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

[math]\displaystyle{ \begin{bmatrix} u & 0 \\ 0 & u^{-1} \end{bmatrix} }[/math]

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

[math]\displaystyle{ \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}). }[/math]

Here, [math]\displaystyle{ e_{ij}(s) }[/math] indicates a matrix whose diagonal block is [math]\displaystyle{ 1 }[/math] and [math]\displaystyle{ ij }[/math]-th entry is [math]\displaystyle{ s }[/math].

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,

[math]\displaystyle{ \operatorname{E}(A) = [\operatorname{GL}(A),\operatorname{GL}(A)] }[/math].

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

[math]\displaystyle{ \operatorname{GL}(2, \mathbb{Z}/2\mathbb{Z}) }[/math]

one has:

[math]\displaystyle{ \operatorname{Alt}(3) \cong [\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}),\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z})] \lt \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), }[/math]

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

See also

• Special linear group#Relations to other subgroups of GL(n, A)