Puneet Varma (Editor)

Whitehead's lemma

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Whitehead's lemma is a technical result in abstract algebra used in algebraic K-theory. It states that a matrix of the form

[ u 0 0 u − 1 ]

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

[ u 0 0 u − 1 ] = e 21 ( u − 1 ) e 12 ( 1 − u ) e 21 ( − 1 ) e 12 ( 1 − u − 1 ) .

Here, e i j ( s ) indicates a matrix whose diagonal block is 1 and i j t h 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,

E ⁡ ( A ) = [ GL ⁡ ( A ) , 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

GL ⁡ ( 2 , Z / 2 Z )

one has:

Alt ⁡ ( 3 ) ≅ [ GL 2 ⁡ ( Z / 2 Z ) , GL 2 ⁡ ( Z / 2 Z ) ] < E 2 ⁡ ( Z / 2 Z ) = SL 2 ⁡ ( Z / 2 Z ) = GL 2 ⁡ ( Z / 2 Z ) ≅ Sym ⁡ ( 3 ) ,

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

References

Whitehead's lemma Wikipedia
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