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. 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

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