Hereditary and maximal crossed product orders

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Abstract

We first deal with classical crossed products Sf * G, where G is a finite group acting on a Dedekind domain S and SG (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 Sf * G is hereditary if and only if S/Jac(S) * G is semi-simple. In particular, the heredity property may hold even when S/SG 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.

Original languageEnglish
Pages (from-to)2733-2742
Number of pages10
JournalProceedings of the American Mathematical Society
Volume135
Issue number9
DOIs
StatePublished - Sep 2007

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics

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