Locally dualisable modular representations and local regularity

TL;DR

This paper explores the structure of the stable module category of finite groups, characterizing dualisable modules via local properties and introducing the concept of local regularity to understand cohomological behavior.

Contribution

It introduces the notion of local regularity in triangulated categories and characterizes dualisable objects in the stable module category using local properties and $ ext{pi}$-points.

Findings

Dualisable modules are characterized by their restriction properties along $ ext{pi}$-points.

Introduction of local regularity as a property controlling cohomological behavior.

Development of the theory relating local regularity to strong generation in tensor triangulated categories.

Abstract

This work concerns the stable module category of a finite group over a field of characteristic dividing the group order. The minimal localising tensor ideals correspond to the non-maximal homogeneous prime ideals in the cohomology ring of the group. Given such a prime ideal, a number of characterisations of the dualisable objects in the corresponding tensor ideal are given. One characterisation of interest is that they are exactly the modules whose restriction along a corresponding $\pi$-point are finite dimensional plus projective. A key insight is the identification of a special property of the stable module category that controls the cohomological behaviour of local dualisable objects. This property, introduced in this work for general triangulated categories and called local regularity, is related to strong generation. A major part of the paper is devoted to developing this notion and investigating its ramifications for various special classes of objects in tensor triangulated categories.