Whitehead's lemma
- <templatestyles src="Module:Hatnote/styles.css"></templatestyles>
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 the identity matrix by elementary transformations (that is, transvections):
Here, indicates a matrix whose diagonal block is
and
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
one has:
where Alt(3) and Sym(3) denote the alternating resp. symmetric group on 3 letters.
See also
References
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
<templatestyles src="Asbox/styles.css"></templatestyles>