The geometry of permutation modules
Representation Theory
2025-07-22 v2 Category Theory
Abstract
We consider the derived category of permutation modules for a finite group, in positive characteristic. We stratify this tensor triangulated category using Brauer quotients. We describe the spectrum of its compact objects, by reducing the problem to elementary abelian groups and then by using a twisted form of cohomology to express the spectrum locally in terms of the graded endomorphism ring of the unit. Together, these results yield a classification of thick and of localizing ideals.
Cite
@article{arxiv.2307.04398,
title = {The geometry of permutation modules},
author = {Paul Balmer and Martin Gallauer},
journal= {arXiv preprint arXiv:2307.04398},
year = {2025}
}
Comments
76 pages; v2: new title, reflecting the fact that this version results from merging part I (originally at arXiv:2210.08311) and part II (v1 at this address); to appear in Inventiones