Homogeneous SK1 of simple graded algebras

TL;DR

This paper investigates the properties of the homogeneous SK1 group in simple graded algebras, establishing a relationship with the base algebra's SK1 and highlighting its non-Morita invariance.

Contribution

It introduces a short exact sequence connecting the homogeneous SK1 of a simple graded algebra to that of its base graded division algebra, revealing new invariance properties.

Findings

Homogeneous SK1 is not generally Morita invariant.

A short exact sequence relates the homogeneous SK1 of the algebra to that of the base algebra.

The paper clarifies the structure of SK1 in the context of simple graded algebras.

Abstract

For a simple graded algebra A=M_n(E) over a graded division algebra E, a short exact sequence relating the reduced Whitehead group of the homogeneous part of A to that of E is established. In particular it is shown that the homogeneous SK1 is not in general Morita invariant.