# What does $K_1(R)$ tell us about $GL_n(R)/E_n(R)$?

Let $D$ be a division ring, and $R=D[t_1,\ldots,t_n]$. If $GL_m(R)$ is the usual group of invertible matrices over $R$, then by $E_m(R)$ I mean the subgroup of $GL_m(R)$ generated by the elementary matrices of the form $I_m+r\epsilon(i,j)$, $r\in U(R)$. It is known that the whitehead group $K_1(R)$ of $R$ is isomorphic to the commutator quotient group of $U(D)$. Is there any way we can extract from this, anything about $GL_m(R)/E_m(R)$?

In general what you are asking about is the problem of $K_1$-stabilization, that is the study of maps $GL(n, R)/E(n, R)\to GL(n+1, R)/E(n+1, R)$. For rings of finite dimension all such maps are bijective starting with some $n$ depending on the dimension.
In this note we shall prove for two types of regular rings $A$ that every element of $GL_r(A[X_1,\ldots,X_n])$ is a product of an element of $E_r(A[X_1,\ldots,X_n])$ (the group of elementary matrices) and an element of $GL_r(A)$, for $r\geq3$ and $n$ arbitrary. This is a kind of $GL_r$-analogue of results of Lindel and Mohan-Kumar and is an extension of a result of Suslin.
• such as $\left(\frac{-1,-1}{\mathbb{Q}}\right)$. Dec 30, 2016 at 15:46