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.

该项目从同伦不变交换代数的角度研究有限群G的模表示论。更具体地说，我们知道，系数在特征 p 的域 k 中的增强群上同调环 B=C*(BG) 始终是 Gorenstein（根据 PI、Dwyer 和 Iyengar 定理），并遵循 Auslander- 准则Buchsbaum-Serre 和有限群同伦论，当且仅当 G 是 p-幂零时，B 是正则的。该提议是为了了解这两个极端之间的群体范围。该方法是考虑奇点类别 Dsg(B)（由 PI 和 Stevenson 定义）。从广义上讲，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，而 TA 是其类似 Tate 的局部化。 PI 和 Benson 完整地分析了循环 Sylow p 子群的情况。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，让我们为对偶性（Koszul、Anderson 和 Tate）提供正式框架，并为处理一般有限群 G 和其他环谱的 B=C*(BG) 进行定位。在有利的情况下，我们将能够证明有限性定理，证明 Dsg (B) 具有对偶性和支持理论，并给出计算方法。希望在其他一些情况下也可以进行完整的显式计算，并且奇点范畴的良好行为将与群的结构特征相关。