Author:Serge Bouc
email: serge.bouc@u-picardie.fr
Address:
CNRS-LAMFA, Universit\'e de Picardie, 33 rue St Leu, 80039 Amiens Cedex 01, France

Author: Radu Stancu
email: radu.stancu@u-picardie.fr
Address:
CNRS-LAMFA, Universit\'e de Picardie, 33 rue St Leu, 80039 Amiens Cedex 01, France

Author: Peter Webb
email: webb@math.umn.edu
Address:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

Title: 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 $\ZZ$.