Large indecomposable modules over local rings

Häßler, Wolfgang; Karr, Ryan; Klingler, Lee; Wiegand, Roger

doi:10.1016/j.jalgebra.2006.05.016

Cited by 6 publications

(4 citation statements)

References 12 publications

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“…For every a ∈ q(F ), we denote by [a] = aq(H red ) ⊂ q(F ) the class containing a. For g ∈ C(H), P ∩ g is the set of prime divisors lying in g. Concerning the distribution of prime divisors in Krull monoids of isomorphism classes of modules we refer to [20,17,41,18,2].…”

Section: The Semigroup H Is Calledmentioning

confidence: 99%

A characterization of length-factorial Krull monoids

Geroldinger,

Zhong

2021

Preprint

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Section: The Semigroup H Is Calledmentioning

confidence: 99%

A characterization of length-factorial Krull monoids

Geroldinger,

Zhong

2021

Preprint

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“…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] …”

Section: Modules With Torsionmentioning

confidence: 99%

Direct-sum behavior of modules over one-dimensional rings

Karr

Wiegand²

2010

Commutative Algebra

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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:

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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],…”

Section: Introductionmentioning

confidence: 99%