Steinberg group (K-theory)

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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.

It is named after Robert Steinberg, and is connected with lower K-groups, notably K2 and K3.


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[edit] 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, hence has a universal central extension).

Concretely, it can also be described by generators and relations.

[edit] Steinberg relations

Elementary matrices—meaning matrices of the form e_{pq}(\lambda) := \mathbf{1} + a_{pq}(\lambda), where \mathbf{1} is the identity matrix, apq(λ) is the matrix with λ 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) && \mbox{for } i \neq k\\ \left[ e_{ij}(\lambda),e_{kl}(\mu) \right] &= \mathbf{1} && \mbox{for } i \neq l, j \neq k\\ \end{align}

The unstable Steinberg group of order r over A, \operatorname{St}_r(A), is defined by the generators xij(λ), 1\leq i,j\leq r, i\neq j, \lambda \in A, subject to the Steinberg relations. The stable Steinberg group, \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\colon\operatorname{St}(A)\to\operatorname{GL}(A).

As the elementary matrices generate the commutator subgroup, this map is onto the commutator subgroup.

[edit] Relation to K-theory

[edit] K1

K1(A) is the cokernel of the map \varphi\colon\operatorname{St}(A)\to \operatorname{GL}(A), as K1 is the abelianization of \operatorname{GL}(A) and \varphi is onto the commutator subgroup.

[edit] K2

K2(A) is the center of the Steinberg group; this was Milnor's definition, and also follows from more general definitions of higher K-groups.

It is also the kernel of the map \varphi\colon\operatorname{St}(A)\to\operatorname{GL}(A), and indeed there is an exact sequence

1\longrightarrow K_2(A) \longrightarrow \operatorname{St}(A) \longrightarrow \operatorname{GL}(A) \longrightarrow K_1(A)\longrightarrow 1.

Equivalently, it is the Schur multiplier of the group of elementary matrices, and thus is also a homology group: K_2(A) = H_2(\operatorname{E}(A),\mathbf{Z}).

[edit] K3

K3 of a ring is H3 of the Steinberg group.

This result is proven is the eponymous paper:

  • S. M. Gersten (Feb., 1973). "K3 of a Ring is H3 of the Steinberg Group". Proceedings of the American Mathematical Society 37 (2): 366–368. doi:10.2307/2039440. JSTOR.