Steinberg group (K-theory)

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

It is named after Robert Steinberg, and it is connected with lower ${\displaystyle K}$-groups, notably ${\displaystyle K_{2}}$ and ${\displaystyle K_{3}}$.

Definition

Abstractly, given a ring ${\displaystyle A}$, the Steinberg group ${\displaystyle \operatorname {St} (A)}$ 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).

Presentation using generators and relations

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.

Interpretation as a fundamental group

The Steinberg group is the fundamental group of the Volodin space, which is the union of classifying spaces of the unipotent subgroups of GL(A).

Relation to K-theory

K₁

${\displaystyle {K_{1}}(A)}$ 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.

K₂

${\displaystyle {K_{2}}(A)}$ 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: ${\displaystyle {K_{2}}(A)={H_{2}}(E(A);\mathbb {Z} )}$.

K₃

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

References

• Gersten, S. M. (1973), "${\displaystyle K_{3}}$ of a Ring is ${\displaystyle H_{3}}$ 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 ${\displaystyle K}$-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, archived from the original on 2012-09-10