Whitehead's lemma

For a lemma on Lie algebras, see Whitehead's lemma (Lie algebras).

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.[1][2] 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.

See also

References

  1. ↑ Milnor, John Willard (1971). Introduction to algebraic K-theory. Annals of Mathematics Studies 72. Princeton, NJ: Princeton University Press. Section 3.1. MR 0349811. Zbl 0237.18005.
  2. ↑ Snaith, V. P. (1994). Explicit Brauer Induction: With Applications to Algebra and Number Theory. Cambridge Studies in Advanced Mathematics 40. Cambridge University Press. p. 164. ISBN 0-521-46015-8. Zbl 0991.20005.