometry

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 Pro

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Resolving G-torsors by abelian base extensions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Resolving G-torsors by abelian base extensions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Resolving G-torsors by abelian base extensions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-18680