Results on $\mathrm{K}_1$ of general quadratic groups

TL;DR

This paper investigates the properties of K_1 groups of general quadratic groups, extending local-global principles and applying them to stable, unstable, and nil K-groups.

Contribution

It introduces graded local-global principles for quadratic groups and applies these to analyze K_1 and Bass' nil groups in new contexts.

Findings

Extended local-global principles for quadratic groups.

Results on stable and unstable K_1 groups.

Applications to Bass' nil groups.

Abstract

In the first part of this article we discuss the relative cases of Quillen-Suslin's local-global principle for the general quadratic (Bak's unitary) groups, and its applications for the (relative) stable and unstable $\mathrm{K}_1$-groups. The second part is dedicated to the graded version of the local-global principle for the general quadratic groups and its application to deduce a result for Bass' nil groups.