Periodicity of hermitian K-theory and Milnor's K-groups

TL;DR

This paper advances the understanding of hermitian K-theory's periodicity, introduces a new filtration of the Witt ring linked to Milnor's K-groups, and connects Milnor's conjecture to higher Witt groups over fields.

Contribution

It improves the periodicity theorem in hermitian K-theory and defines a new filtration of the Witt ring related to Milnor and Quillen K-groups.

Findings

Improved periodicity theorem in hermitian K-theory.

Defined a new filtration of the Witt ring W(A).

Linked Milnor's K-groups mod 2 to higher Witt groups.

Abstract

We use recent results proved by Berrick and the author (math.KT/0509404) to improve the periodicity theorem in hermitian K-theory. We define also a new filtration of the classical Witt ring W(A), built from non degenerate quadratic forms over any commutative ring A where 2 is invertible. This filtration is linked to the Milnor and Quillen K-groups. Using the solution of Milnor's conjecture by Voevodsky, we show that the non triviality of Milnor's K-groups mod. 2 implies also the non triviality of higher Witt groups (when A is a field).