Relative unitary commutator calculus and applications

TL;DR

This paper develops a more general and simplified relative commutator calculus for unitary groups, enabling new results like the mixed commutator formula for form ideals, advancing the algebraic understanding of these structures.

Contribution

It introduces a new, more general relative commutator calculus for unitary groups and applies it to prove the mixed commutator formula, solving open problems.

Findings

Developed a more general relative conjugation and commutator calculus for unitary groups.

Proved the mixed commutator formula for two form ideals of a form ring.

Simplified the existing calculus methods, making them more accessible.

Abstract

This note revisits localisation and patching method in the setting of generalised unitary groups. Introducing certain subgroups of relative elementary unitary groups, we develop relative versions of the conjugation calculus and the commutator calculus in unitary groups, which are both more general, and substantially easier than the ones available in the literature. For the general linear group such relative commutator calculus has been recently developed by the first and the third authors. As an application we prove the mixed commutator formula, for two form ideals of a form ring. This answers two problems posed in a paper by Alexei Stepanov and the second author.