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SEMIGROUPS OF MODULES : A SURVEY

This paper is a survey of some recent developments in the study of direct-sum decompositions of modules. The idea is to examine direct-sum behavior by looking at semigroups of isomorphism classes of modules. To be precise, let R be a ring, and let C be a class of modules closed under isomorphism, finite direct sums, and direct summands. Assume there is a set V(C) ⊆ C of representatives; thus each element M ∈ C is isomorphic to exactly one element [M ] ∈ V(C). We make V(C) into an additive semigroup in the obvious way: [M ] + [N ] = [M ⊕ N ]. This semigroup encodes information about the way modules in C decompose as direct sums of other modules in C. Given a module M , we are particularly interested in the class +(M), which consists of modules that are isomorphic to direct summands of direct sums of finitely many copies of M . We will see, for example, that if R is a (commutative Noetherian) local ring and M is a finitely generated R-module, then V(+(M)) is a finitely generated Krull semigroup (defined in Definition 2.4). In the terminology associated with convex polytopes [5], V(+(M)) is a positive normal affine semigroup. (See definitions and various characterizations in §2.) Conversely, we have a “realization theorem”: given any finitely generated Krull semigroup H, there is a one-dimensional local domain R with a finitely generated R-module M such that V((+M)) ∼= H. These two results show exactly how badly Krull-Schmidt uniqueness can fail for direct-sum decompositions of finitely generated modules over local rings. For example, using the realization theorem, one can find an indecomposable finitely generated module M over a local ring such that M,M , . . . , M (4) have only the obvious direct-sum decompositions, whereas M (5) ∼= V ⊕W , where V and W are non-isomorphic indecomposables. On the other hand, using the fact that V(+(M)) is a Krull semigroup, one can show, for example, that there is no local ring R with indecomposable finitely generated modules M and N such that M (2) ∼= N . More subtle restrictions are mentioned in Section 9.