[Paper Review] Periodicity of hermitian K-theory and Milnor's K-groups

Max Karoubi|ArXiv.org|Sep 20, 2005

Algebraic Geometry and Number Theory4 references3 citations

TL;DR

This paper establishes a periodicity theorem in hermitian K-theory using recent results from Berrick and Karoubi, introducing a new filtration of the Witt ring W(A) over commutative rings with 2 invertible. By linking this filtration to Milnor and Quillen K-groups and leveraging Voevodsky's proof of Milnor's conjecture, the authors show that nontriviality of Milnor K-groups modulo 2 implies nontriviality of higher Witt groups when A is a field.

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).

Motivation & Objective

• To refine the periodicity theorem in hermitian K-theory using recent advances in K-theory.
• To define a new filtration on the classical Witt ring W(A) for commutative rings A where 2 is invertible.
• To establish a structural connection between this filtration and Milnor and Quillen K-groups.
• To demonstrate that nontriviality of Milnor K-groups modulo 2 implies nontriviality of higher Witt groups when A is a field.
• To leverage Voevodsky's proof of Milnor's conjecture to derive implications in hermitian K-theory and Witt group structure.

Proposed method

• Utilizes recent results from Berrick and Karoubi (math.KT/0509404) to strengthen the periodicity theorem in hermitian K-theory.
• Constructs a new filtration on the Witt ring W(A) based on nondegenerate quadratic forms over a commutative ring A with 2 invertible.
• Relies on the algebraic structure of Milnor and Quillen K-groups to relate the filtration layers to K-theoretic invariants.
• Applies Voevodsky's proof of Milnor's conjecture to connect mod 2 Milnor K-groups with Galois cohomology and Witt group structure.
• Uses the isomorphism between Milnor K-theory modulo 2 and Galois cohomology to deduce properties of higher Witt groups.
• Analyzes the implications of nontrivial Milnor K-groups mod 2 on the vanishing or nonvanishing of higher Witt groups in the case of fields.

Experimental results

Research questions

• RQ1 How can the periodicity theorem in hermitian K-theory be improved using recent K-theoretic results?
• RQ2 What is the structure of a new filtration on the Witt ring W(A) defined via nondegenerate quadratic forms over rings with 2 invertible?
• RQ3 How are the layers of this filtration related to Milnor and Quillen K-groups?
• RQ4 What does the nontriviality of Milnor K-groups modulo 2 imply for the structure of higher Witt groups when A is a field?
• RQ5 Can Voevodsky’s proof of Milnor’s conjecture be used to derive new results in hermitian K-theory and Witt group theory?

Key findings

• The paper improves the periodicity theorem in hermitian K-theory by incorporating recent advances in K-theory.
• A new filtration of the Witt ring W(A) is defined using nondegenerate quadratic forms over rings where 2 is invertible.
• This filtration is shown to be deeply connected to Milnor and Quillen K-groups through structural and cohomological links.
• The nontriviality of Milnor K-groups modulo 2 implies the nontriviality of higher Witt groups when A is a field.
• The result follows from applying Voevodsky’s proof of Milnor’s conjecture, which establishes an isomorphism between Milnor K-theory modulo 2 and Galois cohomology.
• The construction provides a new algebraic mechanism to detect nontriviality in higher Witt groups via K-theoretic invariants.