The Nucleus of a Compact Lie Group, and Support of Singularity Categories

TL;DR

This paper extends the concept of the nucleus to compact Lie groups, linking it to the singularities in cohomology schemes and establishing a support theory for singularity categories of ring spectra.

Contribution

It adapts the nucleus notion to compact Lie groups and connects it to the support of singularity categories, confirming a conjecture for finite groups.

Findings

Nucleus describes singularities of cohomology schemes

Support for singularity categories matches the nucleus

Confirms Benson-Greenlees conjecture for finite groups

Abstract

In this paper we adapt the notion of the nucleus defined by Benson, Carlson, and Robinson to compact Lie groups in non-modular characteristic. We show that it describes the singularities of the projective scheme of the cohomology of its classifying space. A notion of support for singularity categories of ring spectra (in the sense of Greenlees and Stevenson) is established, and is shown to be precisely the nucleus in this case, consistent with a conjecture of Benson and Greenlees for finite groups.