- Steinberg group (K-theory)
In
algebraic K-theory , a field ofmathematics , the Steinberg group operatorname{St}(A) of a ring "A", is theuniversal central extension of thecommutator subgroup of the stablegeneral linear group .It is named after
Robert Steinberg , and 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 thecommutator subgroup of the stablegeneral linear group (the commutator subgroup is perfect, hence has a universal central extension).Concretely, it can also be described by
generators and relations .teinberg relations
Elementary matrices —meaning 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 eq q—satisfy the following relations, called the Steinberg relations::egin{align}e_{ij}(lambda) e_{ij}(mu) &= e_{ij}(lambda+mu) \left [ e_{ij}(lambda),e_{jk}(mu) ight] &= e_{ik}(lambda mu) && mbox{for } i eq k\left [ e_{ij}(lambda),e_{kl}(mu) ight] &= mathbf{1} && mbox{for } i eq l, j eq k\end{align}
The unstable Steinberg group of order "r" over "A", operatorname{St}_r(A), is defined by the generatorsx_{ij}(lambda), 1leq i,jleq r, i eq 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) o 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 :varphicolonoperatorname{St}(A) ooperatorname{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 varphicolonoperatorname{St}(A) o 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 varphicolonoperatorname{St}(A) ooperatorname{GL}(A), and indeed there is an
exact sequence :1longrightarrowK_2(A) longrightarrowoperatorname{St}(A) longrightarrowoperatorname{GL}(A) longrightarrowK_1(A)longrightarrow 1.Equivalently, it is the
Schur multiplier of the group ofelementary matrices , and thus is also a homology group: K_2(A) = H_2(operatorname{E}(A),mathbf{Z}).K3
K_3 of a ring is H_3 of the Steinberg group.
This result is proven is the eponymous paper:
* cite journal
title=K_3 of a Ring is H_3 of the Steinberg Group
author=S. M. Gersten
journal=Proceedings of the American Mathematical Society
vol=37
issue=2
date=Feb., 1973
pages=366–368
doi=10.2307/2039440
id=JSTOR stable URL|0002-9939(197302)37%3A2%3C366%3AOARIOT%3E2.0.CO%3B2-Z
month=Feb
year=1973
volume=37
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