Geodesic
Special Groups, Versality and the Grothendieck-Serre Conjecture
Documenta mathematica, Tome 25 (2020), pp. 171-188
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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→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→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.
DOI : 10.4171/dm/743
Classification : 14L05, 20G05, 20G10, 20G15, 20G35
Mots-clés : local ring, essential dimension, algebraic group, torsor, special group, Grothendieck-Serre conjecture 
@article{10_4171_dm_743,
     author = {Dajano Tossici and Zinovy Reichstein},
     title = {Special {Groups,} {Versality} and the {Grothendieck-Serre} {Conjecture}},
     journal = {Documenta mathematica},
     pages = {171--188},
     year = {2020},
     volume = {25},
     doi = {10.4171/dm/743},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/743/}
}
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Dajano Tossici; Zinovy Reichstein. Special Groups, Versality and the Grothendieck-Serre Conjecture. Documenta mathematica, Tome 25 (2020), pp. 171-188. doi: 10.4171/dm/743

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