Intended Frobenius Functor Applications

This dissertation consists of six sections. We shall investigate the application of quasi-Frobenius functors.The first and second sections are introduction and preliminaries respectively.In section 3 we first give the relationship between categories of entwined modules and categories of comodules of corings, then we prove the characterization of quasi-Frobenius functors between categories of entwined modules. In section 4 we give an characterization of quasi-Frobenius functors to graded ring theory and apply our results to the category of graded modules by a G-set. In section 5 we first introduce the notion of corings with a duality: let C be an A-coring, C is flat as A-bimodule, M^C,^CM are locally Noetherian categories, we will say that C has a duality if the basic duality extends to a duality(-)^*:M_f^C(?)^C M_f:^*(-),then we generalize the results of quasi-Frobenius functors between categories of comodules over corings and give the characterization of quasi-Frobenius functors between categories of comodules over corings with a duality. In the last section we study when the forgetful functor from the categories of right G-graded comodules of A-corings C to the categories of right A-modules. Frist we study when the forgetful functor U₁ : gr^C→M^C isFrobenius, and then use the conditions of when U₂ : M^C→M_A is quasi-Frobenius we give the conditions when U : gr^C→M_A is quasi-Frobenius.