Doi–Hopf Modules Associated to Comodule Coalgebras

To any right comodule coalgebra C over a Hopf algebra H we associate a left H-comodule algebra A. Under certain conditions, in particular in the case where H has nonzero integrals, we show that the category of right C, H-comodules is isomorphic to a certain subcategory of the category of Doi–Hopf modules associated to A. As an application, we investigate the connection between C and the smash coproduct C ⋊ H being right semiperfect.