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

Abstract

Let \(R\) be a ring with unit. Passing to the colimit with respect to the standard inclusions \(\mathrm{GL}(n,R) \to \mathrm{GL}(n+1,R)\) (which add a unit vector as new last row and column) yields, by definition, the stable linear group \(\mathrm{GL}(R)\); 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 \(K_1(R) = \mathrm{GL}(R)/E(R)\) of \(R\), giving an elementary description that does not involve elementary matrices explicitly.