Special groups, versality and the Grothendieck-Serre conjecture

DOI10.25537/DM.2020V25.171-188arXiv1912.08109MaRDI QIDQ6331286

Dajano Tossici, Z. Reichstein

Publication date: 17 December 2019

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 $ToX$ 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 $ToSpec(K)$ 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.

Mathematics Subject Classification ID

Linear algebraic groups over arbitrary fields (20G15) Representation theory for linear algebraic groups (20G05) Cohomology theory for linear algebraic groups (20G10) Formal groups, (p)-divisible groups (14L05) Linear algebraic groups over adèles and other rings and schemes (20G35)

This page was built for publication: Special groups, versality and the Grothendieck-Serre conjecture