# Tensor triangular geometry and KK-theory

## Ivo Dell'Ambrogio

This is a first foray of \emph{tensor triangular geometry} \cite{balmer_prime} into the realm of bivariant topological $K$-theory. As a motivation, we first establish a connection between the Balmer spectrum $\spc(\KK^G)$ and a strong form of the Baum-Connes conjecture with coefficients for the group~$G$, as studied in \cite{meyernest-bc}. We then turn to more tractable categories, namely, the thick triangulated subcategory $\mathcal K^G\subset \KK^G$ and the localizing subcategory $\tcat^G\subset \KK^G$ generated by the tensor unit~$\C$. For $G$ finite, we construct for the objects of $\tcat^G$ a support theory in $\spec(R(G))$ with good properties. We see as a consequence that $\spc(\mathcal K^G)$ contains a copy of the Zariski spectrum $\spec(R(G))$ as a retract, where $R(G)=\End_{\KK^G}(\mathbb C)$ is the complex character ring of~$G$. Not surprisingly, we find that $\spc(\mathcal K^{\{1\}})\simeq \spec(\Z)$.

Journal of Homotopy and Related Structures, Vol. 5(2010), No. 1, pp. 319-358