Cohomological finiteness conditions for Mackey and cohomological Mackey functors

Abstract: We study cohomological finiteness conditions for groups associated to Mackey and cohomological Mackey functors, proving that the cohomological dimension associated to cohomological Mackey functors is always equal to the $\mathcal{F}$-cohomological dimension, and characterising the conditions Mackey-$\mathrm{FP}_n$ and cohomological Mackey-$\mathrm{FP}_n$. We show that all finiteness conditions for cohomological Mackey functors are unchanged when considering only the family of $p$-subgroups, and we characterise cohomological Mackey-$\mathrm{FP}_n$ conditions over the finite field $\mathbb{F}_p$.