Torsion and Tensor Products Over Domains and Specializations to Semigroup Rings

Leamer, Micah J.

doi:10.48550/arxiv.1211.2896

Cited by 4 publications

(6 citation statements)

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“…) given by sending x to multiplication by x; see [8,Remark 1.7]. Since the map [A → I A ] establishes a bijection between relative ideals of Γ and fractional monomial ideals of k[Γ], the result follows.…”

Section: Fundamentalsmentioning

confidence: 96%

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Huneke–Wiegand Conjecture for complete intersection numerical semigroup rings

García-Sánchez

Leamer

2013

Journal of Algebra

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We give a positive answer to the Huneke-Wiegand Conjecture for monomial ideals over free numerical semigroup rings, and for two generated monomial ideals over complete intersection numerical semigroup rings.It is often the case that open problems in ring theory remain difficult when specialized to numerical semigroup rings. In such instances it may be beneficial to gain perspective on the problem by trying to tackle its number theoretic analog. Since the integral closure of a numerical semigroup ring is just the polynomial ring in one variable, this perspective seems all the more reasonable for problems where the case for integrally closed rings is much easier to solve.If R is a one-dimensional integrally closed local domain and M is a finitely generated torsion-free R-module, then M is free if and only if M ⊗ R Hom(M, R) is torsion-free. This follows from either [1, 3.3] or from the structure theorem for modules over a principle ideal domains. C. Huneke and R. Wiegand have conjectured that this property holds for all one-dimensional Gorenstein domains.Conjecture 1. [7,[473][474] Let R be a one-dimensional Gorenstein domain and let M be a non-zero finitely generated R-module, which is not projective. Then M ⊗ R Hom R (M, R) has a non-trivial torsion submodule.

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Section: Fundamentalsmentioning

confidence: 96%

“…, x n ) a relative ideal of Γ. It follows from the equivalences in [8,Theorem 1.4] that A is Huneke-Wiegand if and only if there exists a partition {S, S ′ } of {1, . .…”

Section: Fundamentalsmentioning

confidence: 99%

Huneke–Wiegand Conjecture for complete intersection numerical semigroup rings

García-Sánchez

Leamer

2013

Journal of Algebra

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“…The problem is, however, still open in general, and no one has a complete answer to the following Conjecture 1.2, even in the case where R is a complete intersection, or in the rather special case where R is a numerical semigroup ring; see [2,7,8,9]. The reader can consult [5,12] for the recent major progress on numerical semigroup rings. Conjecture 1.2.…”

Section: Introductionmentioning

confidence: 99%

Huneke–Wiegand conjecture and change of rings

et al. 2015

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Let R be a Cohen-Macaulay local ring of dimension one with a canonical module K R . Let I be a faithful ideal of R. We explore the problem of when I ⊗ R I ∨ is torsionfree, where I ∨ = Hom R (I, K R ). We prove that if R has multiplicity at most 6, then I is isomorphic to R or K R as an R-module, once I ⊗ R I ∨ is torsionfree. This result is applied to monomial ideals of numerical semigroup rings. A higher dimensional assertion is also discussed.

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“…Recently, this still-open conjecture has spurred much subsequent work (see [4,5,10,14,16]). Of particular interest is [10], where P. García-Sánchez and M. Leamer study this conjecture in special cases related to numerical monoid algebras.…”

Section: Introductionmentioning

confidence: 99%

“…Given a numerical monoid Γ and s ∈ N \ Γ, they construct a monoid, which we denote as S s Γ , whose elements correspond to arithmetic sequences in Γ of step size s and whose monoid operation is set-wise addition. These monoids, which we refer to as Leamer monoids in honor of [16], reduce a special case of the Huneke-Wiegand conjecture to finding irreducible elements of a certain type. Proposition 1.2 ([10, Corollary 7]).…”

Section: Introductionmentioning

confidence: 99%

Factorization properties of Leamer monoids

et al. 2014

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Abstract. The Huneke-Wiegand conjecture has prompted much recent research in Commutative Algebra. In studying this conjecture for certain classes of rings, García-Sánchez and Leamer construct a monoid S s Γ whose elements correspond to arithmetic sequences in a numerical monoid Γ of step size s. These monoids, which we call Leamer monoids, possess a very interesting factorization theory that is significantly different from the numerical monoids from which they are derived. In this paper, we offer much of the foundational theory of Leamer monoids, including an analysis of their atomic structure, and investigate certain factorization invariants. Furthermore, when S s Γ is an arithmetical Leamer monoid, we give an exact description of its atoms and use this to provide explicit formulae for its Delta set and catenary degree.

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Torsion and Tensor Products Over Domains and Specializations to Semigroup Rings

Cited by 4 publications

References 0 publications

Huneke–Wiegand Conjecture for complete intersection numerical semigroup rings

Huneke–Wiegand Conjecture for complete intersection numerical semigroup rings

Huneke–Wiegand conjecture and change of rings

Factorization properties of Leamer monoids

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