Steinberg group (K-theory) (original) (raw)

From Wikipedia, the free encyclopedia

In algebraic K-theory, a field of mathematics, the Steinberg group St ⁡ ( A ) {\displaystyle \operatorname {St} (A)} {\displaystyle \operatorname {St} (A)} of a ring A {\displaystyle A} {\displaystyle A} is the universal central extension of the commutator subgroup of the stable general linear group of A {\displaystyle A} {\displaystyle A}.

It is named after Robert Steinberg, and it is connected with lower K {\displaystyle K} {\displaystyle K}-groups, notably K 2 {\displaystyle K_{2}} {\displaystyle K_{2}} and K 3 {\displaystyle K_{3}} {\displaystyle K_{3}}.

Abstractly, given a ring A {\displaystyle A} {\displaystyle A}, the Steinberg group St ⁡ ( A ) {\displaystyle \operatorname {St} (A)} {\displaystyle \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

[edit]

A concrete presentation using generators and relations is as follows. Elementary matrices — i.e. matrices of the form e p q ( λ ) := 1 + a p q ( λ ) {\displaystyle {e_{pq}}(\lambda ):=\mathbf {1} +{a_{pq}}(\lambda )} {\displaystyle {e_{pq}}(\lambda ):=\mathbf {1} +{a_{pq}}(\lambda )}, where 1 {\displaystyle \mathbf {1} } {\displaystyle \mathbf {1} } is the identity matrix, a p q ( λ ) {\displaystyle {a_{pq}}(\lambda )} {\displaystyle {a_{pq}}(\lambda )} is the matrix with λ {\displaystyle \lambda } {\displaystyle \lambda } in the ( p , q ) {\displaystyle (p,q)} {\displaystyle (p,q)}-entry and zeros elsewhere, and p ≠ q {\displaystyle p\neq q} {\displaystyle p\neq q} — satisfy the following relations, called the Steinberg relations:

e i j ( λ ) e i j ( μ ) = e i j ( λ + μ ) ; [ e i j ( λ ) , e j k ( μ ) ] = e i k ( λ μ ) , for i ≠ k ; [ e i j ( λ ) , e k l ( μ ) ] = 1 , for i ≠ l and j ≠ k . {\displaystyle {\begin{aligned}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{aligned}}} {\displaystyle {\begin{aligned}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{aligned}}}

The unstable Steinberg group of order r {\displaystyle r} {\displaystyle r} over A {\displaystyle A} {\displaystyle A}, denoted by St r ( A ) {\displaystyle {\operatorname {St} _{r}}(A)} {\displaystyle {\operatorname {St} _{r}}(A)}, is defined by the generators x i j ( λ ) {\displaystyle {x_{ij}}(\lambda )} {\displaystyle {x_{ij}}(\lambda )}, where 1 ≤ i ≠ j ≤ r {\displaystyle 1\leq i\neq j\leq r} {\displaystyle 1\leq i\neq j\leq r} and λ ∈ A {\displaystyle \lambda \in A} {\displaystyle \lambda \in A}, these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by St ⁡ ( A ) {\displaystyle \operatorname {St} (A)} {\displaystyle \operatorname {St} (A)}, is the direct limit of the system St r ( A ) → St r + 1 ( A ) {\displaystyle {\operatorname {St} _{r}}(A)\to {\operatorname {St} _{r+1}}(A)} {\displaystyle {\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 i j ( λ ) ↦ e i j ( λ ) {\displaystyle {x_{ij}}(\lambda )\mapsto {e_{ij}}(\lambda )} {\displaystyle {x_{ij}}(\lambda )\mapsto {e_{ij}}(\lambda )} yields a group homomorphism φ : St ⁡ ( A ) → GL ∞ ( A ) {\displaystyle \varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)} {\displaystyle \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

[edit]

The Steinberg group is the fundamental group of the Volodin space, which is the union of classifying spaces of the unipotent subgroups of GL ⁡ ( A ) {\displaystyle \operatorname {GL} (A)} {\displaystyle \operatorname {GL} (A)}.

Relation to _K_-theory

[edit]

K 1 ( A ) {\displaystyle {K_{1}}(A)} {\displaystyle {K_{1}}(A)} is the cokernel of the map φ : St ⁡ ( A ) → GL ∞ ( A ) {\displaystyle \varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)} {\displaystyle \varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)}, as K 1 {\displaystyle K_{1}} {\displaystyle K_{1}} is the abelianization of GL ∞ ( A ) {\displaystyle {\operatorname {GL} _{\infty }}(A)} {\displaystyle {\operatorname {GL} _{\infty }}(A)} and the mapping φ {\displaystyle \varphi } {\displaystyle \varphi } is surjective onto the commutator subgroup.

K 2 ( A ) {\displaystyle {K_{2}}(A)} {\displaystyle {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 {\displaystyle K} {\displaystyle K}-groups.

It is also the kernel of the mapping φ : St ⁡ ( A ) → GL ∞ ( A ) {\displaystyle \varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)} {\displaystyle \varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)}. Indeed, there is an exact sequence

1 → K 2 ( A ) → St ⁡ ( A ) → GL ∞ ( A ) → K 1 ( A ) → 1. {\displaystyle 1\to {K_{2}}(A)\to \operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)\to {K_{1}}(A)\to 1.} {\displaystyle 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 ) ; Z ) {\displaystyle {K_{2}}(A)={H_{2}}(E(A);\mathbb {Z} )} {\displaystyle {K_{2}}(A)={H_{2}}(E(A);\mathbb {Z} )}.

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