The spectrum of local dualisable modular representations

TL;DR

The paper classifies thick tensor ideals of the localised, dualisable part of the stable module category $StMod(kG)$ at a prime $rak p$ in $Proj H^*(G,k)$, showing a bijection with specialization-closed subsets of $Spec\,H^*(G,k)^{∧}_rak p$ via cohomological support. The authors first solve the elementary abelian case using the BGG correspondence to relate thick subcategories to Thomason subsets of $Spec(H^*(E,k)^{∧}_rak p)$, then extend to general finite groups via Quillen stratification and restriction to elementary abelian subgroups. A recollement with the homotopy category of injectives reduces the local problem to a derived-category framework, enabling a uniform treatment of dualisable objects. The results refine our understanding of local tensor-triangular geometry in modular representation theory, reveal how completion changes the spectrum, and provide a Quillen-type stratification for local dualisable modules analogous to Avrunin–Scott's cohomological stratification. Overall, the work bridges Balmer's spectrum with cohomological and stratification techniques to describe the local structure of stable categories at primes.

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 $kG$.

Theorems & Definitions (32)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Remark 2.6
• Lemma 2.9
• proof