Cohomological Dimension of Mackey Functors for Infinite Groups

Abstract

We consider the cohomology of Mackey functors for infinite groups and define the Mackeycohomological dimension cd M G of a group G. We will relate this dimension to other cohomological dimensions such as the Bredon-cohomological dimension cd F G and the relative cohomological dimension F-cdG. In particular, we show that for virtually torsion free groups the Mackeycohomological dimension is equal to both F-cdG and the virtual cohomological dimension.

Stable finiteness properties of infinite discrete groups

“…Here A is the Burnside ring functor that takes H ∈ F to the Burnside ring A(H) of H. As before, one shows using standard techniques that the invariant cd M (G) coincides with the length of the shortest free (or projective) resolution of the Burnside ring functor in Mack F G. Functors between orbit categories and Mackey categories give rise to induction, coinduction and restriction functors on the level of module categories, satisfying the usual adjointness properties (for example, see [35,Section 2]). One can construct a functor…”

“…and ind π (Z) = A (see [35,Theorem 3.7]). The functor ind π is not exact in general, but does preserve exactness of projective resolutions, which yields that (see [35,Theorem 3.8])…”

“…If G is virtually torsion-free, its virtual cohomological dimension, denoted by vcd(G), is by definition the cohomological dimension of any torsion-free finite index subgroup of G. This notion is well defined due to a result of Serre (for example, see [9, Chapter VIII]). A surprising result proven by Martínez-Pérez and Nucinkis in [35] says that if G is virtually torsion-free one has vcd(G) = cd M (G).…”

A bound for the Bredon cohomological dimension

“…Relative cohomology is in fact a particular case of Bredon cohomology with coe‰cients on a Mackey functor, as shown in [18]. Relative cohomology is in fact a particular case of Bredon cohomology with coe‰cients on a Mackey functor, as shown in [18].…”

“…The transitivity of coinduction (see [18]) implies that The transitivity of coinduction (see [18]) implies that…”