Hereditary and maximal crossed product orders

Abstract:

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 $f:G\times G\rightarrow S^*$ is a 2-cocycle. We prove that $S^f*G$ is hereditary if and only if $S/\operatorname {Jac}(S)^{\bar {f}}*G$ 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.