Stratification in equivariant Kasparov theory
Ivo Dell'Ambrogio, Rubén Martos
TL;DR
The paper develops a countable Balmer–Favi stratification framework for the equivariant bootstrap category Cell(G) within Kasparov theory, proving countable stratification for finite groups whose nontrivial elements have prime order and rational (i.e., Q-coefficients) stratification for all finite G. It constructs enlarged big tt-categories to apply existing strong stratification results and computes Balmer spectra in key cases, linking the spectrum to the representation ring End(1) via the comparison map. A central achievement is showing Spc(Cell(G)^c) or prime-order groups is homeomorphic to Spec(R(G)) and establishing a robust descent strategy (finite étale) to transfer stratification from subgroups to G. The results unify perspectives from tensor triangular geometry with equivariant KK-theory, providing a practical pathway to classify localizing tensor ideals and to understand their geometric content in terms of the group representation theory and cyclic subgroups. The techniques pave the way for further exploration of stratification for broader classes of finite groups and for noncompact settings, with potential connections to the Baum–Connes program via spectrum-based criteria.
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.