Abstract. A commutative Noetherian local ring (R, m, k) is called Dedekindlike provided R is one-dimensional and reduced, the integral closure R is generated by at most 2 elements as an R-module, and m is the Jacobson radical of R. If M is an indecomposable finitely generated module over a Dedekind-like ring R, and if P is a minimal prime ideal of R, it follows from a classification theorem due to L. Klingler and L. Levy that M P must be free of rank 0, 1 or 2.Now suppose (R, m, k) is a one-dimensional Cohen-Macaulay local ring that is not Dedekind-like, and let P 1 , . . . , P t be the minimal prime ideals of R. The main theorem in the paper asserts that, for each non-zero t-tuple (n 1 , . . . , n t ) of non-negative integers, there is an infinite family of pairwise non-isomorphic indecomposable finitely generated R-modules M satisfyingIn 1911, E. Steinitz determined the structure of all finitely generated modules over Dedekind domains. This structure is so simple that one is tempted to try to generalize Steinitz's result to a larger class of commutative rings. Indeed, in a recent series of papers [KL1], [KL2], [KL3] L. Klingler and L. Levy presented a classification, up to isomorphism, of all finitely generated modules over a class of commutative rings they call "Dedekind-like".We recall that a commutative Noetherian local ring (R, m, k) is Dedekind-like [KL1, Definition 2.5] provided R is one-dimensional and reduced, the integral closure R is generated by at most 2 elements as an R-module, and m is the Jacobson radical of R. (In [KL2, (1.1.3)] a further requirement is imposed: If R/m is a field, then it is a separable extension of k. Klingler and Levy prove their classification theorem only under this additional hypothesis. In the present paper, however, we do not require that Dedekind-like rings satisfy this separability condition.) Although Dedekind-like rings are very close to their normalizations, their module structure is much more complicated than that of Dedekind domains. Klingler and Levy dash any hope of a further extension of their classification theorem by showing that, if R is not a homomorphic image of a Dedekind-like ring or a special type
For commutative, Noetherian, local ring R of dimension one, we show that, if R is not a homomorphic image of a Dedekind-like ring, then R has indecomposable finitely generated modules that are free of arbitrary rank at each minimal prime. For Cohen-Macaulay ring R, this theorem was proved in [W. Hassler, R. Karr, L. Klingler, R. Wiegand, Indecomposable modules of large rank over Cohen-Macaulay local rings, Trans. Amer. Math. Soc., in press]; in this paper we handle the general case.
Let R be a reduced, one-dimensional Noetherian local ring whose integral closure R is finitely generated over R. Since R is a direct product of finitely many principal ideal domains (one for each minimal prime ideal of R), the indecomposable finitely generated R-modules are easily described, and every finitely generated R-module is uniquely a direct sum of indecomposable modules. In this article we will see how little of this good behavior trickles down to R. Indeed, there are relatively few situations where one can describe all of the indecomposable R-modules, or even the torsion-free ones. Moreover, a given finitely generated module can have many different representations as a direct sum of indecomposable modules. Finite Cohen-Macaulay typeIf R is a one-dimensional reduced Noetherian local ring, the maximal CohenMacaulay R-modules (those with depth 1) are exactly the non-zero finitely generated torsion-free modules. One says that R has finite Cohen-Macaulay type provided there are, up to isomorphism, only finitely many indecomposable maximal Cohen-Macaulay modules. The following theorem classifies these rings:
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.