From Wikipedia, the free encyclopedia
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)}
1
{\displaystyle 1}
i
j
t
h
{\displaystyle ij^{th}}
s
{\displaystyle s}
The name "Whitehead's lemma" also refers to the closely related result[ 1] 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)=[\operatorname {GL} (A),\operatorname {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
)
{\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 {GL} _{2}(\mathbb {Z} /2\mathbb {Z} ),\operatorname {GL} _{2}(\mathbb {Z} /2\mathbb {Z} )]<\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
^ J. Milnor , Introduction to algebraic K -theory, Annals of Mathematics Studies 72, Princeton University Press, 1971. Section 3.1.
![{\displaystyle \operatorname {E} (A)=[\operatorname {GL} (A),\operatorname {GL} (A)]}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/7f46da4b6b93a3e69bcc12eef1d2a6a47da66007)
![{\displaystyle \operatorname {Alt} (3)\cong [\operatorname {GL} _{2}(\mathbb {Z} /2\mathbb {Z} ),\operatorname {GL} _{2}(\mathbb {Z} /2\mathbb {Z} )]<\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),}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/bc2418d5d063a8f39416e16e188d9d34cc2172ca)
