  Let A be an H-Galois extension of B. 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]. Given an injective left C-comodule V, we denote the
cotensor product of M and V by N. Our aim is to investigate the relationship
between the cotensor product of HH.(A,M) and V, on the one hand, and HH.(B,N)
on the other hand. The roots of this problem can be found in Lorenz's work on
the descent of Hochschild homology of centrally Galois extensions, where
HH.(A)^G, the subspace of invariant cycles with respect to the action of the
Galois group, and HH.(B) are shown to be isomorphic.
