Resolving G-torsors by abelian base extensions
By V. Chernousov, P. Gille, Zinovy Reichstein. 2004
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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 α ∈ H 1 ( K , G ) there exists an abelian field extension L / K such that α L ∈ H 1 ( L , G ) is represented by a G-torsor over a projective variety. From this we deduce that α L has trivial fixed 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 Konigsmann, answer a question of Tits and establish an important case of Serre's Conjecture II for the group E 8 .
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/f34ea9790db35007f0c112f27eb0ae1f556a25ed
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In Algebra and Number Theory
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Cohomological Invariants: Exceptional Groups and Spin Groups (2009, background): “Indeed, the paper [ CGR06 ] gives a finite constant subgroup S of G such that the exponent of C is relatively prime to |S| and the map H1(k,S) ! H1(k,G) is surjective”