Cyclic homology of Brzeziński’s crossed products and of braided Hopf crossed products

Let kk be a field, AA a unitary associative kk-algebra and VV a kk-vector space endowed with a distinguished element 1V1V. We obtain a mixed complex, simpler than the canonical one, that gives the Hochschild, cyclic, negative and periodic homologies of a crossed product E≔A#fVE≔A#fV, in the sense of Brzeziński. We actually work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra KK of AA that satisfies suitable hypothesis and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homologies of EE relative to KK. Then, when EE is a cleft braided Hopf crossed product, we obtain a simpler mixed complex, that also gives the Hochschild, cyclic, negative and periodic homologies of EE.