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Cohomological invariants for orthogonal involutions on degree 8 algebras

Published online by Cambridge University Press:  05 July 2011

Anne Quéguiner-Mathieu  and
Jean-Pierre Tignol
Anne Quéguiner-Mathieu
Affiliation:
LAGA - UMR 7539 du CNRS, Université Paris 13, F-93430 Villetaneuse and UPEC, F-94010 Créteil, [email protected]
Jean-Pierre Tignol
Affiliation:
Zukunftskolleg, Universität Konstanz, D-78457 Konstanz, Germany and ICTEAM Institute, Université catholique de Louvain, B-1348 Louvain-la-Neuve, [email protected]
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Abstract

Using triality, we define a relative Arason invariant for orthogonal involutions on a -possibly division- central simple algebra of degree 8. This invariant detects hyperbolicity, but it does not detect isomorphism. We produce explicit examples, in index 4 and 8, of non isomorphic involutions with trivial relative Arason invariant.

Keywords

Cohomological invariantorthogonal groupalgebra with involutionClifford algebrahalf-neighbortriality
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 2 , April 2012 , pp. 333 - 358
Copyright
Copyright © ISOPP 2011

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References

1.Amitsur, S. A., Rowen, L. H. and Tignol, J.-P., Division algebras of degree 4 and 8 with involution, Israel J. Math. 33 (1979), no. 2, 133–148.CrossRefGoogle Scholar
2.Arason, J. Kr., Cohomologische Invarianten quadratischer Formen, J. Algebra 36 (1975), no. 3, 448–491.CrossRefGoogle Scholar
3.Bayer-Fluckiger, E., Shapiro, D. B. and Tignol, J.-P., Hyperbolic involutions, Math. Z. 214 (1993), no. 3, 461–476.CrossRefGoogle Scholar
4.Bayer-Fluckiger, E., Parimala, R. and Quéguiner-Mathieu, A., Pfister involutions, Proc. Indian Acad. Sci. Math. Sci. 113 (2003), no. 4, 365–377.Google Scholar
5.Becher, K. J., A proof of the Pfister factor conjecture, Invent. Math. 173 (2008), no. 1, 1–6.CrossRefGoogle Scholar
6.Berhuy, G., Cohomological invariants of quaternionic skew-Hermitian forms, Arch. Math. (Basel) 88 (2007), no. 5, 434–447.CrossRefGoogle Scholar
7.Bruhat, F. and Tits, J., Groupes algébriques sur un corps local. Chapitre III. Compléments et applications à la cohomologie galoisienne. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 3, 671–698.Google Scholar
8.Dejaiffe, I., Formes antihermitiennes devenant hyperboliques sur un corps de déploiement, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 2, 105–108.CrossRefGoogle Scholar
9.Elman, R. et al. , Witt rings and Brauer groups under multiquadratic extensions. I, Amer. J. Math. 105 (1983), no. 5, 1119–1170.CrossRefGoogle Scholar
10.Elman, R., Karpenko, N. and Merkurjev, A., The algebraic and geometric theory of quadratic forms, Amer. Math. Soc., Providence, RI, 2008.CrossRefGoogle Scholar
11.Garibaldi, S., Merkurjev, A. and Serre, J.-P., Cohomological invariants in Galois cohomology, Amer. Math. Soc., Providence, RI, 2003.CrossRefGoogle Scholar
12.Garibaldi, S., Orthogonal involutions on algebras of degree 16 and the Killing form of E 8, in Quadratic forms—algebra, arithmetic, and geometry, 131–162, Contemp. Math. 493, Amer. Math. Soc., Providence, RI.Google Scholar
13.Gille, P., Invariants cohomologiques de Rost en caractéristique positive, K-Theory 21 (2000), no. 1, 57–100.Google Scholar
14.Hoffmann, D. W., Similarity of quadratic forms and half-neighbors, J. Algebra 204 (1998), no. 1, 255–280.CrossRefGoogle Scholar
15.Hoffmann, D. W. and Tignol, J.-P., On 14-dimensional quadratic forms in I 3, 8-dimensional forms in I 2, and the common value property, Doc. Math. 3 (1998), 189–214 (electronic).CrossRefGoogle Scholar
16.Izhboldin, O.T. and Karpenko, N. A., Some new examples in the theory of quadratic forms, Math. Z. 234 (2000), 647–695.CrossRefGoogle Scholar
17.Jacobson, N., Finite-dimensional division algebras over fields, Springer, Berlin, 1996.CrossRefGoogle Scholar
18.Kahn, B., Un invariant de degré 3 des algébres centrales simples d'exposant 2, J. Ramanujan Math. Soc. 25 (2010) 355–358.Google Scholar
19.Karpenko, N. A., Torsion in CH2 of Severi-Brauer varieties and indecomposability of generic algebras, Manuscripta Math. 88 (1995), no. 1, 109–117.Google Scholar
20.Knus, M.-A. et al. , The book of involutions, Amer. Math. Soc., Providence, RI, 1998.CrossRefGoogle Scholar
21.Laghribi, A., Isotropie de certaines formes quadratiques de dimensions 7 et 8 sur le corps des fonctions d'une quadrique, Duke Math. J. 85 (1996), no. 2, 397–410.CrossRefGoogle Scholar
22.Lam, T. Y., Introduction to quadratic forms over fields, Amer. Math. Soc., Providence, RI, 2005.Google Scholar
23.Masquelein, A., Quéguiner-Mathieu, A. and Tignol, J.-P., Quadratic forms of dimension 8 with trivial discriminant and Clifford algebra of index 4, Arch. Math. (Basel) 93 (2009), no. 2, 129–138.CrossRefGoogle Scholar
24.Parimala, R., Sridharan, R. and Suresh, V., Hermitian analogue of a theorem of Springer, J. Algebra 243 (2001), no. 2, 780–789.Google Scholar
25.Peyre, E., Galois cohomology in degree three and homogeneous varieties, K-Theory 15 (1998), no. 2, 99–145.CrossRefGoogle Scholar
26.Quéguiner-Mathieu, A. and Tignol, J.-P., Discriminant and Clifford algebras, Math. Z. 240 (2002), no. 2, 345–384.CrossRefGoogle Scholar
27.Quéguiner-Mathieu, A. and Tignol, J.-P., Algebras with involution that become hyperbolic over the function field of a conic, Israel J. Math. 180 (2010), 317–344.Google Scholar
28.Quéguiner-Mathieu, A. and Tignol, J.-P., Arason invariant for algebras with orthogonal involutions, in preparation.Google Scholar
29.Sivatski, A. S., Applications of Clifford algebras to involutions and quadratic forms, Comm. Algebra 33 (2005), no. 3, 937–951.Google Scholar
30.Tao, D., The generalized even Clifford algebra, J. Algebra 172 (1995), no. 1, 184–204.CrossRefGoogle Scholar
31.Tignol, J.-P., Corps à involution neutralisés par une extension abélienne élémentaire, in The Brauer group (Sém., Les Plans-sur-Bex, 1980), 1–34, Lecture Notes in Math. 844, Springer, Berlin.Google Scholar
32.Tignol, J.-P., A Cassels-Pfister theorem for involutions on central simple algebras, J. Algebra 181 (1996), no. 3, 857–875.CrossRefGoogle Scholar
33.Tignol, J.-P., Cohomological invariants of central simple algebras with involution, Quadratic forms, linear algebraic groups, and cohomology, 137–171, Dev. Math., 18, Springer, New York, 2010.Google Scholar