Steinberg group (K-theory)

In algebraic K-theory, a field of mathematics, the Steinberg group formula_1 of a ring formula_2 is the universal central extension of the commutator subgroup of the stable general linear group of formula_2.
It is named after Robert Steinberg, and it is connected with lower formula_4-groups, notably formula_5 and formula_6.
Definition.
Abstractly, given a ring formula_2, the Steinberg group formula_1 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).
Presentation using generators and relations.
A concrete presentation using generators and relations is as follows. Elementary matrices — i.e. matrices of the form formula_9, where formula_10 is the identity matrix, formula_11 is the matrix with formula_12 in the formula_13-entry and zeros elsewhere, and formula_14 — satisfy the following relations, called the Steinberg relations:
The unstable Steinberg group of order formula_16 over formula_2, denoted by formula_18, is defined by the generators formula_19, where formula_20 and formula_21, these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by formula_1, is the direct limit of the system formula_23. It can also be thought of as the Steinberg group of infinite order.
Mapping formula_24 yields a group homomorphism formula_25. As the elementary matrices generate the commutator subgroup, this mapping is surjective onto the commutator subgroup.
Interpretation as a fundamental group.
The Steinberg group is the fundamental group of the Volodin space, which is the union of classifying spaces of the unipotent subgroups of GL("A").
Relation to "K"-theory.
"K"1.
formula_26 is the cokernel of the map formula_25, as formula_28 is the abelianization of formula_29 and the mapping formula_30 is surjective onto the commutator subgroup.
"K"2.
formula_31 is the center of the Steinberg group. This was Milnor's definition, and it also follows from more general definitions of higher formula_4-groups.
It is also the kernel of the mapping formula_25. Indeed, there is an exact sequence
Equivalently, it is the Schur multiplier of the group of elementary matrices, so it is also a homology group: formula_35.
"K"3.
 showed that formula_36.