On strongly graded Gorestein orders

Abstract

Let \(G\) be a finite group and let \(\Lambda=\oplus_{g \in G}\Lambda_{g}\) be a strongly \(G\)-graded \(R\)-algebra, where \(R\) is a commutative ring  with unity. We prove that if \(R\) is a Dedekind domain with quotient field \(K\), \(\Lambda\) is an \(R\)-order in a separable \(K\)-algebra such that the algebra \(\Lambda_{1}\) is a Gorenstein \(R\)-order, then \(\Lambda\) is also a Gorenstein \(R\)-order. Moreover, we prove that the induction functor \(ind:\ Mod\Lambda_{H}\rightarrow\ Mod\Lambda \) defined in Section 3, for a subgroup \(H\) of \(G\), commutes with the standard duality functor.