Huneke-Wiegand conjecture for complete intersection numerical semigroup rings.
type: Item
instance of: scholarly article
title: Huneke-Wiegand conjecture for complete intersection numerical semigroup rings.
zbMATH Open document ID: 1301.20065
author: Pedro A. García Sánchez
author: Micah J. Leamer
published in: Journal of Algebra
publication date: 2014-03-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1211.4554
review text: With regards to the study of torsion in tensor products of modules, \textit{C. Huneke} and \textit{R. Wiegand} [Math. Ann. 299, No. 3, 449-476 (1994; Zbl 0803.13008)] formulated the following conjecture: if \(R\) is a one-dimensional Gorenstein local domain and \(M\) is a non-zero finitely generated \(R\)-module such that \(M\) and \(M\otimes\Hom_R(M,R)\) are torsion-free, then \(M\) is free. It has been shown that a positive answer to the conjecture would imply that the same statement holds for Gorenstein local domains of any dimension, and it would further prove a conjecture of Auslander and Reiten for such rings (cf. \textit{O. Celikbas} and \textit{R. Takahashi} [J. Algebra 382, 100-114 (2013; Zbl 1342.13022)]). In the paper under review, the authors analyze the problem in the case where \(R\) is a complete intersection numerical semigroup ring and \(M\) is a monomial ideal of \(R\). Let \(\mathcal S\) be a numerical semigroup, i.e. a cofinite submonoid of \((\mathbb N,+)\), and consider the semigroup ring \(R=k\mathcal S=kt^s:s\in \mathcal S\). Some ring-theoretic properties of \(R\) (e.g. Gorensteinness) have a clear semigroup-theoretic characterization, and several classical definitions for rings, such as complete intersection and embedding dimension, are extended to numerical semigroups by means of this correspondence. \textit{C. Delorme} [Ann. Sci. Éc. Norm. Supér. (4) 9, 145-154 (1976; Zbl 0325.20065)] provides a recursive characterization of the complete intersection property, in terms of an operation on numerical semigroups that is now known as gluing: the gluing of two complete intersection numerical semigroups is a complete intersection, and every complete intersection numerical semigroup other than \(\mathbb N\) is a gluing of two complete intersections with smaller embedding dimension. By studying the behavior of fractional ideals under gluing and applying some previous results of the second author [Torsion and tensor products over domains an…
reviewed by: Alessio Sammartano
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 13C40
zbMATH DE Number: 6266757
zbMATH Keywords: Huneke-Wiegand conjecture
zbMATH Keywords: numerical semigroup rings
zbMATH Keywords: complete intersections
zbMATH Keywords: free numerical semigroups
zbMATH Keywords: gluings
Wikidata QID: Q122888574
describes a project that uses: numericalsgps
MaRDI profile type: Publication
OpenAlex ID: W2963466265
arXiv ID: 1211.4554
cites work: Modules over unramified regular local rings
cites work: Semi-groupes d'entiers et application aux branches
cites work: Auslander-Reiten conjecture and Auslander-Reiten duality
cites work: numericalsgps, a GAP package for numerical semigroups
cites work: Sous-monoïdes d'intersection complète de $N$
cites work: The number of generators for an ideal and its dual in a numerical semigroup
cites work: Tensor products of modules and the rigidity of Tor
cites work: Torsion and tensor products over domains and specializations to semigroup rings
cites work: Numerical semigroups.
DOI: 10.1016/J.JALGEBRA.2013.06.007
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description: scientific article