Stratifying modular representations of finite groups

TL;DR

This paper classifies certain subcategories of the stable module category of finite groups, proving the telescope conjecture and providing new proofs of key theorems, with applications to differential graded modules.

Contribution

It introduces a classification of localising subcategories closed under tensor product, proving the telescope conjecture in this setting and offering new proofs of existing theorems.

Findings

Classification of localising subcategories for finite group modules

Proof of the telescope conjecture in this context

New proofs of the tensor product theorem and classification of thick subcategories

Abstract

We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.