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

本课题从同伦不变交换代数的角度研究有限群G的模表示理论。更具体地说，已知在特征为p的域k中具有系数的增强群上同调环B=C*(BG)总是Gorenstein（由PI、Dwyer和Iyengar定理），并且根据Auslander-Buchsbaum-Serre判据和有限群的同伦理论，当且仅当G是p幂零时，B是正则的。我们的建议是了解这两个极端之间的群体范围。该方法考虑奇异类Dsg(B)（由PI和Stevenson定义）。广义地说，该方法是证明Dsg(B)等价于TA的有界派生范畴，其中A是B的Koszul对偶，TA是TA的类态定位。用PI和Benson完全分析了循环Sylow p-亚群的情况。在这种简单的情况下，可以使用显式计算，但除了极少数特殊情况外，这是不可能的。该项目旨在开发环谱的结构工具，为一般有限群G和其他环谱的对偶性（Koszul， Anderson和Tate）和局部化处理B=C*(BG)提供正式框架。在有利的情况下，我们将能够证明有限定理，表明Dsg (B)具有对偶性和支持理论，并给出计算方法。人们希望在其他一些情况下也可以进行完整的显式计算，并且奇点类别的良好行为将与该组的结构特征有关。

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.