The Balmer spectrum of integral permutation modules

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.