EPSRC Reference: 
EP/W036320/1 
Title: 
Koszul duality and the singularity category for the enhanced group cohomology ring 
Principal Investigator: 
Greenlees, Professor J 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of Warwick 
Scheme: 
Standard Research 
Starts: 
01 October 2022 
Ends: 
30 September 2026 
Value (£): 
461,981

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
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 AuslanderBuchsbaumSerre and homotopy theory of finite groups, B is regular if and only if G is pnilpotent. 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 Tatelike localization of it.
The case of a cyclic Sylow psubgroup 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.

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Summary 

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Organisation Website: 
http://www.warwick.ac.uk 