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Clifford modules and invariants of quadratic forms

Published online by Cambridge University Press:  22 March 2011

Max Karoubi  and
Jean-Pierre Serre
Max Karoubi
Affiliation:
Université Denis Diderot- Paris 7, UFR de Mathématiques, Case 7012, 175, rue du Chevaleret, 75205 Paris cedex 13, France, max.karoubi@gmail.com
Jean-Pierre Serre
Affiliation:
Collège de France, Place Marcelin Berthelot, 75005 Paris, France, serre@noos.fr
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Abstract

We construct new invariants of quadratic forms over commutative rings, using ideas from Topology. More precisely, we define a hermitian analog of the Bott class with target algebraic K-theory, based on the classification of Clifford modules. These invariants of quadratic forms go beyond the classical invariants defined via the Clifford algebra. An appendix by J.-P. Serre, of independent interest, describes the “square root” of the Bott class in the general framework of lambda rings.

Keywords

Quadratic formsClifford modulesBott classalgebraic K-theorylambda rings
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 1 , February 2012 , pp. 1 - 44
Copyright
Copyright © ISOPP 2011

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References

1.Atiyah, M.F.. Power operations in K-theory. Quart. J. Math. Oxford Ser. 17:165–193, 1966.CrossRefGoogle Scholar
2.Atiyah, M.F., Bott, R. and Shapiro, A.. Clifford modules. Topology 3:3–38, 1964.CrossRefGoogle Scholar
3.Atiyah, M.F., Macdonald, I.G.. Introduction to Commutative Algebra. Addison-Wesley, 1969.Google Scholar
4.Auslander, M., Goldman, O.. The Brauer group of a commutative ring. Transactions of the Amer. Math. Soc. 97, N°3: 367–409, 1969.CrossRefGoogle Scholar
5.Bass, H.. K-theory and stable algebra. Publ. Math. IHES, tome 22:5–60, 1964.CrossRefGoogle Scholar
6.Bass, H.. Clifford algebras and spinor norms over a commutative ring. Amer. J. Math. 96:156–206, 1974.CrossRefGoogle Scholar
7.Berrick, A. J.. Characterisation of plus-constructive fibrations. Advances in Mathematics 48: 172–176 (1983).CrossRefGoogle Scholar
8.Bott, R.. A note on the KO-theory of sphere bundles. Bull. Amer. Math. Soc. 68:395–400, 1962.CrossRefGoogle Scholar
9.Caenepeel, S.. A cohomological interpretation of the Brauer-Wall group. Algebra and Geometry, Santiago de Compostela:31–46, 1989.Google Scholar
10.Donovan, P. and Karoubi, M.. Graded Brauer groups and K-theory with local coefficients. Inst. Hautes Etudes Sci. Publ. Math. 38:5–25, 1970.CrossRefGoogle Scholar
11.Hirzebruch, F.. New Topological Methods in Algebraic Geometry. Springer (1962).Google Scholar
12.Hausmann, C. and Husemoller, D.. Acyclic maps. L'enseignement Mathématique 25:53–75, 1979.Google Scholar
13.Karoubi, M.. Le théorème fondamental de la K-théorie hermitienne. Ann. of Math. (2) 112 (2): 259–282, 1980.CrossRefGoogle Scholar
14.Karoubi, M.. K-theory, an Introduction. Grundlehren der math. Wiss. 226. New edition in Classics in Mathematics. Springer-verlag, 2008.Google Scholar
15.Karoubi, M.. Homologie cyclique et K-théorie. Astérisque 149. Société Mathématique de France, 1987.Google Scholar
16.Karoubi, M.. Twisted K-theory, old and new, in K-theory and noncommutative geometry, 117–149, EMS Sér. Congr. Rep., Eur. Math. Soc. Zürich, 2008.Google Scholar
17.Knus, M.A. et Ojanguren, M.. Théorie de la descente et algèbres d'Azumaya. Springer Lecture Notes in Mathematics 389, Springer-Verlag, 1974.Google Scholar
18.Knus, M.A., Merkurjev, A., Rost, M., Tignol, J.-P.. The book of Involutions. American Math. Society Publications 44, 1998.Google Scholar
19.Serre, J.-P.. Cours d'Arithmétique. Presses Universitaires de France, Paris (1970).Google Scholar
20.Wall, C.T.C.. Graded Brauer groups. J. reine angew. Math. 213:187–199,1963.Google Scholar