# 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 $GL_m(R)/E_m(R)$?

## 1 Answer

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.

The first relevant link is "On the stabilization of general linear group" by L. Vaserstein. I believe that any paper on the subject cites this one, so you can explore them by following the link.

For the particular problem about polynomials over a division ring, take a look at "The general linear group of polynomial rings over regular rings" by T. Vorst. Here's the abstract:

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.

• @ Andrei Smolensky Thanks, this is exactly what I needed. I am having a bit of trouble accessing the paper by Vorst, is there any chance you know what the two types of regular ring mentioned in the abstract actually are? Dec 30, 2016 at 15:19
• i) complete regular equicharacteristic local ring; ii) regular and of essentially finite type over a perfect field; And now I start to wonder whether division algebras fall into one of these two types... Dec 30, 2016 at 15:29
• Well I am mostly interested in rings that are finite dimensional over their center (which in my case would be a perfect field). That said, I don't think it is of essentially finite type Dec 30, 2016 at 15:39
• such as $\left(\frac{-1,-1}{\mathbb{Q}}\right)$. Dec 30, 2016 at 15:46