Package Deal Theorems and Splitting Orders in Dimension 1 

by Lawrence S. Levy and Charles J. Odenthal 

KEYWORDS 
        localization, completion, normalization, splitting order, package deal 
DATE 
        10/13/95 
STATUS 
        Accepted. To appear in Trans. Amer. Math. Soc. 
COMMENT 
        Typset in AMS-Latex 2.09 

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 $\Lambda_m$, at maximal ideals of 
$R$, is the family of all localizations $M_m$ 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.