Hereditary and maximal crossed product orders

Abstract

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