The cohomology structure of an algebra entwined with a coalgebra

Two cochain complexes are constructed for an algebra A and a coalgebra C en twined with each other via the map psi : C x A --> A x C. One complex is as sociated to an A-bimodule, the other to a C-bicomodule. In the former case the resulting complex can be considered as a psi -twisted Hochschild comple x of A, while for the latter one obtains a psi -twist of the Cartier comple x of C. The notion of a weak comp algebra is introduced by weakening the ax ioms of the Gerstenhaber comp algebra. It is shown that such a weak comp al gebra is a cochain complex with two cup products that descend to the cohomo logy. It is also shown that the complexes associated to an entwining struct ure 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 en twining structures is outlined. (C) 2001 Academic Press.