Periodicity of hermitian K -groups

TLDR: In this article, it was shown that the periodicity of the algebraic K-groups for any ring implies periodicity for the hermitian K-group, analogous to orthogonal and symplectic topological K-theory.

Abstract: Bott periodicity for the unitary and symplectic groups is fundamental to topological K-theory. Analogous to unitary topological K-theory, for algebraic Kgroups 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 etale hermitian K-groups by proving a hermitian version of Thomason’s etale 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.