The spectrum of the equivariant stable homotopy category of a finite group

TL;DR

This paper characterizes the prime ideal spectrum of the equivariant stable homotopy category for finite groups, providing a complete description for square-free groups and advancing understanding for all finite groups.

Contribution

It offers a complete set-theoretic description of the spectrum for all finite groups and advances the understanding of its topology, especially for square-free groups.

Findings

Complete description of the spectrum for all finite groups.

Progress in understanding the topology for groups of square-free order.

Reduction of unresolved topology questions to p-groups and a new chromatic blue-shift phenomenon.

Abstract

We study the spectrum of prime ideals in the tensor-triangulated category of compact equivariant spectra over a finite group. We completely describe this spectrum as a set for all finite groups. We also make significant progress in determining its topology and obtain a complete answer for groups of square-free order. For general finite groups, we describe the topology up to an unresolved indeterminacy, which we reduce to the case of p-groups. We then translate the remaining unresolved question into a new chromatic blue-shift phenomenon for Tate cohomology. Finally, we draw conclusions on the classification of thick tensor ideals.