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.

Steinberg 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 $p\neq q$—satisfy the following relations, called the Steinberg relations:

${\begin{aligned}e_{ij}(\lambda )e_{ij}(\mu )&=e_{ij}(\lambda +\mu )\\\left[e_{ij}(\lambda ),e_{jk}(\mu )\right]&=e_{ik}(\lambda \mu )&&{\mbox{for }}i\neq k\\\left[e_{ij}(\lambda ),e_{kl}(\mu )\right]&=\mathbf {1} &&{\mbox{for }}i\neq l,j\neq k\\\end{aligned}}$

The unstable Steinberg group of order r over A, $\operatorname {St} _{r}(A)$, is defined by the generators $x_{ij}(\lambda )$, $1\leq i,j\leq r,i\neq j,\lambda \in A$, subject to the Steinberg relations. The stable Steinberg group, $\operatorname {St} (A)$, is the direct limit of the system $\operatorname {St} _{r}(A)\to \operatorname {St} _{r+1}(A)$. 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

$\varphi \colon \operatorname {St} (A)\to \operatorname {GL} (A).$

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 $\varphi \colon \operatorname {St} (A)\to \operatorname {GL} (A)$, 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 $\varphi \colon \operatorname {St} (A)\to \operatorname {GL} (A)$, and indeed there is an exact sequence

$1\longrightarrow K_{2}(A)\longrightarrow \operatorname {St} (A)\longrightarrow \operatorname {GL} (A)\longrightarrow K_{1}(A)\longrightarrow 1.$

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

Gersten (1973) showed that $K_{3}$ of a ring is $H_{3}$ of the Steinberg group.

References

• Gersten, S. M. (1973), "$K_{3}$ of a Ring is $H_{3}$ of the Steinberg Group", Proceedings of the American Mathematical Society, 37 (2), American Mathematical Society: 366–368, doi:10.2307/2039440, JSTOR 2039440
• Milnor, John Willard (1971), Introduction to algebraic K-theory, Annals of Mathematics Studies, vol. 72, Princeton University Press, MR 0349811
• Steinberg, Robert (1968), Lectures on Chevalley groups, Yale University, New Haven, Conn., MR 0466335