A matrix description for $K_1$ of graded rings

TL;DR

This paper develops a matrix-based framework for computing the $K_1$ groups of graded rings, extending classical algebraic $K$-theory to the graded setting with new computational tools.

Contribution

It introduces a matrix description for the $K_1^{gr}$ of graded rings, enabling explicit calculations and extending the classical $K$-theory exact sequence to the graded context.

Findings

Matrix description of $K_1^{gr}$ facilitates computations.

$K_1^{gr}$ is a $bZ[ ext{grading group}]$-module.

Exact sequence for graded $K_1$ groups is established.

Abstract

The current paper is dedicated to the study of the classical $K_1$ groups of graded rings. Let $A$ be a $\Gamma$ graded ring with identity $1$, where the grading $\Gamma$ is an abelian group. We associate a category with suspension to the $\Gamma$ graded ring $A$. This allows us to construct the group valued functor $K_1$ of graded rings. It will be denoted by $K_1^{gr}$. It is not only an abelian group but also a $\mathbb Z[\Gamma]$-module. From the construction, it follows that there exists "locally" a matrix description of $K_1^{gr}$ of graded rings. The matrix description makes it possible to compute $K_1^{gr}$ of various types of graded rings. The $K_1^{gr}$ satisfies the well known $K$-theory exact sequence $$ K_{1}^{gr}(A,I)\to K_1^{gr}(A)\to K_1^{gr}(A/I) $$ for any graded ideal $I$ of $A$. The above is used to compute $K^{gr}_1$ of cross products.