On the separability of the restriction functor

Abstract

Let \(G\) be a group, \(\Lambda=\bigoplus_{\sigma \in G}\Lambda_{\sigma}\) a strongly graded ring by \(G\), \(H\) a subgroup of \(G\) and \(\Lambda_{H}=\bigoplus_{\sigma \in H}\Lambda_{\sigma}\). We give a necessary and sufficient condition for the ring \(\Lambda/\Lambda_{H}\) to be separable, generalizing the corresponding result for the ring extension \(\Lambda/\Lambda_{1}\). As a consequence of this result we give a condition for \(\Lambda\) to be a hereditary order in case \(\Lambda\) is a strongly graded by finite group \(R\)-order in a separable \(K\)-algebra, for \(R\) a Dedekind domain with quotient field \(K\).