Stratifying integral representations of finite groups

TL;DR

This paper classifies localizing tensor ideals in the integral stable module category for finite groups, extending known modular results and establishing new structural and support theories in tensor-triangular geometry.

Contribution

It provides a comprehensive classification of integral stable module categories for finite groups, generalizing modular results and introducing novel descent techniques.

Findings

Classification of localizing tensor ideals for any finite group

Verification of the generalized telescope conjecture in this setting

A tensor product formula for integral cohomological support

Abstract

We classify the localizing tensor ideals of the integral stable module category for any finite group $G$. This results in a generic classification of $\mathbb{Z}[G]$-lattices of finite and infinite rank and globalizes the modular case established in celebrated work of Benson, Iyengar, and Krause. Further consequences include a verification of the generalized telescope conjecture in this context, a tensor product formula for integral cohomological support, as well as a generalization of Quillen's stratification theorem for group cohomology. Our proof makes use of novel descent techniques for stratification in tensor-triangular geometry that are of independent interest.