#
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