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, Volume 93, Issue 1, pp 247–266 | Cite as

Galois cohomology of special orthogonal groups

  • Ryan Garibaldi
  • Jean-Pierre Tignol
  • Adrian R. Wadsworth


If (A,σ) is a central simple algebra of even degree with orthogonal involution, then for the map of Galois cohomology sets fromH 1(F,SO(A,σ)) to the 2-torsion in the Brauer group ofF, we describe fully the image of a given element ofH 1(F,SO(A,σ)) and prove that this description is correct in two different ways. As an easy consequence, we derive a result of Bartels [Bar, Satz 3].

Subject Classifications

12G05 20G15 16K20 11E88 


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  1. [All]
    Allen, H.P.: Hermitian forms II. J. of Algebra10 (1968), 503–515zbMATHCrossRefGoogle Scholar
  2. [Ap]
    Arason, J. Kr.: Cohomologische invarianten quadratischer Formen. J. of Algebra36 (1975), 448–491zbMATHCrossRefGoogle Scholar
  3. [Bap]
    Bartels, H.-J.: Invarianten hermitescher Formen über Scheifkörpern. Math. Ann.215 (1975), 269–288zbMATHCrossRefGoogle Scholar
  4. [B-F]
    Bayer-Fluckiger, E.: Multiplicateurs de similitudes. C. R. Acad. Sci. Paris (I)319 (1994), 1151–1153zbMATHGoogle Scholar
  5. [B-FP]
    Bayer-Fluckiger, E., and Parimala, R.: Galois cohomology of the classical groups over fields of cohomological dimension ≤2. Invent. Math.122 (1995), 195–229zbMATHCrossRefGoogle Scholar
  6. [B-FST]
    Bayer-Fluckiger, E., Shapiro, D.B., and Tignol, J.-P.: Hyperbolic involutions. Math. Zeit.214 (1993), 461–476zbMATHCrossRefGoogle Scholar
  7. [Bla]
    Blanchet, A.: Function fields of generalized Brauer-Severi varieties. Comm. in Algebra19 (1991), 97–118zbMATHCrossRefGoogle Scholar
  8. [Bor]
    Borel, A.: Linear algebraic groups. second edition, Springer, Graduate Texts in Mathematics126 (1991)Google Scholar
  9. [Brbk]
    Bourbaki, N.: Algèbre, Ch. 9: Formes sesquilinéaires et formes quadratiques. Hermann, Paris (1959)Google Scholar
  10. [Chev]
    Chevalley, C.: The algebraic theory of spinors. Columbia University Press (1954)Google Scholar
  11. [Drxl]
    Draxl, P.: Skew fields. London Mathematical Society (1983)Google Scholar
  12. [HKRT]
    Haile, D., Knus, M.-A., Rost, M., and Tignol, J.-P.: Algebras of odd degree with involution, trace forms and dihedral extensions. to appear in Israel J. Math.Google Scholar
  13. [Jac]
    Jacobson, N.: Clifford algebras for algebras with involution of typeD. J. of Algebra1 (1964), 288–300zbMATHCrossRefGoogle Scholar
  14. [Kn]
    Kneser, M.: Lectures on Galois cohomology of classical groups. Tata Lecture Notes in Mathematics47 (1969)Google Scholar
  15. [KMRT]
    Knus, M.-A., Merkurjev, A.S., Rost, M., and Tignol, J.-P.: The book of involutions. in preparationGoogle Scholar
  16. [KPS]
    Knus, M.-A., Parimala, R., and Sridharan, R.: On the discriminant of an involution. Bull. Soc. Math. Belgique (A)43 (1991), 89–98Google Scholar
  17. [Lam]
    Lam, T.-Y.: The algebraic theory of quadratic forms. Benjamin (1973)Google Scholar
  18. [Mer]
    Merkurjev, A.S.: Galois cohomology of orthogonal groups. handwritten notes dated 05.07.93Google Scholar
  19. [MPW1]
    Merkurjev, A.S., Panin, I.A., and Wadsworth, A.R.: Index reduction formulas for twisted flag varieties, I.K-Theory10 (1996), 517–596zbMATHCrossRefGoogle Scholar
  20. [MPW2]
    Merkurjev, A.S., Panin, I.A., and Wadsworth, A.R.: Index reduction formulas for twisted flag varieties, II. preprint (1996)Google Scholar
  21. [MT]
    Merkurjev, A.S., and Tignol, J.-P.: The multipliers of similitudes and the Brauer group of homogeneous varieties. J. reine angew. Math.461 (1995), 13–47zbMATHGoogle Scholar
  22. [Sch]
    Scharlau, W.: Quadratic and hermitian forms. Springer (1985)Google Scholar
  23. [SvdB]
    Schofield, A., and van den Bergh, M.: The index of a Brauer class on a Brauer-Severi variety. Trans. AMS333 (1992), 729–739zbMATHCrossRefGoogle Scholar
  24. [SeCG]
    Serre, J.-P.: Cohomologie Galoisienne. Springer, fifth edition, Lecture Notes in Mathematics5 (1994)Google Scholar
  25. [SeLF]
    Serre, J.-P.: Local fields. Springer, Graduate Texts in Mathematics67 (1979) (English translation ofCorps locaux)Google Scholar
  26. [Sp]
    Springer, T.A.: On the equivalence of quadratic forms. Proc. Kon. Ned. Akad. Wet.62 (1959), 241–253Google Scholar
  27. [Tao]
    Tao, D.: The generalized even Clifford algebra. J. Algebra172 (1995), 184–204zbMATHCrossRefGoogle Scholar
  28. [Tits]
    Tits, J.: Formes quadratiques, groupes orthogonaux et algèbres de Clifford. Invent. Math.5 (1968), 19–41zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Ryan Garibaldi
    • 1
  • Jean-Pierre Tignol
    • 2
  • Adrian R. Wadsworth
    • 1
  1. 1.Dept. of MathematicsUniversity of California, San DiegoLa JollaUSA
  2. 2.Institut de MathématiqueUniversité Catholique de LouvainLouvain-La-NeuveBelgium

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