Algebra/Topology Seminar

Speaker: Sam Miller

Title: Endotrivial complexes and the geometry of permutation modules

Abstract: Let G be a finite group and k a field of characteristic p > 0. The recent work of Balmer and Gallauer has illuminated much about the bounded homotopy category of p-permutation modules. This tensor-triangulated category is fundamentally linked to numerous structures and topics, including Broue's abelian defect group conjecture, Voevodsky's Artin motives, and cohomological Mackey functors. The Balmer spectrum of this category is controlled by the Balmer spectrum of derived module categories of "p-local subgroups" via modular fixed points functors, an example of "local-to-global" behavior in modular representation theory.

In this talk, I will motivate this category and discuss its tt-geometry. I will also describe the surprising classification of invertible objects, which I call "endotrivial complexes," and discuss ongoing work, joint with Balmer and Gallauer, that connects the Picard group and the tt-geometry of the category.