Hochschild and Cyclic Homology of Centrally Hopf-galois Extensions

Let B ⊆ A be an H-Galois extension. If M is a Hopf bimodule then HH∗(A, M), the Hochschild homology of A with coefficients in M , is a right comodule over the coalgebra CH = H/[H,H]. Given an injective left CHcomodule V , our aim is to investigate the relationship between HH∗(A, M) CHV and HH∗(B, M CHV ). The roots of this problem can be found in [Lo2], where HH∗(A,A) G and HH∗(B,B) are shown to be isomorphic, for any centrally G-Galois extension. To approach the above mentioned problem we construct a spectral sequence TorH p (HHq(B, M CHV )) =⇒ HHp+q(A, M) CHV, where RH denotes a certain subalgebra of H. In the case when H is a finitedimensional commutative Hopf algebra over a field of characteristic zero we show that the above spectral sequence collapses. Thus its edge maps induce isomorphisms K ⊗RH HH∗(B, M CHV ) ∼= HH∗(A, M) CHV, that generalize the isomorphisms in [Lo2]. In the last part of the paper, for a centrally Hopf-Galois extension B ⊆ A, we apply the foregoing results to compute the subspace of H-coinvariant elements in HH∗(A, M), where H ′ is the abelianization of H. A similar result is derived for cyclic homology of A.