\(H\)-module endomorphism rings
Let \(H\) be a finite-dimensional Hopf algebra and \(A/B\) an \(H\)-Galois extension. Then for any right \(A\)-module \(M\) there is a ring isomorphism \(\text{End}_A(M\otimes_BA)\simeq\text{End}_B(M)\# H\). \(\text{End}_B(M)/\text{End}_A(M)\) is an \(H^*\)-extension. If moreover \(H\) is unimodular, \(\text{End}_B(M)/\text{End}_A(M)\) is Galois if and only if \(M\otimes_BA\) weakly divides \(M\), and \(\text{End}_B(M)\) has a trace 1 element if and only if \(M\) weakly divides \(M\otimes_BA\) (the unimodularity is not actually needed for the last assertion). In the case where \(M\) and \(M\otimes_BA\) weakly divide each other, the rings \(\text{End}_A(M)\) and \(\text{End}_A(M\otimes_BA)\) are Morita equivalent. A Higman criterion for existence of trace 1 elements in the \(H\)-module algebra \(\text{End}_B(M)\) is established. Some results of Dade (for the graded case) and Schneider are obtained as applications. Finally the stable Clifford theory for graded rings is extended to the case of Hopf extensions.
scientific article; zbMATH DE number 810234