Equivalence theorems and Hopf-Galois extensions
This work studies equivalence theorems for categories \({\mathcal M}(H)^D_A\) of right (\(D\)-\(A\))-Hopf modules. The authors get a characterization of the Hopf-Galois extensions. The paper has three main parts. In section two the notions which are going to be used are defined and some of the theorems relating the category of modules over a ring and the category of modules over its endomorphism ring are proved. Some of the results in this section are not new, but their inclusion in the paper makes it much more readable. When \(P\) is an \(R\) module and \(T=\text{End}(P)\) then the functors \(\text{Hom}_R(P,-)\) and \(-\otimes_TP\) are studied, in particular in the case where \(P\) is a quasiprogenerator. In the third section appear the main results which are obtained as applications of the theorems in section 2 to the case of an \(H\)-comodule algebra and \(H\)-module algebra. In the last section there are some applications to the finite dimensional case. The paper is almost self contained, which does not mean easy. The results of section 2, which make the work selfcontained, stand for themselves, and could have various other applications.
scientific article; zbMATH DE number 1053883
scientific article