EXACT SEQUENCES OF WITT GROUPS

ABSTRACT

An exact sequence of Witt groups, motivated by exact sequences obtained by Lewis and by Parimala, Sridharan and Suresh, is constructed. The behavior of the maps involved in these sequences with respect to isotropy is completely determined in the case of division algebras. In particular, the kernels of the maps involved in the previous sequences are explicitly given, leading to a new proof of their exactness. Similar exact sequences of equivariant Witt groups are constructed. As an application, relations between the cardinality of certain Witt groups are obtained.

Key Words:

• Central simple algebra with involution
• Equivariant Witt group
• Exact sequence of Witt groups
• Hermitian form
• Isotropy
• Morita equivalence
• Witt group

2000 Mathematics Subject Classification:

• Primary 11E39
• Secondary 11E81

ACKNOWLEDGMENTS

We thank Eva Bayer-Fluckiger, Detlev Hoffmann, and Emmanuel Lequeu for useful conversations about this work and for their comments on earlier versions of this paper. We also thank David Lewis for many helpful comments. We would also like to thank the referee whose suggestions helped improve a previous version of this paper. This research was supported by the RT Network “K-theory, Linear Algebraic Groups and Related Structures” (contract HPRN-CT-2002-00287).

Notes

Communicated by A. Prestel.