Projective dimension in filtrated K-theory

TL;DR

This paper characterizes modules with finite projective resolutions in filtrated K-theory over finite spaces, showing most have projective dimension at most 2, with applications to C*-algebras and a universal coefficient theorem.

Contribution

It provides a characterization of modules with finite projective dimension in filtrated K-theory and applies this to C*-algebras, establishing bounds and counterexamples.

Findings

Filtrated K-theory modules over spaces with up to four points have projective dimension ≤ 2.

A universal coefficient theorem for rational equivariant KK-theory is derived for these spaces.

Counterexample with projective dimension 3 over a five-point space.

Abstract

Under mild assumptions, we characterise modules with projective resolutions of length n in the target category of filtrated K-theory over a finite topological space in terms of two conditions involving certain Tor-groups. We show that the filtrated K-theory of any separable C*-algebra over any topological space with at most four points has projective dimension 2 or less. We observe that this implies a universal coefficient theorem for rational equivariant KK-theory over these spaces. As a contrasting example, we find a separable C*-algebra in the bootstrap class over a certain five-point space, the filtrated K-theory of which has projective dimension 3. Finally, as an application of our investigations, we exhibit Cuntz-Krieger algebras which have projective dimension 2 in filtrated K-theory over their respective primitive spectrum.