Abstract
A linear algebraic group G defined over a field k is called special if every G-torsor over every field extension of k is trivial. In 1958 Grothendieck classified special groups in the case where the base field is algebraically closed. In this paper we describe the derived subgroup and the coradical of a special reductive group over an arbitrary field k. We also classify special semisimple groups, special reductive groups of inner type, and special quasisplit reductive groups over an arbitrary field k. Finally, we give an application to a conjecture of Serre.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck, M. Raynaud, J.-P. Serre, Schémas en Groupes, Fasc. 2a, Exp. 5 et 6, Institut des Hautes Études Scientifiques, Paris, 1963/64.
M. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass., Ont., 1969.
E. Bayer-Fluckiger, H. W. Lenstra, Forms in odd degree extensions and self-dual normal bases, Amer. J. Math. 112 (1990), 359-373.
G. Berhuy, C. Frings, J.-P. Tignol, Galois cohomology of the classical groups over imperfect fields, J. Pure Appl. Algebra 211 (2007), 307-341.
J. Black, Zero cycles of degree one on principal homogeneous spaces, J. Algebra 334 (2011), 232-246.
M. Borovoi, Z. Reichstein, Toric friendly groups, Algebra Number Theory 5 (2011), no. 3, 361-378.
V. Chernousov, P. Gille, A. Pianzola, Torsors over the punctured affine line, Amer. J. Math. 134 (2012), 1541-1583.
J.-L. Colliot-Thélène, J.-J. Sansuc, Principal homogeneous spaces under asque tori: applications, J. Algebra 106 (1987), 148-205.
J.-L. Colliot-Thélène, J.-J. Sansuc, The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group) in: Algebraic Groups and Homogeneous Spaces, Tata Inst. Fund. Res., Narosa, Bew Delhi, 2007, pp. 113-186.
M. Florence, Zéro-cycles de degré un sur les espaces homogènes, Int. Math. Res. Not. 54 (2004), 2897-2914.
P. Gille, La R-équivalence sur les groupes algébriques réductifs définis sur un corps global, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 199-235.
P. Gille, T. Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, Cambridge, 2006.
A. Grothendieck, Torsion homologique et sections rationnelles, in: Séminaire Claude Chevalley “Anneaux de Chow et Applications”, Paris, 1958, pp. 5-1-5–29.
M.-A. Knus, A. S. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions, American Mathematical Society, Providence, RI, 1998.
R. Lötscher, Essential dimension of involutions and subalgebras, Israel J. Math. 192 (2012), 325-346.
A. S. Merkurjev, Essential dimension: a survey, Transform. Groups 18 (2013), 415-481.
D.-T. Nguyen, On the essential dimension of unipotent algebraic groups, J. Pure Appl. Algebra 217 (2013), 432-448.
R. Parimala, Homogeneous varieties—zero-cycles of degree one versus rational points, Asian J. Math. 9 (2005), 251-256.
A. Pfister, Quadratic Forms with Applications to Algebraic Geometry and Topology, Cambridge University Press, Cambridge, 1995.
J.-J. Sansuc, roupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. reine angew. Math. 327 (1981), 12-80.
A. Schofield, M. Van den Bergh, The index of a Brauer class on a Brauer–Severi variety, Trans. Amer. Math. Soc. 333 (1992), 729-739.
J.-P. Serre, Espaces fibrés algébriques, in: Séminaire Claude Chevalley “An-neaux de Chow et Applications”, Paris, 1958, pp. 1-1–1–37.
J.-P. Serre, Cohomologie Galoisienne, 5th ed., Springer-Verlag, Berlin, 1994.
J.-P. Serre, Cohomologie galoisienne: progrés et problèmes, Astérisque 227 (1995), 229-257.
J.-P. Serre, Exposés de Séminaires (1950-1999), Société Mathématique de France, Paris, 2001.
T. A. Springer, Linear Algebraic Groups, 2nd ed., Birkhäuser Boston, Boston, MA, 2009.
B. Totaro, Splitting fields for E8-torsors, Duke Math. J. 121 (2004), 425-455.
V. E. Voskresenskiĭ, Algebraic Groups and Their Birational Invariants, American Mathematical Society, Providence, RI, 1998.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
HURUGUEN, M. SPECIAL REDUCTIVE GROUPS OVER AN ARBITRARY FIELD. Transformation Groups 21, 1079–1104 (2016). https://doi.org/10.1007/s00031-016-9378-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-016-9378-5