The Balmer spectrum of integral permutation modules

TL;DR

This paper generalizes the understanding of the tensor-triangular spectrum of permutation modules from fields to Noetherian rings, providing a detailed description and topological analysis, especially for elementary abelian groups.

Contribution

It extends the tt-geometry analysis of permutation modules to a broader base ring and develops the theory of twisted cohomology for elementary abelian groups.

Findings

The tt-spectrum can be described as a set and its topology reduced to elementary abelian cases.

Under certain conditions, the tt-spectrum forms a Dirac scheme for elementary abelian p-groups.

The study generalizes previous results from fields to Noetherian rings.

Abstract

We extend the analysis of Balmer and Gallauer on the tt-geometry of the small derived category of permutation modules for a finite group over a field to the setting of a commutative Noetherian base. In this general context, we provide a description of the tt-spectrum as a set and reduce the study of its topology to the elementary abelian case. Under certain mild additional assumptions on the ground ring, we further develop their theory of twisted cohomology, which enables us to realize the tt-spectrum as a Dirac scheme when restricted to elementary abelian $p$-groups.