Hopf Algebra Equivariant Cyclic Homology and Cyclic Homology of Crossed Product Algebras

Abstract

We introduce the cylindrical module $A♮\mathcal{H}$, where $\mathcal{H}$ is a Hopf algebra and $A$ is a Hopf module algebra over $\mathcal{H}$. We show that there exists an isomorphism between ${\mathsf{C}}_{\bullet }(A^{op}⋊{\mathcal{H}}^{cop})$ the cyclic module of the crossed product algebra $A^{op}⋊{\mathcal{H}}^{cop}$, and $\Delta (A♮\mathcal{H})$, the cyclic module related to the diagonal of $A♮\mathcal{H}$. If $S$, the antipode of $\mathcal{H}$, is invertible it follows that ${\mathsf{C}}_{\bullet }(A⋊\mathcal{H})\simeq \Delta (A^{op}♮{\mathcal{H}}^{cop})$. When $S$ is invertible, we approximate $H{C}_{\bullet }(A⋊\mathcal{H})$ by a spectral sequence and give an interpretation of ${\mathsf{E}}^0,{\mathsf{E}}^1$ and ${\mathsf{E}}^2$ terms of this spectral sequence.

Final version, to appear in "Crelle's Journal"