  Let $\Tt$ be an aperiodic and repetitive tiling of $\RM^d$ with finite local
complexity. We present a spectral sequence that converges to the $K$-theory of
$\Tt$ with $E_2$-page given by a new cohomology that will be called PV in
reference to the Pimsner-Voiculescu exact sequence. It is a generalization of
the Serre spectral sequence. The PV cohomology of $\Tt$ generalizes the
cohomology of the base space of a fibration with local coefficients in the
$K$-theory of its fiber. We prove that it is isomorphic to the \v{C}ech
cohomology of the hull of $\Tt$ (a compactification of the family of its
translates).
