Stratification in equivariant Kasparov theory

In this paper, 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 [Formula: see text]. 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 [Formula: see text]-categorical enhancements and adapting ideas of Bunke–Engel–Land.