Steinberg group (K-theory)

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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[edit]

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[edit]

 K_{1} [edit]

 {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} [edit]

 {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} [edit]

Gersten (1973) showed that  {K_{3}}(A) = {H_{3}}(\operatorname{St}(A);\mathbb{Z}) .

References[edit]

  • 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