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

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

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

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

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

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

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,

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

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

${\displaystyle \operatorname {GL} (2,\mathbb {Z} /2\mathbb {Z} )}$

one has:

${\displaystyle \operatorname {Alt} (3)\cong [\operatorname {GL} _{2}(\mathbb {Z} /2\mathbb {Z} ),\operatorname {GL} _{2}(\mathbb {Z} /2\mathbb {Z} )]<\operatorname {E} _{2}(\mathbb {Z} /2\mathbb {Z} )=\operatorname {SL} _{2}(\mathbb {Z} /2\mathbb {Z} )=\operatorname {GL} _{2}(\mathbb {Z} /2\mathbb {Z} )\cong \operatorname {Sym} (3),}$

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