The homotopy groups of a G-spectrum form a (graded) G-Mackey functor, so calculations of Mackey functors and (derived) maps between them occur throughout stable equivariant homotopy theory. A natural question is to ask: how complicated is the homological algebra of this category? A good measure of this complexity is the global dimension (also called the projective dimension or homological dimension). For finite groups the global dimension of G-Mackey functors is infinite. If we work rationally, however, the global dimension is zero and hence the category is semi-simple. Thus for finite groups, we only see the two extremes.
In this talk we work rationally and generalise in two directions. Firstly, we generalise the group to the compact Lie and profinite case. Secondly, we return to the case of finite groups but now work with the incomplete Mackey functors from Kedziorek’s talk. We show that arbitrary global dimension occurs in all of these cases and give topological methods for calculating the global dimension.
