Special Groups, Versality and the Grothendieck-Serre Conjecture

  • Zinovy Reichstein

    Department of Mathematics, University of British Columbia, Vancouver, BC V6R1Z2, Canada
  • Dajano Tossici

    Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence, France
Special Groups, Versality and the Grothendieck-Serre Conjecture cover
Download PDF

This article is published open access.

Abstract

Let kk be a base field and GG be an algebraic group over kk. J.-P. Serre defined GG to be special if every GG-torsor TXT \to X is locally trivial in the Zariski topology for every reduced algebraic variety XX defined over kk. In recent papers an a priori weaker condition is used: GG is called special if every GG-torsor TSpec(K)T \to \operatorname{Spec}(K) is split for every field KK containing kk. 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.

Cite this article

Zinovy Reichstein, Dajano Tossici, Special Groups, Versality and the Grothendieck-Serre Conjecture. Doc. Math. 25 (2020), pp. 171–188

DOI 10.4171/DM/743