Steinberg group (K-theory) explained

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).

Presentation using generators and relations

A concrete presentation using generators and relations is as follows. Elementary matrices — i.e. matrices of the form

{e _{pq} }(\lambda) := \mathbf + (\lambda) , where 1 is the identity matrix, {a _{pq} }(\lambda) is the matrix with λ in the (p,q) -entry and zeros elsewhere, and p ≠ q — satisfy the following relations, called the Steinberg relations:\begin{align} e _{ij} ( λ ) e _{ij} (\mu) & = e _{ij} ( λ+ \mu); && \\ \left[ e _{ij} ( λ ),e _{jk} (\mu) \right] & = e _{ik} ( λ \mu), && for i ≠ k; \\ \left[ e _{ij} ( λ ),e _{kl} (\mu) \right] & = 1 , && for i ≠ l and j ≠ 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 ≠ j \leq r and λ \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 (A) . It can also be thought of as the Steinberg group of infinite order.

Mapping

{x _{ij} }(\lambda) \mapsto (\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.

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

\operatorname{GL}(A) .

Relation to K-theory

K₁

{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.

K₂

{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 sequence1 \to {K _{2} }(A) \to \operatorname(A) \to (A) \to (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) = (E(A);\mathbb) .

K₃

showed that

{K _{3} }(A) = (\operatorname(A);\mathbb)