(Co)homology of crossed products by weak Hopf algebras

We obtain a mixed complex simpler than the canonical one that computes the cyclic type homologies of a crossed product with invertible cocycle A×_ρ^fH, of a weak module algebra A by a weak Hopf algebra H. This complex is endowed with a filtration. The spectral sequence of this filtration generalizes the spectral sequence obtained in [12]. When f takes its values in a separable subalgebra of A that satisfies suitable conditions, the above mentioned mixed complex is provided with another filtration, whose spectral sequence generalize the Feigin-Tsygan spectral sequence.