Coactions on Hochschild homology of Hopf–Galois extensions and their coinvariants

Abstract

Let B ⊆ A be an H -Galois extension, where H is a Hopf algebra over a field K . 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 C H = H / [ H , H ] . Given an injective left C H -comodule V , our aim is to understand the relationship between HH ∗ ( A , M ) □ C H V and HH ∗ ( B , M □ C H V ) . The roots of this problem can be found in Lorenz (1994) [15], 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, in the case when A is a faithfully flat B -module and H satisfies some technical conditions, we construct a spectral sequence Tor p R H ( K , HH q ( B , M □ C H V ) ) ⟹ HH p + q ( A , M ) □ C H V , where R H denotes the subalgebra of cocommutative elements in H . We also find conditions on H such that the edge maps of the above spectral sequence yield isomorphisms K ⊗ R H HH ∗ ( B , M □ C H V ) ≅ HH ∗ ( A , M ) □ C H V . In the last part of the paper we define centrally Hopf–Galois extensions and we show that for such an extension B ⊆ A , the R H -action on HH ∗ ( B , M □ C H V ) is trivial. As an application, we compute the subspace of H -coinvariant elements in HH ∗ ( A , M ) . A similar result is derived for HC ∗ ( A ) , the cyclic homology of A .