Hereditary and maximal crossed product orders
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- by Amiram Braun, Yuval Ginosar and Amit Levy PDF
- Proc. Amer. Math. Soc. 135 (2007), 2733-2742 Request permission
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.References
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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
- MR Author ID: 349785
- Email: ginosar@math.haifa.ac.il
- Amit Levy
- Affiliation: Department of Mathematics, University of Haifa, Haifa 31905, Israel
- Email: amitlevy1@gmail.com
- Received by editor(s): June 1, 2006
- Published electronically: May 8, 2007
- Communicated by: Martin Lorenz
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 2733-2742
- MSC (2000): Primary 16H05, 16E60, 16E65
- DOI: https://doi.org/10.1090/S0002-9939-07-08977-0
- MathSciNet review: 2317946