Stability for odd unitary $K_1$

TL;DR

This paper introduces a new algebraic framework for odd unitary groups using odd form rings, proving stability theorems for odd unitary K_1 without relying on linear K-theory, and establishing stabilization results for related groups.

Contribution

It provides a novel algebraic approach to odd unitary groups and proves stability theorems independently of linear K-theory results under stable rank conditions.

Findings

Proved stability theorems for odd unitary K_1 using algebraic methods.

Established stabilization results for projective and general unitary groups.

Developed a new purely algebraic approach with odd form rings.

Abstract

We give a new purely algebraic approach to odd unitary groups using odd form rings. Using these objects, we prove the stability theorems for odd unitary $K_1$-functor without using the corresponding result from linear $K$-theory under the ordinary stable rank condition. Moreover, we prove a natural stabilization result for projective unitary groups and various general unitary groups.