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Whitehead's lemma

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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

[ u 0 0 u − 1 ] {\displaystyle {\begin{bmatrix}u&0\\0&u^{-1}\end{bmatrix}}}

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 ) . {\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, e i j ( s ) {\displaystyle e_{ij}(s)} indicates a matrix whose diagonal block is 1 {\displaystyle 1} and i j {\displaystyle ij} -th entry is s {\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. In symbols,

E ⁡ ( A ) = [ GL ⁡ ( A ) , GL ⁡ ( A ) ] {\displaystyle \operatorname {E} (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 ) {\displaystyle \operatorname {GL} (2,\mathbb {Z} /2\mathbb {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 ) , {\displaystyle \operatorname {Alt} (3)\cong <\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. Vol. 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. Vol. 40. Cambridge University Press. p. 164. ISBN 0-521-46015-8. Zbl 0991.20005.


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