Package deal theorems and splitting orders in dimension 1

Abstract

Let Λ \Lambda be a module-finite algebra over a commutative noetherian ring R R of Krull dimension 1. We determine when a collection of finitely generated modules over the localizations Λ m \Lambda _{\mathbf {m}} , at maximal ideals of R R , is the family of all localizations M m M_{\mathbf {m}} of a finitely generated Λ \Lambda -module M M . When R R is semilocal we also determine which finitely generated modules over the J ( R ) J(R) -adic completion of Λ \Lambda are completions of finitely generated Λ \Lambda -modules. If Λ \Lambda is an R 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 Λ \Lambda that can replace maximal orders in many situations in which maximal orders do not exist.