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[1] 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).

See also

References

  1. ^ J. Milnor, Introduction to algebraic K -theory, Annals of Mathematics Studies 72, Princeton University Press, 1971. Section 3.1.