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, American Mathematical Society, 37 (2): 366–368, doi:10.2307/2039440, JSTOR 2039440

• Milnor, John Willard (1971), Introduction to Algebraic $K$-theory, Annals of Mathematics Studies, 72, Princeton University Press, MR 0349811

• Steinberg, Robert (1968), Lectures on Chevalley Groups, Yale University, New Haven, Conn., MR 0466335