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:

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

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