Special groups, versality and the Grothendieck-Serre conjecture

Zinovy Reichstein, Dajano Tossici

Published 2019-12-17Version 1

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 $X \to Y$ is locally trivial in the Zariski topology for every $k$-scheme $Y$. In recent papers an a priori weaker condition was used: $G$ is called special if every $G$-torsor $X \rightarrow \operatorname{Spec}(K)$ is split for every field extension $K/k$. 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.