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.Comment: 16 pages, 2 figures, revised and final version, results unchange
