Steinberg group (K-theory)

In algebraic K-theory, a field of mathematics, the Steinberg group $operatorname\{St\}(A)$ of a ring "A", is the universal central extension of the commutator subgroup of the stable general linear group.

It is named after Robert Steinberg, and is connected with lower K-groups, notably $K\_2$ and $K\_3$.

Definition

Abstractly, given a ring "A", the Steinberg group $operatorname\{St\}(A)$ is the universal central extension of the commutator subgroup of the stable general linear group (the commutator subgroup is perfect, hence has a universal central extension).

Concretely, it can also be described by generators and relations.

teinberg relations

Elementary matrices—meaning matrices of the form $e\_\{pq\}(lambda)\; :=\; mathbf\{1\}\; +\; a\_\{pq\}(lambda)$, where $mathbf\{1\}$ is the identity matrix, $a\_\{pq\}(lambda)$ is the matrix with $lambda$ in the $(p,q)$ entry and zeros elsewhere, and $p\; eq\; q$—satisfy the following relations, called the Steinberg relations:

:

The unstable Steinberg group of order "r" over "A", $operatorname\{St\}\_r(A)$, is defined by the generators$x\_\{ij\}(lambda)$, $1leq\; i,jleq\; r,\; i\; eq\; j,\; lambda\; in\; A$, subject to the Steinberg relations. The stable Steinberg group, $operatorname\{St\}(A)$, is the direct limit of the system $operatorname\{St\}\_r(A)\; o\; operatorname\{St\}\_\{r+1\}(A)$. It can also be thought of as the Steinberg group of infinite order.

Mapping $x\_\{ij\}(lambda)\; mapsto\; e\_\{ij\}(lambda)$ yields a group homomorphism

:$varphicolonoperatorname\{St\}(A)\; ooperatorname\{GL\}(A).$

As the elementary matrices generate the commutator subgroup, this map is onto the commutator subgroup.

Relation to K-theory

K₁

$K\_1(A)$ is the cokernel of the map $varphicolonoperatorname\{St\}(A)\; o\; operatorname\{GL\}(A)$, as $K\_1$ is the abelianization of $operatorname\{GL\}(A)$ and $varphi$ is onto the commutator subgroup.

K₂

$K\_2(A)$ is the center of the Steinberg group; this was Milnor's definition, and also follows from more general definitions of higher K-groups.

It is also the kernel of the map $varphicolonoperatorname\{St\}(A)\; ooperatorname\{GL\}(A)$, and indeed there is an exact sequence:$1longrightarrowK\_2(A)\; longrightarrowoperatorname\{St\}(A)\; longrightarrowoperatorname\{GL\}(A)\; longrightarrowK\_1(A)longrightarrow\; 1.$

Equivalently, it is the Schur multiplier of the group of elementary matrices, and thus is also a homology group: $K\_2(A)\; =\; H\_2(operatorname\{E\}(A),mathbf\{Z\})$.

K₃

$K\_3$ of a ring is $H\_3$ of the Steinberg group.

This result is proven is the eponymous paper:
* cite journal
title=$K\_3$ of a Ring is $H\_3$ of the Steinberg Group
author=S. M. Gersten
journal=Proceedings of the American Mathematical Society
vol=37
issue=2
date=Feb., 1973
pages=366–368
doi=10.2307/2039440
id=JSTOR stable URL|0002-9939(197302)37%3A2%3C366%3AOARIOT%3E2.0.CO%3B2-Z
month=Feb
year=1973
volume=37