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 of $A$.

It is named after Robert Steinberg, and it 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 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 ${e_{pq}}(\lambda ):=\mathbf {1} +{a_{pq}}(\lambda )$, 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:

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

The unstable Steinberg group of order $r$ over $A$, denoted by ${\operatorname {St} _{r}}(A)$, is defined by the generators ${x_{ij}}(\lambda )$, where $1\leq i\neq j\leq r$ and $\lambda \in A$, these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by $\operatorname {St} (A)$, 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 $\varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)$. As the elementary matrices generate the commutator subgroup, this mapping is surjective onto the commutator subgroup.

Relation to $K$-Theory

$K_{1}$

${K_{1}}(A)$ is the cokernel of the map $\varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)$, as $K_{1}$ is the abelianization of ${\operatorname {GL} _{\infty }}(A)$ and the mapping $\varphi$ is surjective onto the commutator subgroup.

$K_{2}$

${K_{2}}(A)$ is the center of the Steinberg group. This was Milnor's definition, and it also follows from more general definitions of higher $K$-groups.

It is also the kernel of the mapping $\varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)$. Indeed, there is an exact sequence

$1\to {K_{2}}(A)\to \operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)\to {K_{1}}(A)\to 1.$

Equivalently, it is the Schur multiplier of the group of elementary matrices, so it is also a homology group: ${K_{2}}(A)={H_{2}}(E(A);\mathbb {Z} )$.

$K_{3}$

Gersten (1973) showed that ${K_{3}}(A)={H_{3}}(\operatorname {St} (A);\mathbb {Z} )$.

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