Stratification in equivariant Kasparov theory

TL;DR

This paper investigates the classification of subcategories in equivariant KK-theory of C*-algebras using geometric stratification methods, proving conjectures for certain finite groups and computing spectra.

Contribution

It introduces a new notion of stratification for equivariant KK-theory, proves it for groups with prime order elements, and computes the Balmer spectrum in these cases.

Findings

Conjecture holds for groups with all nontrivial elements of prime order.

Conjecture verified rationally for all finite groups.

Computed the Balmer spectrum of compact objects in these cases.

Abstract

We study stratification, that is the classification of localizing tensor ideal subcategories by geometric means, in the context of Kasparov's equivariant KK-theory of C*-algebras. We introduce a straightforward countable analog of the notion of stratification by Balmer-Favi supports and conjecture that it holds for the equivariant bootstrap subcategory of every finite group G. We prove this conjecture for groups whose nontrivial elements all have prime order, and we verify it rationally for arbitrary finite groups. In all these cases we also compute the Balmer spectrum of compact objects. In our proofs we use larger versions of the equivariant Kasparov categories which admit not only countable coproducts but all small ones; they are constructed in an Appendix using infinity-categorical enhancements and adapting ideas of Bunke-Engel-Land.