Wolfgang Hassler

Direct-sum decompositions over one-dimensional Cohen-Macaulay local rings

1 Alberto Facchini, Dipartimento di Matematica Pura e Applicata, Universita di Padova, Via Belzoni 7, I-35131 Padova, Italy, facchini@math.unipd.it 2 Wolfgang Hassler, Institut fur Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universitat Graz, Heinrichstrase 36/IV, A-8010 Graz, Austria, wolfgang.hassler@uni-graz.at 3 Lee Klingler, Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431-6498, klingler@fau.edu 4 Roger Wiegand, Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0323, rwiegand@math.unl.edu

Chains of Factorizations and Factorizations with Successive Lengths

ABSTRACT Let H be a commutative cancellative monoid. H is called atomic if every nonunit a ∈ H decomposes (in general in a highly nonunique way) into a product of irreducible elements (atoms) u i of H. The integer n is called the length of (†), and L(a) = {n ∈ ℕ | a decomposes into n irreducible elements of H} is called the set of lengths of a. Two integers k < l are called successive lengths of a if L(a) ∩ {m ∈ ℕ | k ≤ m ≤ l} = {k, l}. For a ∈ H we denote by Z n (a) the set of factorizations of a with length n. Suppose now that H is one of the following monoids: i. A congruence monoid in a Dedekind domain with finite residue fields. ii. H = D\ {0}, where D is a Noetherian domain having the following properties: is a Krull domain with finite divisor class group, and is a finite ring. iii. H = D\ {0}, where D is a one-dimensional Noetherian domain with finite normalization and finite Picard group (but possibly infinite residue fields). Let a ∈ H. In the present article, we investigate the structure of concatenating chains in Z n (a) as well as the relation between Z k (a) and Z l (a) if k and l are successive lengths of a. The work continues earlier investigations in Foroutan (2003), Foroutan and Geroldinger (2004), and Hassler (to appear).

Indecomposable modules of large rank over Cohen-Macaulay local rings

A commutative Noetherian local ring (R, m, k) is called Dedekind-like 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 satisfying MP i ≅ (Rp i ) (n i ) for each i.