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Zero-cycles on projective varieties and the norm principle

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  23 December 2009

Philippe Gille  and
Nikita Semenov
Philippe Gille
Affiliation:
UMR 8553 du CNRS, École Normale Supérieure, 45 Rue d’Ulm, 75005 Paris, France (email: Pilippe.Gille@ens.fr)
Nikita Semenov
Affiliation:
Mathematisches Institut, Universität München, Theresienstrasse 39, 80333 München, Germany (email: semenov@mathematik.uni-muenchen.de)
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Abstract

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Using the Gille–Merkurjev norm principle we compute in a uniform way the image of the degree map for quadrics (Springer’s theorem), for twisted forms of maximal orthogonal Grassmannians (a theorem of Bayer-Fluckiger and Lenstra), and for E6- (a theorem of Rost) and E7-varieties.

Keywords

norm principlealgebraic groupszero-cycles

MSC classification

Secondary: 20G15: Linear algebraic groups over arbitrary fields
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 2 , March 2010 , pp. 457 - 464
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Barquero, P. and Merkurjev, A., Norm principle for reductive algebraic groups, in Algebra, arithmetic and geometry, Part I, II, Mumbai, 2000, Tata Institute of Fundamental Research Studies in Mathematics, vol. 16 (Tata Institute of Fundamental Research, Bombay, 2002), 123–137.Google Scholar
[2]Bayer-Fluckiger, E. and Lenstra, H. W. Jr., Forms in odd degree extensions and self-dual normal bases, Amer. J. Math. 112 (1990), 359–373.CrossRefGoogle Scholar
[3]Ferrar, J. C., Strictly regular elements in Freudenthal triple systems, Trans. Amer. Math. Soc. 174 (1972), 313–331.CrossRefGoogle Scholar
[4]Garibaldi, S., Structurable algebras and groups of type E6 and E7, J. Algebra 236 (2001), 651–691.CrossRefGoogle Scholar
[5]Garibaldi, S., Groups of type E7 over arbitrary fields, Comm. Algebra 29 (2001), 2689–2710.CrossRefGoogle Scholar
[6]Gille, Ph., La R-équivalence sur les groupes algébriques réductifs définis sur un corps global, Publ. Math. Inst. Hautes Études Sci. 86 (1997), 199–235.CrossRefGoogle Scholar
[7]Gille, Ph., Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique ⩽2, Compositio Math. 125 (2001), 283–325.CrossRefGoogle Scholar
[8]Merkurjev, A., A norm principle for algebraic groups, St. Petersburg Math. J. 7 (1996), 243–264.Google Scholar
[9]Petrov, V. and Semenov, N., Higher Tits indices of algebraic groups, Appendix by M. Florence (2007), available from http://arxiv.org/abs/0706.2827.Google Scholar
[10]Petrov, V., Semenov, N. and Zainoulline, K., Zero-cycles on a Cayley plane, Canad. Math. Bull. 51 (2008), 114–124.CrossRefGoogle Scholar
[11]Tits, J., Classification of algebraic semisimple groups, in Algebraic groups and discontinuous subgroups, Proceedings of Symposia in Pure Mathematics, Boulder, CO, 1965 (American Mathematical Society, Providence, RI, 1966), 33–62.CrossRefGoogle Scholar
[12]Tits, J., Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque, J. Reine Angew. Math. 247 (1971), 196–220.Google Scholar
[13]Tits, J., Strongly inner anisotropic forms of simple algebraic groups, J. Algebra 131 (1990), 648–677.CrossRefGoogle Scholar