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}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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}).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Here,
indicates a matrix whose diagonal block is
and
-th entry is
.
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,
.
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} )}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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),}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where Alt(3) and Sym(3) denote the alternating resp. symmetric group on 3 letters.
See also
References