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

Let B⊆AB⊆A be an HH-Galois extension, where HH is a Hopf algebra over a field KK. If MM is a Hopf bimodule then HH∗(A,M), the Hochschild homology of AA with coefficients in MM, is a right comodule over the coalgebra CH=H/[H,H]CH=H/[H,H]. Given an injective left CHCH-comodule VV, our aim is to understand the relationship between HH∗(A,M)□CHV and HH∗(B,M□CHV). 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 GG-Galois extension. To approach the above mentioned problem, in the case when AA is a faithfully flat BB-module and HH satisfies some technical conditions, we construct a spectral sequence TorpRH(K,HHq(B,M□CHV))⟹HHp+q(A,M)□CHV, where RHRH denotes the subalgebra of cocommutative elements in HH. We also find conditions on HH such that the edge maps of the above spectral sequence yield isomorphisms K⊗RHHH∗(B,M□CHV)≅HH∗(A,M)□CHV. In the last part of the paper we define centrally Hopf–Galois extensions and we show that for such an extension B⊆AB⊆A, the RHRH-action on HH∗(B,M□CHV) is trivial. As an application, we compute the subspace of HH-coinvariant elements in HH∗(A,M)HH∗(A,M). A similar result is derived for HC∗(A)HC∗(A), the cyclic homology of AA.