Koszul duality and the singularity category for the enhanced group cohomology ring

The project studies modular representation theory of finite groups G from the point of view of homotopy invariant commutative algebra.

More specifically it is known that the enhanced group cohomology ring B=C*(BG) with coefficients in a field k of characteristic p is always Gorenstein (by the theorem of the PI, Dwyer and Iyengar), and following the criterion of Auslander-Buchsbaum-Serre and homotopy theory of finite groups, B is regular if and only if G is p-nilpotent. The proposal is to understand the spectrum of groups along the range between these two extremes.

The method is to consider the singularity category Dsg(B) (as defined by the PI and Stevenson). In broad terms the method is to show that Dsg(B) is equivalent to the bounded derived category of TA where A is the Koszul dual of B and TA is a Tate-like localization of it.
The case of a cyclic Sylow p-subgroup was completely analysed by the PI and Benson. In such a simple case, one can use explicit calculation, but this will be impossible except for a very few special cases. The project is to develop structural tools for ring spectra that let us provide a formal framework for duality (Koszul, Anderson and Tate) and localization for treating B=C*(BG) for general finite groups G and other ring spectra. In favourable cases we will be able to prove finiteness theorems, showing that Dsg (B) has duality and a theory of support, and to give methods of calculation. It is hoped that complete explicit calculations will also be possible in some other cases, and that the good behaviour of the singularity category will be related to structural features of the group.