Hereditary and maximal crossed product orders

We first deal with classical crossed products S^{f *} G, where G is a finite group acting on a Dedekind domain S and S^G (the G-invariant elements in S) a DVR, admitting a separable residue fields extension. Here f: G × G → S^* is a 2-cocycle. We prove that S^f * G is hereditary if and only if S/Jac(S)^f̄ * G is semi-simple. In particular, the heredity property may hold even when S/S^G is not tamely ramified (contradicting standard textbook references). For an arbitrary Krull domain S, we use the above to prove that under the same separability assumption, S ^f * G is a maximal order if and only if its height one prime ideals are extended from S. We generalize these results by dropping the residual separability assumptions. An application to non-commutative unique factorization rings is also presented.