(Co)homology of partial smash products

Abstract

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