Steinberg group (K-theory)
Abstractly, given a ring , the Steinberg group 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 , where is the identity matrix, is the matrix with in the -entry and zeros elsewhere, and — satisfy the following relations, called the Steinberg relations:
The unstable Steinberg group of order over , denoted by , is defined by the generators , where and , these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by , is the direct limit of the system . It can also be thought of as the Steinberg group of infinite order.
Relation to -Theory
is the cokernel of the map , as is the abelianization of and the mapping is surjective onto the commutator subgroup.
is the center of the Steinberg group. This was Milnor's definition, and it also follows from more general definitions of higher -groups.
It is also the kernel of the mapping . Indeed, there is an exact sequence
Gersten (1973) showed that .
- Gersten, S. M. (1973), " of a Ring is of the Steinberg Group", Proceedings of the American Mathematical Society (American Mathematical Society) 37 (2): 366–368, doi:10.2307/2039440, JSTOR 2039440
- Milnor, John Willard (1971), Introduction to Algebraic -theory, Annals of Mathematics Studies 72, Princeton University Press, MR 0349811