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Resolving G-torsors by abelian base extensions
- Publication Year :
- 2004
- Publisher :
- arXiv, 2004.
-
Abstract
- 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.<br />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
- Subjects :
- Abelian variety
Unramified cover
Abelian extension
Torsor
Elementary abelian group
11E72, 14L30, 14E20
Group Theory (math.GR)
Non-abelian cohomology
01 natural sciences
Algebraic closure
Linear algebraic group
Algebraic element
Cohomological dimension
Combinatorics
Mathematics - Algebraic Geometry
Brauer group
G-cover
0103 physical sciences
FOS: Mathematics
Genus field
0101 mathematics
Algebraic Geometry (math.AG)
Mathematics
Discrete mathematics
Algebra and Number Theory
Fixed point obstruction
010102 general mathematics
Group action
Algebraic extension
16. Peace & justice
Field extension
010307 mathematical physics
Mathematics - Group Theory
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....9414f54a4fed1b8a9fa61fba319960cc
- Full Text :
- https://doi.org/10.48550/arxiv.math/0404392