(Co)Homology of Partial Smash Products

Abstract: Given a cocommutative Hopf algebra $\mathcal{H}$ over a commutative ring $K$ and a symmetric partial action of $\mathcal{H}$ on a $K$-algebra $A,$ we obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the smash product $A # \mathcal{H},$ involving the Hochschild homology of $A$ and the partial homology of $\mathcal{H}.$ An analogous third quadrant cohomological spectral sequence is also obtained. The definition of the partial (co)homology of $\mathcal{H}$ under consideration is based on the category of the partial representations of $\mathcal{H}.$ A specific partial representation of $\mathcal{H}$ on a subalgebra $\mathcal{B}$ of the partial ``Hopf" algebra $\mathcal{H}_{par} $ is involved in the definition and we construct a projective resolution of $\mathcal{B}.$