On the Balmer spectrum for compact Lie groups

TL;DR

This paper characterizes the Balmer spectrum of finite G-spectra for compact Lie groups, extending previous work, and provides a complete classification of thick tensor-ideals for abelian groups and partial results for others.

Contribution

It offers a comprehensive description of the Balmer spectrum for compact Lie groups and classifies thick tensor-ideals, extending finite group results to the Lie group setting.

Findings

Complete description of the spectrum for abelian compact Lie groups.

Topology of the spectrum determined by subgroup inclusion relations.

Classification of thick tensor-ideals for most primes in general compact Lie groups.

Abstract

We study the Balmer spectrum of the category of finite G-spectra for a compact Lie group G, extending the work for finite G by Strickland, Balmer-Sanders, Barthel-Hausmann-Naumann-Nikolaus-Noel-Stapleton and others. We give a description of the underlying set of the spectrum and show that the Balmer topology is completely determined by the inclusions between the prime ideals and the topology on the space of closed subgroups of G. Using this, we obtain a complete description of this topology for all abelian compact Lie groups and consequently a complete classification of thick tensor-ideals. For general compact Lie groups we obtain such a classification away from a finite set of primes p.