In algebraic K-theory, a field of mathematics, the Steinberg group
St
(
A
)
of a ring
A
is the universal central extension of the commutator subgroup of the stable general linear group of
A
.
It is named after Robert Steinberg, and it is connected with lower
K
-groups, notably
K
2
and
K
3
.
Abstractly, given a ring
A
, the Steinberg group
St
(
A
)
is the universal central extension of the commutator subgroup of the stable general linear group (the commutator subgroup is perfect and so has a universal central extension).
Concretely, it can be described using generators and relations.
Elementary matrices — i.e. matrices of the form
e
p
q
(
λ
)
:=
1
+
a
p
q
(
λ
)
, where
1
is the identity matrix,
a
p
q
(
λ
)
is the matrix with
λ
in the
(
p
,
q
)
-entry and zeros elsewhere, and
p
≠
q
— satisfy the following relations, called the Steinberg relations:
e
i
j
(
λ
)
e
i
j
(
μ
)
=
e
i
j
(
λ
+
μ
)
;
[
e
i
j
(
λ
)
,
e
j
k
(
μ
)
]
=
e
i
k
(
λ
μ
)
,
for
i
≠
k
;
[
e
i
j
(
λ
)
,
e
k
l
(
μ
)
]
=
1
,
for
i
≠
l
and
j
≠
k
.
The unstable Steinberg group of order
r
over
A
, denoted by
St
r
(
A
)
, is defined by the generators
x
i
j
(
λ
)
, where
1
≤
i
≠
j
≤
r
and
λ
∈
A
, these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by
St
(
A
)
, is the direct limit of the system
St
r
(
A
)
→
St
r
+
1
(
A
)
. It can also be thought of as the Steinberg group of infinite order.
Mapping
x
i
j
(
λ
)
↦
e
i
j
(
λ
)
yields a group homomorphism
φ
:
St
(
A
)
→
GL
∞
(
A
)
. As the elementary matrices generate the commutator subgroup, this mapping is surjective onto the commutator subgroup.
K
1
(
A
)
is the cokernel of the map
φ
:
St
(
A
)
→
GL
∞
(
A
)
, as
K
1
is the abelianization of
GL
∞
(
A
)
and the mapping
φ
is surjective onto the commutator subgroup.
K
2
(
A
)
is the center of the Steinberg group. This was Milnor's definition, and it also follows from more general definitions of higher
K
-groups.
It is also the kernel of the mapping
φ
:
St
(
A
)
→
GL
∞
(
A
)
. Indeed, there is an exact sequence
1
→
K
2
(
A
)
→
St
(
A
)
→
GL
∞
(
A
)
→
K
1
(
A
)
→
1.
Equivalently, it is the Schur multiplier of the group of elementary matrices, so it is also a homology group:
K
2
(
A
)
=
H
2
(
E
(
A
)
;
Z
)
.
Gersten (1973) showed that
K
3
(
A
)
=
H
3
(
St
(
A
)
;
Z
)
.
