Elementary matricies (Ex)

Let $R$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}R$ be an associative ring with unit and recall that an elementary matrix $E$$E$ over $R$$R$ is a square matrix of the form

$\displaystyle E = \text{Id}_n + a e_{ij}$ $$\displaystyle E = \text{Id}_n + a e_{ij}$$

where $\text{Id}_n$$\text{Id}_n$ is the $n \times n$$n \times n$ identity matrix, $a \in R$$a \in R$ and $e_{ij}$$e_{ij}$ is the matrix with zeros in all places except $(i, j)$$(i, j)$ where it is $1$$1$ and we have $i \neq j$$i \neq j$. Clearly each elementary matrix is invertible and so defines an element $E \in GL(R)$$E \in GL(R)$ where

$\displaystyle GL(R) = GL_{\infty}(R) : = \text{lim}_{n \to \infty}GL_n(R)$ $$\displaystyle GL(R) = GL_{\infty}(R) : = \text{lim}_{n \to \infty}GL_n(R)$$

is the limit of the invertible matricies.

Exercise 0.1. Prove that $E(R)=[GL(R),GL(R)]$$E(R)=[GL(R),GL(R)]$, where $E(R)\subset GL(R)$$E(R)\subset GL(R)$ is the subgroup generated by all elements in $GL(R)$$GL(R)$ which are represented by elementary matrices.

Hint 0.2. For $A\in GL(m,R)$$A\in GL(m,R)$ and $B\in GL(n,R)$$B\in GL(n,R)$ write the matrix

$\displaystyle \left( \begin{array}{cc} ABA^{-1}B^{-1} & 0\\ 0 & I \end{array} \right)$ $$\displaystyle \left( \begin{array}{cc} ABA^{-1}B^{-1} & 0\\ 0 & I \end{array} \right)$$

as a product of elementary matrices

$\displaystyle \left( \begin{array}{cc} I & X\\ 0 & I \end{array} \right) = \prod_{i=1}^{m}\prod_{j=1}^{n} (I+x_{ij}E_{i,j+m})$ $$\displaystyle \left( \begin{array}{cc} I & X\\ 0 & I \end{array} \right) = \prod_{i=1}^{m}\prod_{j=1}^{n} (I+x_{ij}E_{i,j+m})$$

where $X=(x_{ij})$$X=(x_{ij})$ is an $m\times n$$m\times n$ matrix.

Recall that $K_1(R)$$K_1(R)$ is defined to be the abelian group

$\displaystyle K_1(R) : = GL(R)_{ab} = GL(R)/[GL(R), GL(R)].$ $$\displaystyle K_1(R) : = GL(R)_{ab} = GL(R)/[GL(R), GL(R)].$$

Exercise 0.3. Prove that $K_1(\Zz) = \{ \pm 1 \}$$K_1(\Zz) = \{ \pm 1 \}$.