Ph. Gille | Searcharxiv

arXiv2007

Reduction of structure for torsors over semilocal rings

Let G be a reductive affine group scheme defined over a semilocal ring k. Assume that either G is semisimple or k is normal and noetherian. We show that G has a finite k-subgroup S such that the natural map H^1(R, S) --> H^1(R, G) is surjective for every semilocal ring R containing k. In other words, G-torsors over Spec(R) admit reduction of structure to S. We also show that the natural map H^1(X, S) --> H^1(X, G) is surjective in several other contexts, under suitable assumptions on the base ring k, the scheme X/k and the group scheme G/k. These results have already been used to study loop algebras as well as essential dimension of connected algebraic groups in prime characteristic. Additional applications are presented at the end of this paper.

math.AG↗

arXiv2006

Resolving G-torsors by abelian base extensions

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.

math.AG↗