Proof of the Whitehead Lemma

The Whitehead lemma is one of the foundational results in algebraic K-theory: it states that the group of elementary matrices $E(R)$ equals the commutator subgroup $[GL(R), GL(R)]$ of the stable general linear group. This justifies the definition $K_1(R) = GL(R)/E(R)$ as an abelianization.

Statement

Theorem1.3Whitehead Lemma

For any ring $R$ , the subgroup $E(R)$ of elementary matrices equals the commutator subgroup of $GL(R)$ :

$E(R) = [GL(R), GL(R)].$

In particular, $K_1(R) = GL(R)/E(R) = GL(R)^{\mathrm{ab}}$ is the abelianization of $GL(R)$ .

Proof

The proof has two parts: showing $E(R) \subseteq [GL(R), GL(R)]$ and then $[GL(R), GL(R)] \subseteq E(R)$ .

Part 1: $E(R) \subseteq [GL(R), GL(R)]$ .

It suffices to show each elementary matrix $e_{ij}(\lambda)$ is a commutator in $GL(R)$ (after stabilization). We use the identity:

$e_{ij}(\lambda) = [e_{ik}(\lambda), e_{kj}(1)]$

for distinct indices $i, j, k$ . This is simply the Steinberg relation: $[e_{ik}(\lambda), e_{kj}(1)] = e_{ij}(\lambda \cdot 1) = e_{ij}(\lambda)$ . Since elementary matrices are elements of $GL(R)$ , this shows $e_{ij}(\lambda) \in [GL(R), GL(R)]$ .

Part 2: $[GL(R), GL(R)] \subseteq E(R)$ .

This is the substantial direction. We need to show that for any $A, B \in GL_n(R)$ , the commutator $ABA^{-1}B^{-1}$ lies in $E(R)$ (after stabilization).

Step 2a: The "whitehead trick." For any $A \in GL_n(R)$ , consider the block matrix

$\begin{pmatrix} A & 0 \\ 0 & A^{-1} \end{pmatrix} \in GL_{2n}(R).$

We claim this lies in $E_{2n}(R)$ . Indeed, we have the factorization:

$\begin{pmatrix} A & 0 \\ 0 & A^{-1} \end{pmatrix} = \begin{pmatrix} I & A \\ 0 & I \end{pmatrix} \begin{pmatrix} I & 0 \\ -A^{-1} & I \end{pmatrix} \begin{pmatrix} I & A \\ 0 & I \end{pmatrix} \begin{pmatrix} 0 & -I \\ I & 0 \end{pmatrix}.$

Each of the first three factors is a product of elementary matrices (block upper/lower triangular with identity diagonal blocks). The fourth factor $\begin{pmatrix} 0 & -I \\ I & 0 \end{pmatrix}$ is also a product of elementary matrices:

$\begin{pmatrix} 0 & -I \\ I & 0 \end{pmatrix} = \begin{pmatrix} I & -I \\ 0 & I \end{pmatrix}\begin{pmatrix} I & 0 \\ I & I \end{pmatrix}\begin{pmatrix} I & -I \\ 0 & I \end{pmatrix}.$

Thus $\begin{pmatrix} A & 0 \\ 0 & A^{-1} \end{pmatrix} \in E_{2n}(R)$ .

Step 2b: Express the commutator. For $A, B \in GL_n(R)$ , consider the commutator in $GL_{2n}(R)$ :

$\begin{pmatrix} ABA^{-1}B^{-1} & 0 \\ 0 & I \end{pmatrix} = \begin{pmatrix} AB & 0 \\ 0 & (AB)^{-1} \end{pmatrix} \cdot \begin{pmatrix} B^{-1} & 0 \\ 0 & B \end{pmatrix} \cdot \begin{pmatrix} A^{-1} & 0 \\ 0 & A \end{pmatrix}.$

Each factor on the right is of the form $\begin{pmatrix} C & 0 \\ 0 & C^{-1} \end{pmatrix}$ , which by Step 2a lies in $E_{2n}(R)$ .

Therefore $\begin{pmatrix} ABA^{-1}B^{-1} & 0 \\ 0 & I \end{pmatrix} \in E_{2n}(R)$ , which means $ABA^{-1}B^{-1} \in E(R)$ in the stable limit.

Conclusion. Combining Parts 1 and 2, $E(R) = [GL(R), GL(R)]$ , and $K_1(R) = GL(R)/E(R)$ is the abelianization. $\square$

■

Refined versions

RemarkUnstable Whitehead lemma

The stable version uses the limit $GL(R) = \varinjlim GL_n(R)$ . For the unstable version, one asks: for which $n$ does $E_n(R) = [GL_n(R), GL_n(R)]$ ?

For a commutative ring $R$ with stable range $\operatorname{sr}(R) = d$ , the identity $E_n(R) = [GL_n(R), GL_n(R)]$ holds for $n \geq d + 1$ . Recall:

$\operatorname{sr}(\mathbb{Z}) = 2$ , so $E_n(\mathbb{Z}) = SL_n(\mathbb{Z})$ for $n \geq 3$ .
$\operatorname{sr}(k[x_1, \ldots, x_d]) = d + 1$ for a field $k$ .
$\operatorname{sr}(\mathcal{O}_F) = 2$ for a number ring $\mathcal{O}_F$ .

The case $n = 2$ is delicate: $SL_2(\mathbb{Z})$ is not generated by elementary matrices alone (it is a free product $\mathbb{Z}/4 *_{\mathbb{Z}/2} \mathbb{Z}/6$ ), but $SL_2(\mathbb{Z}) = E_2(\mathbb{Z})$ does hold.

ExampleApplications to SK₁

The Whitehead lemma implies $K_1(R) = GL(R)^{\mathrm{ab}}$ . For commutative $R$ , this gives the short exact sequence

$1 \to SK_1(R) \to K_1(R) \xrightarrow{\det} R^{\times} \to 1$

where $SK_1(R) = SL(R)/E(R)$ .

Commutative local rings: $SK_1(R) = 0$ (every matrix in $SL_n(R)$ is elementary for $n$ large enough). This uses the fact that the stable range of a local ring is 1.
Division rings: For a division ring $D$ , $K_1(D) = D^{\times} / [D^{\times}, D^{\times}]$ , the abelianization of the multiplicative group. For the quaternion algebra $D = \mathbb{H}$ , $K_1(\mathbb{H}) \cong \mathbb{R}_{>0}$ since $\mathbb{H}^{\times}/[\mathbb{H}^{\times}, \mathbb{H}^{\times}] \cong \mathbb{R}_{>0}$ via the reduced norm.
Non-commutative examples: $SK_1$ can detect "non-commutativity." For certain central simple algebras $A$ over a number field, $SK_1(A)$ is finite and computed by the Wang--Platonov theorem.