On the projective dimensions of Mackey functors

TL;DR

This paper investigates the projective dimensions of Mackey and cohomological Mackey functors, revealing conditions under which they are Gorenstein or have finite global dimension, and providing new proofs and completing previous work.

Contribution

It establishes new criteria for the Gorenstein property and finite global dimension of cohomological Mackey functors, and offers a novel proof of Greenlees' theorem.

Findings

Cohomological Mackey functors are Gorenstein iff Sylow p-subgroups are cyclic or dihedral.

Finite global dimension occurs when group order is invertible or Sylow subgroups are cyclic of order 2.

Only projective Mackey functors have finite projective dimension over a field.

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 $\ZZ$.