Hopfological invariants for tame subextensions

Let H be a finite dimensional Hopf algebra over a field K. In this paper, we study when an H-extension becomes a tame H-extension by calculating Hopfological homology and Hopf-cyclic homology. In the (derived) category of H’-comodules for a Hopf algebra H’, we take Hopf subalgebra H of H’ and a certain order A of H. We see the behavior of Hopfological homology for a tame A-subextension S/R in terms of the surjectivity of trace map and of cyclic modules, which induce Hopf-cyclic homology, for Hopf-Galois extensions with H in terms of relative Hopf modules.