Tensor triangular geometry and KK-theory

TL;DR

This paper explores the structure of equivariant KK-theory using tensor triangular geometry, revealing connections between the spectrum of certain subcategories and the Zariski spectrum of the complex character ring for finite groups.

Contribution

It establishes a relationship between the spectrum of KK^G subcategories and the Zariski spectrum of the character ring, and provides a criterion for universal support data in tensor triangulated categories.

Findings

Spectrum of KK^G contains a canonical copy of the Zariski spectrum of the character ring.

For trivial G, the spectrum inclusion is a homeomorphism.

Provides a criterion for support theories to be universal support data.

Abstract

We present some results on equivariant KK-theory in the context of tensor triangular geometry. More specifically, for G a finite group, we show that the spectrum of the tensor triangulated subcategory of KK^G generated by the tensor unit contains as a retract a canonical copy of the Zariski spectrum of the complex character ring of G. For G trivial, this inclusion is a homeomorphism. We also prove a general criterion for a support theory on a compactly generated tensor triangulated category to provide the universal support datum, in the sense of Paul Balmer, on its subcategory of compact objects.