Direct-Sum Decompositions over Local Rings,

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 R-modules. By passing to the m-adic completion , 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 R-module M a full submonoid Λ(M) of n, where n is the number of distinct indecomposable direct summands of RM. 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 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.