Periodicity of hermitian K-groups

TL;DR

This paper establishes that periodicity in algebraic K-groups implies similar periodicity in hermitian K-groups, extending topological K-theory concepts to algebraic settings and providing bounds and relations to etale hermitian K-theory.

Contribution

It proves that algebraic K-group periodicity implies hermitian K-group periodicity and extends higher KSC theories, with applications to various algebraic and topological rings.

Findings

Periodic algebraic K-groups imply hermitian K-groups periodicity.

Provides bounds for hermitian K-groups in terms of algebraic K-groups.

Relates periodicity to etale hermitian K-theory via a descent theorem.

Abstract

Bott periodicity for the unitary, orthogonal and symplectic groups is fundamental to topological K-theory. Analogous to unitary topological K-theory, for algebraic K-groups with finite coefficients similar periodicity 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 its 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 the higher algebraic K-groups. We also relate periodicity to etale hermitian K-groups by proving ahermitian 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.