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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Package deal theorems and splitting orders
in dimension 1


Authors: Lawrence S. Levy and Charles J. Odenthal
Journal: Trans. Amer. Math. Soc. 348 (1996), 3457-3503
MSC (1991): Primary 16P40, 16P50, 16W60; Secondary 13E05, 13B30, 13J10
MathSciNet review: 1351493
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Abstract: Let $\Lambda $ be a module-finite algebra over a commutative noetherian ring $R$ of Krull dimension 1. We determine when a collection of finitely generated modules over the localizations $\Lambda _{\mathbf {m}}$, at maximal ideals of $R$, is the family of all localizations $M_{\mathbf {m}}$ of a finitely generated $\Lambda $-module $M$. When $R$ is semilocal we also determine which finitely generated modules over the $J(R)$-adic completion of $\Lambda $ are completions of finitely generated $\Lambda $-modules.

If $\Lambda $ is an $R$-order in a semisimple artinian ring, but not contained in a maximal such order, several of the basic tools of integral representation theory behave differently than in the classical situation. The theme of this paper is to develop ways of dealing with this, as in the case of localizations and completions mentioned above. In addition, we introduce a type of order called a ``splitting order'' of $\Lambda $ that can replace maximal orders in many situations in which maximal orders do not exist.


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Additional Information

Lawrence S. Levy
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388
Email: levy@math.wisc.edu

Charles J. Odenthal
Affiliation: Department of Mathematics, University of Toledo, Toledo, Ohio 43606-3390
Email: codentha@math.utoledo.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01620-0
PII: S 0002-9947(96)01620-0
Keywords: Localization, completion, normalization, splitting order, package deal
Received by editor(s): April 11, 1994
Received by editor(s) in revised form: September 25, 1995
Additional Notes: Levy’s research was partially supported by NSF and NSA grants.
Article copyright: © Copyright 1996 American Mathematical Society