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.

It is named after Robert Steinberg, and is connected with lower K-groups, notably $K_2$ and $K_3$ .

1 Definition
- 1.1 Steinberg relations
2 Relation to K-theory
3 References

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.

Steinberg relations

Elementary matrices—meaning matrices of the form $e_{pq}(\lambda)�:= \mathbf{1} %2B a_{pq}(\lambda)$ , 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:

$\begin{align} e_{ij}(\lambda) e_{ij}(\mu) &= e_{ij}(\lambda%2B\mu) \\ \left[ e_{ij}(\lambda),e_{jk}(\mu) \right] &= e_{ik}(\lambda \mu) && \mbox{for } i \neq k\\ \left[ e_{ij}(\lambda),e_{kl}(\mu) \right] &= \mathbf{1} && \mbox{for } i \neq l, j \neq k\\ \end{align}$

The unstable Steinberg group of order r over A, $\operatorname{St}_r(A)$ , is defined by the generators $x_{ij}(\lambda)$ , $1\leq i,j\leq r, i\neq j, \lambda \in A$ , subject to the Steinberg relations. The stable Steinberg group, $\operatorname{St}(A)$ , is the direct limit of the system $\operatorname{St}_r(A) \to \operatorname{St}_{r%2B1}(A)$ . 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.

Relation to K-theory

K₁

$K_1(A)$ is the cokernel of the map $\varphi\colon\operatorname{St}(A)\to \operatorname{GL}(A)$ , as $K_1$ is the abelianization of $\operatorname{GL}(A)$ and $\varphi$ is onto the commutator subgroup.

K₂

$K_2(A)$ 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: $K_2(A) = H_2(\operatorname{E}(A),\mathbf{Z})$ .

K₃

Gersten (1973) showed that $K_3$ of a ring is $H_3$ of the Steinberg group.

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, http://www.math.ucla.edu/~rst/

Steinberg group (K-theory)

Contents

Definition

Steinberg relations

Relation to K-theory

K1

K2

K3

References

K₁

K₂

K₃