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 ) .
