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 b R, 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 Nn, where n is the number of distinct indecomposable direct summands of b R ⊗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 Nn 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. Suppose we wish to determine how badly the Krull-Schmidt uniqueness theorem fails for direct-sum decompositions of finitely generated modules over a given ring. One approach— the one we will follow here—is to associate to a given finitely generated module M the monoid +(M) consisting of isomorphism classes of modules that are direct summands of direct sums of finitely many copies of M . (For example, M might be the direct sum of a bunch of indecomposable modules demonstrating failure of the Krull-Schmidt theorem.) The structure of these monoids tells us exactly how badly the Krull-Schmidt theorem can fail. Suppose, for example that we determine that +(M) ∼= Λ := {(a, b, c) ∈ N | a + 4b = 5c}, with M corresponding to (1, 1, 1). The minimal elements of Λ, namely, (1, 1, 1), (0, 5, 4) and (5, 0, 1) correspond to the indecomposable modules in +(M). In particular, M is indecomposable. Moreover, writing aM for the direct sum of a copies of M , we see that 2M , 3M and 4M all have unique representations as direct sums of indecomposables. However, 5M = P ⊕Q, where P and Q are the indecomposable modules corresponding to the other 1991 Mathematics Subject Classification. Primary 13C14, 13C20; Secondary 13B35, 13J15,.