Steinberg group (K-theory)

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In algebraic K-theory, a field of mathematics, the Steinberg group of a ring is the universal central extension of the commutator subgroup of the stable general linear group of .

It is named after Robert Steinberg, and it is connected with lower -groups, notably and .

Definition

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.

Steinberg 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.

Mapping yields a group homomorphism . As the elementary matrices generate the commutator subgroup, this mapping is surjective onto the commutator subgroup.

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

Equivalently, it is the Schur multiplier of the group of elementary matrices, so it is also a homology group: .

Template:Harvtxt showed that .

References

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