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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

is equivalent to identity by elementary transformations (here "elementary matrices" means "transvections"):

Here, indicates a matrix whose diagonal block is and entry is .

It also refers to the closely related result[1] that the derived group of the stable general linear group is the group generated by elementary matrices. 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 one has:


References

  1. ^ J. Milnor, Introduction to algebraic K -theory, Annals of Mathematics Studies 72, Princeton University Press, 1971. Section 3.1.