Coactions on Spaces of Morphisms

We study certain comodule structures on spaces of linear morphisms between H -comodules, where H is a Hopf algebra over the field k . We apply the results to show that H has non-zero integrals if and only if there exists a non-zero finite dimensional injective right H -comodule. Using this approach, we prove an extension of a result of Sullivan, by showing that if H is involutory and has non-zero integrals, and there exists an injective indecomposable right comodule whose dimension is not a multiple of char( k ), then H is cosemisimple. Also we prove without using character theory that if H is cosemisimple and M is an absolutely irreducible right H -comodule, then char( k ) does not divide dim( M ).