Indecomposable modules of large rank over Cohen-Macaulay local rings

Abstract

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

Factorization Theory and Decompositions of Modules

“…. , r, s}, there exist, by [HRKW08], indecomposable finitely generated R-modules M i and N i of rank (0, 0, i) and (i, i, 0), respectively. (These modules are not necessarily torsion-free.)…”

Direct-sum behavior of modules over one-dimensional rings

“…If R has two minimal primes P 1 and P 2 , the rank of the R-module M is the pair (r 1 , r 2 ), where r i is the dimension of M P i as a vector space over R Pi . In a series of papers [22,20,21] …”

Constructing big indecomposable modules

“…Indeed, they build indecomposable modules of arbitrarily large rank over hypersurface singularities with positive dimension which are not A 1 -singularities ( [10]), and over rings which are not homomorphic images of Dedekind-like rings ( [7], [8], [9]). While over hypersurfaces, the constructed indecomposable modules are free of constant rank on the punctured spectrum, in the case of rings which are not homomorphic images of Dedekind-like rings, the indecomposable modules are only known to be locally free at a finite set of primes.…”

“…Our interest was raised by work of W. Hassler, R. Karr, L. Klinger and R. Wiegand. Indeed, they build indecomposable modules of arbitrarily large rank over hypersurface singularities with positive dimension which are not A 1 -singularities ( [6]), and over rings which are not homomorphic images of Dedekind-like rings ( [7],…”