Hopf–hochschild (co)homology of Module Algebras

Our goal in this paper is to define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called “module algebras” (Definition 2.1). Our motivation lies in the following problem: for an algebra A which admits a module structure over an arbitrary bialgebra B compatible with its product structure, the Hochschild or the cyclic bicomplexes associated with this algebra need not be differential graded B–modules. The obstruction which prevents these complexes from being B–linear is trivial whenever the bialgebra B is cocommutative, as in the case of group rings and universal enveloping algebras. Yet the same obstruction is far from being trivial if the underlying bialgebra is non-cocommutative. In the sequel, we will investigate how much of the Hochschild homology is retained after dividing this obstruction out. To this end, we will construct a new differential graded B–module QCH∗(A,B, V ) (Proposition 2.10 and Definition 2.11) for a B–module algebra A and a B–equivariant A–bimodule V (Definition 2.2). We will define HH ∗ (A,B, V ) the Hopf–Hochschild homology of A with coefficients in V as the homology of the complex k⊗ B QCH∗(A,B, V ). We would like to point out that the same strategy worked remarkably well in the case of cyclic cohomology of module coalgebras. In [13] we show that if we start with the cocyclic bicomplex of a module coalgebra twisted by a stable anti-Yetter–Drinfeld module, dividing the analogous obstruction results in the Hopf cyclic complex of [10] which was an extension of the Hopf cyclic cohomology of Connes and Moscovici [7].