  Let A a k-algebra, H a Hopf algebra, E = A#H a general crossed product and M
an E-bimodule. We obtain a complex simpler than the canonical one, giving the
Hochschild homology of E with coefficients in M. This complex is eqquiped with
a natural filtration. We prove that the associated spectral sequence coincides
with that obtained by either, the Hochschild-Serre direct method or the
Cartan-Leray-Grothendieck method. We also get similar results for the
cohomology.
