Cosemisimple hopf algebras coacting on coalgebras

Abstract

Let H be a cosemisimple Hopf algebra over a field k, and C an H‐comodule coalgebra. We study Morita‐Takeuchi contexts connecting the categories of (C, H)‐comodules, C^coH-comodules and graded S(C)-comodules, where S(C) denotes the coalgebra of semicoinvariants, graded by the grouplikes of H. Our results generalize results obtained recently for graded coalgebras (cf. [8, 15, 16]), and provide a dual version of a Morita context due to Cohen and Fishman (cf. [4]). Some applications to the structure of injective and simple objects are given. If H is finite dimensional and cosemisimple and if C/C^coH is a Galois coextension, then the category of (C, H)-comodules is equivalent to a category of graded comodules. We also prove a Maschke type theorem for comodules