The cohomology structure of an algebra entwined with a coalgebra

Two cochain complexes are constructed for an algebra A and a coalgebra C entwined with each other via the map ψ : C ⊗ A → A ⊗ C. One complex is associated to an A-bimodule, the other to a C-bicomodule. In the former case the resulting complex can be considered as a ψ-twisted Hochschild complex of A, while for the latter one obtains a ψ-twist of the Cartier complex of C. The notion of a weak comp algebra is introduced by weakening the axioms of the Gerstenhaber comp algebra. It is shown that such a weak comp algebra is a cochain complex with two cup products that descend to the cohomology. It is also shown that the complexes associated to an entwining structure and A or C are examples of a weak comp algebra. Finally both complexes are combined in a double complex whose role in the deformation theory of entwining structures is outlined.