Whitehead's lemma



Whitehead's lemma is a technical result in abstract algebra used in algebraic K-theory. It states that a matrix of the form


 * $$\begin{bmatrix}

u & 0 \\ 0 & u^{-1} \end{bmatrix}$$

is equivalent to the identity matrix by elementary transformations (that is, transvections):


 * $$\begin{bmatrix}

u & 0 \\ 0 & u^{-1} \end{bmatrix} = e_{21}(u^{-1}) e_{12}(1-u) e_{21}(-1) e_{12}(1-u^{-1}).$$

Here, $$e_{ij}(s)$$ indicates a matrix whose diagonal block is $$1$$ and $$ij$$-th entry is $$s$$.

The name "Whitehead's lemma" also refers to the closely related result that the derived group of the stable general linear group is the group generated by elementary matrices. In symbols,
 * $$\operatorname{E}(A) = [\operatorname{GL}(A),\operatorname{GL}(A)]$$.

This holds for the stable group (the direct limit of matrices of finite size) over any ring, but not in general for the unstable groups, even over a field. For instance for
 * $$\operatorname{GL}(2, \mathbb{Z}/2\mathbb{Z})$$

one has:
 * $$\operatorname{Alt}(3) \cong [\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}),\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z})] < \operatorname{E}_2(\mathbb{Z}/2\mathbb{Z}) = \operatorname{SL}_2(\mathbb{Z}/2\mathbb{Z}) = \operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}) \cong \operatorname{Sym}(3),$$

where Alt(3) and Sym(3) denote the alternating resp. symmetric group on 3 letters.