Rational torus-equivariant stable homotopy IV: thick tensor ideals and the Balmer spectrum for finite spectra

TL;DR

This paper classifies thick tensor ideals in rational torus-equivariant spectra, linking them to geometric isotropy and describing the Balmer spectrum as closed subgroups under cotoral inclusion, with implications for general groups.

Contribution

It provides a complete classification of thick tensor ideals and describes the Balmer spectrum for rational torus-equivariant spectra, extending to general groups.

Findings

Thick tensor ideals are determined by geometric isotropy.

The Balmer spectrum corresponds to closed subgroups under cotoral inclusion.

Results extend to toral spectra for general groups.

Abstract

We classify thick tensor ideals of finite objects in the category of rational torus-equivariant spectra, showing that they are completely determined by geometric isotropy. This is essentially equivalent to showing that the Balmer spectrum is the set of closed subgroups under cotoral inclusion. Corresponding statements are deduced for toral spectra for general groups.