Abstract: This is a joint work with Jean Savinien, which appeared in Erg. Th. Dyn. Syst., 29, (2009), 997--1031.) A spectral sequence is proposed to compute the $K$-groups of a crossed product $A\rtimes{\mathbb Z}^d$ by a ${\mathbb Z}^d$-action. This spectral sequence generalizes the Pimsner-Voiculescu exact sequence when $d>1$. It is also dual to the Atiyah-Hirzebruch spectral sequence for CW-complexes. The construction applies to the computation of the K-groups of a tiling space.