Coactions on Hochschild homology of Hopf-Galois extensions and their coinvariants. (Q964521)

Let \(H\) be a Hopf algebra over a field \(k\), let \(R_H\) be the subalgebra of cocommutative elements of \(H\), and let \(C_H=H/[H,H]\) be the quotient coalgebra, where \([H,H]\) is the subspace of \(H\) spanned by all the commutators. Consider a faithfully flat \(H\)-Galois extension \(B\subset A\), an \(A\)-bimodule \(M\) which is also a right \(H\)-comodule, and the \(A\)-module structure maps are both \(H\)-collinear. Also take an injective left \(C_H\)-comodule \(V\). Under certain technical conditions on \(H\), the authors construct a spectral sequence \[ \text{Tor}^{R_H}_p(k,\text{HH}_q(B,M\square_{C_H}V))\Rightarrow\text{HH}_{p+q}(A,M)\square_{C_H}V, \] where \(\text{HH}_*\) denotes the Hochschild homology. Conditions on \(H\) are found such that the edge maps of this spectral sequence yield isomorphisms \(k\otimes_{R_H}\text{HH}_*(B,M\square_{C_H}V)\simeq\text{HH}_*(A,M)\square_{C_H}V\). Centrally Hopf-Galois extensions are defined, and if \(B\subset A\) is such an extension, it is shown that the \(R_H\)-action on \(\text{HH}_*(B,M\square_{C_H}V)\) is trivial. This result is applied to compute the \(H\)-coinvariants of \(\text{HH}_*(A,M)\).