On the Projective Dimensions of Mackey Functors

Abstract

We examine the projective dimensions of Mackey functors and cohomological Mackey functors. We show over a field of characteristic p that cohomological Mackey functors are Gorenstein if and only if Sylow p-subgroups are cyclic or dihedral, and they have finite global dimension if and only if the group order is invertible or Sylow subgroups are cyclic of order 2. By contrast, we show that the only Mackey functors of finite projective dimension over a field are projective. This allows us to give a new proof of a theorem of Greenlees on the projective dimension of Mackey functors over a Dedekind domain. We conclude by completing work of Arnold on the global dimension of cohomological Mackey functors over Z.

Trace maps for Mackey algebras

“…With this tensor product, the category is a closed symmetric monoidal category with the Burnside functor as unit. So, using the formalism of May (Sections 2 and 4 of [11]) where the dualizable Mackey functors are exactly the finitely generated projective Mackey functors, Bouc has defined the notion of Burnside dimension and Burnside trace for these Mackey functors (Section 2 of [4]). Let M be a finitely generated projective Mackey functor.…”

“…In this paper we propose a systematic approach to this question: by using the so-called Burnside Trace, introduced by Serge Bouc (Section 2 of [4]), we reduce the question of the existence of such a bilinear form on the Mackey algebra to the question of the existence of a family of symmetric, associative, non-degenerate bilinear forms on Burnside algebras with an extra property. Here we denote by RB(H) the usual Burnside algebra of the group H.…”

Bredon Cohomology of Polyhedral Products

“…Unfortunately, both parts are not easy to obtain in general. Much recent researches, such as [5] and [11], focus on G = C p n , the cyclic group of prime power order, and on a more rigid subcategory of C G such as category of Mackey functors.…”

Resolutions by permutation modules

“…The related problem of resolutions (1) that are not only exact but remain exact under all fixed-point functors has been recently discussed in [BSW17]. Allowing p-permutation modules P i (that is, direct summands of permutation modules), Bouc-Stancu-Webb prove that such resolutions exist for all M if and only if G has a Sylow subgroup that is either cyclic or dihedral (for p = 2).…”