The spectrum of local dualisable modular representations

TL;DR

This paper investigates the structure of dualisable objects in the local stable module category of a finite group, providing a detailed spectral analysis related to the group's cohomology spectrum.

Contribution

It computes the spectrum of dualisable objects in the local stable module category at each prime in the cohomology spectrum of a finite group.

Findings

Spectrum characterized for dualisable objects at each prime.

Provides explicit description of local dualisable subcategories.

Advances understanding of tensor triangulated categories in modular representation theory.

Abstract

For a point $\mathfrak{p}$ in the spectrum of the cohomology ring of a finite group $G$ over a field $k$, we calculate the spectrum for the subcategory of dualisable objects inside the tensor triangulated category of $\mathfrak{p}$-local and $\mathfrak{p}$-torsion objects in the (big) stable module category of the group algebra $kG$.