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