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Periodicity of hermitian K-groups

Published online by Cambridge University Press:  16 May 2011

A. J. Berrick ,
M. Karoubi  and
P. A. Østvær
A. J. Berrick
Affiliation:
Department of Mathematics, National University of Singapore, Singaporeberrick@math.nus.edu.sg
M. Karoubi
Affiliation:
UFR de Mathématiques, Université Paris 7, Francemax.karoubi@gmail.com
P. A. Østvær
Affiliation:
Department of Mathematics, University of Oslo, Norwaypaularne@math.uio.no
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Abstract

Bott periodicity for the unitary and symplectic groups is fundamental to topological K-theory. Analogous to unitary topological K-theory, for algebraic K-groups with finite coefficients, similar results are consequences of the Milnor and Bloch-Kato conjectures, affirmed by Voevodsky, Rost and others. More generally, we prove that periodicity of the algebraic K-groups for any ring implies periodicity for the hermitian K-groups, analogous to orthogonal and symplectic topological K-theory.

The proofs use in an essential way higher KSC-theories, extending those of Anderson and Green. They also provide an upper bound for the higher hermitian K-groups in terms of higher algebraic K-groups.

We also relate periodicity to étale hermitian K-groups by proving a hermitian version of Thomason's étale descent theorem. The results are illustrated in detail for local fields, rings of integers in number fields, smooth complex algebraic varieties, rings of continuous functions on compact spaces, and group rings.

Keywords

Bott periodicityhigher hermitian K-groupsKSC-theoriesétale cohomology
Type
Research Article
Information
Journal of K-Theory , Volume 7 , Issue 3 , June 2011 , pp. 429 - 493
Copyright
Copyright © ISOPP 2011

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