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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 .

Contents

DefinitionEdit

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).

Presentation using generators and relationsEdit

A concrete presentation using generators and relations is as follows. 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.

Interpretation as a fundamental groupEdit

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-theoryEdit

K1Edit

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

K2Edit

  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:  .

K3Edit

Gersten (1973) showed that  .

ReferencesEdit

  • 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