Geodesic
Special Groups, Versality and the Grothendieck-Serre Conjecture
Documenta mathematica, Tome 25 (2020), pp. 171-188.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 $T \to X$ 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 $T \to \operatorname{Spec}(K)$ 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.
Classification : 20G05, 14L05, 20G10, 20G15, 20G35
Keywords: algebraic group, torsor, special group, local ring, Grothendieck-Serre conjecture, essential dimension 
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     author = {Reichstein, Zinovy and Tossici, Dajano},
     title = {Special {Groups,} {Versality} and the {Grothendieck-Serre} {Conjecture}},
     journal = {Documenta mathematica},
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     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2020__25__a64/}
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Reichstein, Zinovy; Tossici, Dajano. Special Groups, Versality and the Grothendieck-Serre Conjecture. Documenta mathematica, Tome 25 (2020), pp. 171-188. http://geodesic.mathdoc.fr/item/DOCMA_2020__25__a64/