Steinberg group (K-theory)

A concrete presentation using generators and relations is as follows. Elementary matrices — i.e. matrices of the form ${\displaystyle {e_{pq}}(\lambda ):=\mathbf {1} +{a_{pq}}(\lambda )}$, where ${\displaystyle \mathbf {1} }$ is the identity matrix, ${\displaystyle {a_{pq}}(\lambda )}$ is the matrix with ${\displaystyle \lambda }$ in the ${\displaystyle (p,q)}$-entry and zeros elsewhere, and ${\displaystyle p\neq q}$ — satisfy the following relations, called the Steinberg relations:

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

The unstable Steinberg group of order ${\displaystyle r}$ over ${\displaystyle A}$, denoted by ${\displaystyle {\operatorname {St} _{r}}(A)}$, is defined by the generators ${\displaystyle {x_{ij}}(\lambda )}$, where ${\displaystyle 1\leq i\neq j\leq r}$ and ${\displaystyle \lambda \in A}$, these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by ${\displaystyle \operatorname {St} (A)}$, is the direct limit of the system ${\displaystyle {\operatorname {St} _{r}}(A)\to {\operatorname {St} _{r+1}}(A)}$. It can also be thought of as the Steinberg group of infinite order.

Mapping ${\displaystyle {x_{ij}}(\lambda )\mapsto {e_{ij}}(\lambda )}$ yields a group homomorphism ${\displaystyle \varphi$ :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)} . As the elementary matrices generate the commutator subgroup, this mapping is surjective onto the commutator subgroup.

The Steinberg group is the fundamental group of the Volodin space, which is the union of classifying spaces of the unipotent subgroups of ${\displaystyle \operatorname {GL} (A)}$.

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

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

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

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

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

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