An elementary description of $K_1(R)$ without elementary matrices

TL;DR

This paper provides an elementary description of the algebraic K-group $K_1(R)$ that avoids explicit use of elementary matrices by considering colimits of general linear groups with different inclusions.

Contribution

It introduces a novel approach to defining $K_1(R)$ through colimits of $GL(n,R)$ along two types of standard inclusions, bypassing elementary matrices.

Findings

$K_1(R)$ can be obtained from colimits of $GL(n,R)$ with different inclusions.

The approach simplifies the understanding of $K_1(R)$ without elementary matrices.

Provides an elementary and unified perspective on the algebraic $K$-theory of rings.

Abstract

Let $R$ be a ring with unit. Passing to the colimit with respect to the standard inclusions $GL(n,R) \to GL(n+1,R)$ (which add a unit vector as new last row and column) yields, by definition, the stable linear group $GL(R)$; the same result is obtained, up to isomorphism, when using the "opposite" inclusions (which add a unit vector as new first row and column). In this note it is shown that passing to the colimit along both these families of inclusions simultaneously recovers the algebraic $K$-group $K_1(R) = GL(R)/E(R)$ of~$R$, giving an elementary description that does not involve elementary matrices explicitly.