Huneke-Wiegand conjecture for complete intersection numerical semigroup rings.
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 and specializations to semigroup rings. \url{arXiv:1211.2896} (2012)], the authors provide two cases in which the Huneke-Wiegand conjecture is satisfied. The first main result is an affirmative answer to the conjecture when \(R\) is a complete intersection numerical semigroup ring and \(M\) is an ideal of \(R\) generated by two monomials. The second main result of the paper concerns free numerical semigroups, a subclass of the complete intersections that was introduced by \textit{J. Bertin} and \textit{F. Carbonne} [J. Algebra 49, 81-95 (1977; Zbl 0498.14016)] in the study of analytic branches of plane curves. The authors prove that the Huneke-Wiegand conjecture holds for free numerical semigroup rings and arbitrary monomial curves.
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