On homological coherence of discrete groups. (English) Zbl 1057.22013
Given a ring with unit \(A\), a left \(A\)-module is said to be coherent if it has a resolution by finitely generated projective \(A\)-modules. It is said to have finite homological dimension if such a resolution can be chosen to be finite. Let \(\Gamma\) be a discrete group, \(R\) a Noetherian ring and \(R(\Gamma)\) the corresponding group ring, we say roughly that the \(R\)-module \(F\) is \(\Gamma\)-filtered if there exists a set of \(R\)-submodules ordered by the power set of the group \(\Gamma \). If this order is \(\Gamma\)-equivariant we say that \(F\) is a \(\Gamma\)-filtered \(R\)-module.
The main result of this paper is the following: Theorem: Let \(R\) be a Noetherian ring and \(\Gamma\) be a discrete group of finite asymptotic dimension. Then: 1. Lean \(R[\Gamma]\)-modules have projective resolutions of finite type. 2. All \(R[\Gamma]\)-modules with admissible presentations are lean. 3. If in addition \(R\) has finite homological dimension then lean \(R[\Gamma]\)-modules also have finite homological dimension.
The notion of lean module rests upon the notion of boundedly controlled module homomorphisms, so a middle step leads to prove that every \(R[\Gamma]\)-homomorphism between a lean \(R[\Gamma]\)-module and an equivariant \(\Gamma\)-filtered module is boundedly controlled as a homomorphism between filtered \(R\)-modules. The authors use the main theorem to compute the asymptotic dimension of a simply connected nilpotent Lie group with left invariant Riemannian metric.
The main result of this paper is the following: Theorem: Let \(R\) be a Noetherian ring and \(\Gamma\) be a discrete group of finite asymptotic dimension. Then: 1. Lean \(R[\Gamma]\)-modules have projective resolutions of finite type. 2. All \(R[\Gamma]\)-modules with admissible presentations are lean. 3. If in addition \(R\) has finite homological dimension then lean \(R[\Gamma]\)-modules also have finite homological dimension.
The notion of lean module rests upon the notion of boundedly controlled module homomorphisms, so a middle step leads to prove that every \(R[\Gamma]\)-homomorphism between a lean \(R[\Gamma]\)-module and an equivariant \(\Gamma\)-filtered module is boundedly controlled as a homomorphism between filtered \(R\)-modules. The authors use the main theorem to compute the asymptotic dimension of a simply connected nilpotent Lie group with left invariant Riemannian metric.
Reviewer: Juan Antonio Pérez (Zacatecas)
MSC:
| 22E40 | Discrete subgroups of Lie groups |
| 20J06 | Cohomology of groups |
| 22E25 | Nilpotent and solvable Lie groups |
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