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

增强群上同调环的 Koszul 对偶性和奇点范畴

基本信息

批准号：
EP/W036320/1
负责人：
John Greenlees
金额：
$ 58.87万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2023
资助国家：
英国
起止时间：
2023 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW036320%2F1
关键词：
Koszul duality singularity category enhanced

项目摘要

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.

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，特征为p的域k中系数的增强群上同调环B=C*(BG)总是Gorenstein的(根据PI，Dwyer和Iyengar定理)，并且遵循Auslander-Buchsbaum-Serre判据和有限群的同伦理论，B是正则的当且仅当G是p-幂零的。我们的建议是了解这两个极端之间的范围内的群体光谱。方法是考虑奇点范畴DSG(B)(由PI和Stevenson定义)。广义DSG(B)等价于TA的有界导范畴，其中A是B的Koszul对偶，TA是B的Tate-Like局部化。利用PI和Benson完全分析了循环Sylow p-子群的情形。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。该项目是开发环谱的结构工具，使我们能够提供一个形式框架的对偶(Koszul，Anderson和Tate)和局部化处理B=C*(BG)的一般有限群G和其他环谱。在有利的情况下，我们将能够证明有限定理，表明DSG(B)具有对偶性和支撑性理论，并给出计算方法。希望完整的显式计算在其他一些情况下也是可能的，奇点类别的良好行为将与该群体的结构特征有关。