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[edit]

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[edit]

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[edit]

[edit]

is the cokernel of the map , as is the abelianization of and the mapping is surjective onto the commutator subgroup.

[edit]

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: .

[edit]

Gersten (1973) showed that .

References[edit]

  • 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