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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Hereditary and maximal crossed product orders


Authors: Amiram Braun, Yuval Ginosar and Amit Levy
Journal: Proc. Amer. Math. Soc. 135 (2007), 2733-2742
MSC (2000): Primary 16H05, 16E60, 16E65
Published electronically: May 8, 2007
MathSciNet review: 2317946
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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/{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.


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Additional Information

Amiram Braun
Affiliation: Department of Mathematics, University of Haifa, Haifa 31905, Israel
Email: abraun@math.haifa.ac.il

Yuval Ginosar
Affiliation: Department of Mathematics, University of Haifa, Haifa 31905, Israel
Email: ginosar@math.haifa.ac.il

Amit Levy
Affiliation: Department of Mathematics, University of Haifa, Haifa 31905, Israel
Email: amitlevy1@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-07-08977-0
Received by editor(s): June 1, 2006
Published electronically: May 8, 2007
Communicated by: Martin Lorenz
Article copyright: © Copyright 2007 American Mathematical Society