Whitehead's lemma: Difference between revisions
Appearance
Content deleted Content added
cite Snaith |
Bluelinking 1 books for verifiability.) #IABot (v2.1alpha3 |
||
Line 16: | Line 16: | ||
Here, <math>e_{ij}(s)</math> indicates a matrix whose diagonal block is <math>1</math> and <math>ij^{th}</math> entry is <math>s</math>. |
Here, <math>e_{ij}(s)</math> indicates a matrix whose diagonal block is <math>1</math> and <math>ij^{th}</math> entry is <math>s</math>. |
||
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]].<ref name=Mil31>{{cite book | last1=Milnor | first1=John Willard | author1-link= John Milnor | title=Introduction to algebraic K-theory | publisher=[[Princeton University Press]] | location=Princeton, NJ | mr=0349811 | year=1971 | zbl=0237.18005 | series=Annals of Mathematics Studies | volume=72 | at=Section 3.1 }}</ref><ref name=Sn164>{{cite book | title=Explicit Brauer Induction: With Applications to Algebra and Number Theory | volume=40 | series=Cambridge Studies in Advanced Mathematics | first=V. P. | last=Snaith | authorlink= | publisher=[[Cambridge University Press]] | year=1994 | isbn=0-521-46015-8 | zbl=0991.20005 | page=164 }}</ref> In symbols, |
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]].<ref name=Mil31>{{cite book | last1=Milnor | first1=John Willard | author1-link= John Milnor | title=Introduction to algebraic K-theory | publisher=[[Princeton University Press]] | location=Princeton, NJ | mr=0349811 | year=1971 | zbl=0237.18005 | series=Annals of Mathematics Studies | volume=72 | at=Section 3.1 }}</ref><ref name=Sn164>{{cite book | title=Explicit Brauer Induction: With Applications to Algebra and Number Theory | volume=40 | series=Cambridge Studies in Advanced Mathematics | first=V. P. | last=Snaith | authorlink= | publisher=[[Cambridge University Press]] | year=1994 | isbn=0-521-46015-8 | zbl=0991.20005 | page=[https://archive.org/details/explicitbrauerin0000snai/page/164 164] | url=https://archive.org/details/explicitbrauerin0000snai/page/164 }}</ref> In symbols, |
||
:<math>\operatorname{E}(A) = [\operatorname{GL}(A),\operatorname{GL}(A)]</math>. |
:<math>\operatorname{E}(A) = [\operatorname{GL}(A),\operatorname{GL}(A)]</math>. |
||
Revision as of 09:50, 18 December 2019
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
- ^ 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.
- ^ 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.