Let be a base field and be an algebraic group over . J.-P. Serre defined to be special if every -torsor is locally trivial in the Zariski topology for every reduced algebraic variety defined over . In recent papers an a priori weaker condition is used: is called special if every -torsor is split for every field containing . 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. 171–188DOI 10.4171/DM/743