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

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

2000-11-29

arxiv.org/abs/math/0011248v6

Mathematics

K-Theory and Homology

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

Scientific paper

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

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