Special Groups, Versality and the Grothendieck-Serre Conjecture

Abstract

Let k be a base field and G be an algebraic group over k. J.-P. Serre defined G to be special if every G-torsor T → X is locally trivial in the Zariski topology for every reduced algebraic variety X defined over k. In recent papers an a priori weaker condition is used: G is called special if every G-torsor T → Spec(K) is split for every field K containing k. We show that these two definitions are equivalent. We also generalize this fact and propose a strengthened version of the Grothendieck-Serre conjecture based on the notion of essential dimension.