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Cohomological dimension of Mackey functors for infinite groups. (English) Zbl 1126.20034

Let \(G\) be a discrete grop. There are various notions of dimension associated to \(G\) which we now recall: let \(\mathcal F\) be the family of finite subgroups of \(G\). First, the authors extend the existing notion of Mackey functor for finite groups to the notion of a Mackey functor of infinite groups. Then, they observe that the category of Mackey functors is an Abelian category with enough projectives, hence the notion of cohomology of Mackey functors is defined and, therefore, the notion of cohomological dimension, denoted by \(\text{cd}_{\mathcal M_\mathcal F}G\). Next, using Bredon cohomology for \(G\) one defines the notion of cohomological dimension, denoted by \(\text{cd}_{\mathcal F}G\). The third notion is that of virtual cohomological dimension: let \(G\) be a virtually torsion free group and \(H\subseteq G\), be a torsion free subgroup of finite index, then the ‘virtual cohomological dimension’ (vcd) of \(G\) is defined as the cohomological dimension of \(H\) over \(\mathbb{Z}\). The fourth notion is that of ‘relative cohomological dimension’ (\(\mathcal F\text{-cd\,}G\)). A ‘relative’ projective resolution \(P_*\twoheadrightarrow\mathbb{Z}\) is an exact sequence of \(\mathbb{Z} G\)-modules, which splits when restricted to each finite subgroup of \(G\) and where all \(P_i\) are direct summands of direct sums of modules induced up from finite groups. Then, \(\mathcal F\text{-cd\,}G\) is defined as the length of the shortest relative projective resolution of the trivial \(\mathbb{Z} G\)-module \(\mathbb{Z}\).
The authors prove the following sequence of inequalities: \[ \text{cd}_\mathbb{Q} G\leq \mathcal F\text{-cd\,}G\leq\text{cd}_{\mathcal M_\mathcal F}G\leq\text{cd}_{\mathcal F}G. \] They also prove that when \(G\) is a virtually torsion free group then \(\text{vcd\,}G=\mathcal F\text{-cd\,}G=\text{cd}_{\mathcal M_\mathcal F}G\).

MSC:

20J05 Homological methods in group theory
18G20 Homological dimension (category-theoretic aspects)
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
55N25 Homology with local coefficients, equivariant cohomology
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