Direct-sum decompositions over local rings

JOURNAL OF ALGEBRA

卷:240

Article

Wiegand, R

关键词: Krull-Schmidt theorem;    divisor class group;    semilocal ring;    affine monoid;   

DOI  :  10.1006/jabr.2000.8657

来源: Elsevier

【 摘 要 】

Let (R,m) be a local ring (commutative and Noetherian). If R is complete (or, more generally, Henselian), one has the Krull-Schmidt uniqueness theorem for direct sums of indecomposable finitely generated A-modules. By passing to the m-adic completion A, we can get a measure of how badly the Krull-Schmidt theorem can fail for a more general local ring. We assign to each finitely generated A-module M a full submonoid Lambda (M) of N(n), where n is the number of distinct indecomposable direct summands of (R) over cap circle times (R) M. This monoid is naturally isomorphic to the monoid +(M) of isomorphism classes of modules that are direct summands of direct sums of finitely many copies of M. The main theorem of this paper states that every full submonoid of N(n) arises in this fashion. Moreover, the local ring R realizing a given full submonoid can always be taken to be a two-dimensional unique factorization domain. The theorem has two non-commutative consequences: (1) a new proof of a recent theorem of Facchini and Herbera characterizing the monoid of isomorphism classes of finitely generated projective right modules over a non-commutative semilocal ring, and (2) a characterization of the monoids +(N), where N is an Artinian right module over an arbitrary ring. (C) 2001 Academic Press.

【 授权许可】

Free