Gorenstein Numerical Semigroup Rings

• Gorenstein numerical semigroup rings are one-dimensional Cohen–Macaulay rings defined by symmetric numerical semigroups with cyclic canonical modules.
• Their structure relies on numerical invariants such as the Frobenius number and conductor, offering explicit classifications and ideal enumerations.
• They have practical applications in analyzing toric ideals, Ulrich and Arf rings, and homological structures using techniques like gluing and numerical duplication.

A Gorenstein numerical semigroup ring is a one-dimensional Cohen–Macaulay semigroup ring $R = k[H]$, where $H$ is a numerical semigroup (an additive, cofinite submonoid of $\mathbb{N}$ containing $0$), such that $R$ is Gorenstein. By a fundamental result of Kunz and Herzog, this occurs if and only if $H$ is symmetric: for $F(H)$ the Frobenius number of $H$, $H$ is symmetric if for every $n \in \mathbb{Z}$ exactly one of $n$ or $F(H) - n$ lies in $H$. Gorenstein numerical semigroup rings are a central object in the study of one-dimensional Gorenstein domains, Ulrich and Arf rings, and have deep connections with the structure of graded ideals, Betti numbers, and semigroup combinatorics.

1. Symmetry of Numerical Semigroups and the Gorenstein Property

Symmetry of a numerical semigroup $H \subset \mathbb{N}$ is characterized by the property that the set of gaps $G(H) = \mathbb{N} \setminus H$ satisfies: for $F = F(H)$, $n \notin H$ if and only if $F-n \notin H$. The genus $g(H)=|G(H)|$ satisfies $H$ symmetric $\iff$ $g(H) = (F(H)+1)/2$ (Süer et al., 2020). The semigroup ring $R = k[H]$ is Gorenstein if and only if $H$ is symmetric. The canonical module then takes a cyclic form and $R$ is of Cohen–Macaulay type $1$ (Endo, 2023). For analytic localizations (e.g., $k[[H]]$), this equivalence persists.

2. Conductor, Frobenius Number, and Classification

The Frobenius number $F(H) = \max(\mathbb{Z} \setminus H)$ and the conductor $c(H) = F(H)+1$ are numerical invariants intimately tied to the Gorenstein property. Symmetry is manifest in the Apéry set structure and in formulas for genus, conductor, and pseudo-Frobenius numbers. In Gorenstein semigroup rings, the key properties include:

• $H$ symmetric $\iff$ the pseudo-Frobenius set $\mathrm{PF}(H) = \{F(H)\}$,
• The canonical ideal is principal,
• The trace of the canonical module equals $R$,
• The residue $\operatorname{res}(H)$ vanishes, measuring "distance" to Gorenstein (Herzog et al., 2020).

3. Enumeration of Gorenstein Ideals and Explicit Structure

A distinctive feature of Gorenstein numerical semigroup rings is a sharp enumeration and description of non-principal graded ideals $I \subset R$ with $R/I$ also Gorenstein. The number of such ideals equals the conductor $c(H) = F(H) + 1$ (Endo, 2023). Explicitly, the set of such ideals is: $$\left\{\,A_m,\,t^m A_m :\, m \in \mathbb{N} \setminus H \,\right\},$$ where $A_m = R:Rt^m = (1, t^m)^{-1}$, and $m$ runs over the set of gaps of $H$ ("$\mathbb{N}\setminus H$"), establishing a direct bijection between such ideals and the gaps. This structural result also underpins the module-theoretic study of Ulrich and Arf ideals in this context.

4. Algebraic and Homological Structure

Gorenstein numerical semigroup rings display diverse algebraic and homological behaviors. Their toric ideals (the defining ideals in power series or polynomial rings) and their minimal graded free resolutions reflect the combinatorics of $H$:

• For embedding dimension $4$, the minimal number of generators is $3$ or $5$ (Bresinsky's Theorem) (Watanabe, 2018, Eto et al., 2021).
• For Sally-type semigroups, the presentation ideal $I_k$ is determinantal, and Betti numbers can be computed explicitly using Eagon–Northcott complexes (Singh et al., 19 Dec 2025).
• Certain rings exhibit transcendental Betti series, as shown for the semigroup $\langle 36, 48, ..., 135\rangle$ (Löfwall et al., 2012).
• The combinatorics of Apéry sets, Young diagram decompositions, and inverse polynomial annihilators provide systematic tools for analyzing the structure and detecting the Gorenstein property (Süer et al., 2020, Eto et al., 2021).

5. Special Constructions: Sally-type, Gluing, and Numerical Duplication

Numerous families and constructions enrich the landscape of Gorenstein numerical semigroups:

• Sally-type semigroups are generated by subsets of $[e,2e-1]$ with gaps introduced ("Sally-type"), and the Gorenstein property is characterized by explicit conditions (e.g., for $S_k^e(j)$, Gorenstein iff $j=k$ when $1\leq k<e/2$) (Singh et al., 19 Dec 2025, Dubey et al., 15 Jul 2025).
• Gluing of numerical semigroups is a powerful operation: nice or star gluings of symmetric semigroups yield new symmetric (hence Gorenstein) semigroups, preserving the Cohen–Macaulay or Gorenstein property in associated coordinate rings, projective closures, and tangent cones (Singh et al., 2023).
• Numerical duplication produces new semigroups $T = S\Join^b E$ by duplicating $S$ along an ideal $E$ and odd integer $b$, with their tangent cones Gorenstein under explicit Apéry-set and module-theoretic conditions (such as $E$ being a canonical ideal and $S$ being $M$-pure and symmetric) (D'Anna et al., 2018).

6. Trace, Residue, and Nearly Gorenstein Rings

The trace ideal of the canonical module, $\mathrm{Tr}(\omega_R)$, and the residue $$\operatorname{res}(H) = \dim_k R/\mathrm{Tr}(\omega_R)$$, provide a fine measure of proximity to the Gorenstein property. In Gorenstein (i.e., symmetric) semigroups, the trace is the unit ideal and the residue is $0$; for nearly Gorenstein rings, the residue is $1$ and the trace is $m$-primary. In 3-generated cases, $\operatorname{res}(H)$ has the explicit formula $d_1 d_2 d_3$, with the $d_i$ reflecting the structure constants of the semigroup (Herzog et al., 2020).

7. Connections with Arf Rings, Ulrich Ideals, and Open Problems

Gorenstein numerical semigroup rings form a subclass within generalized Gorenstein and Arf rings. In the Gorenstein case, the Arf property is characterized by minimal multiplicity and further conditions on the semigroup data. Ulrich ideals and their quotients play a role in enumerating Gorenstein ideals and connecting module-theoretic properties to the combinatorics of $H$ (Celikbas et al., 2018, Endo, 2023). Open problems remain regarding the classification of Betti series (rational versus transcendental), the structure of higher codimension Gorenstein toric ideals, and the precise relationships among various canonical invariants in families generated by gluings, duplications, or other semigroup operations.

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References:

• (Endo, 2023, Singh et al., 19 Dec 2025, Süer et al., 2020, Celikbas et al., 2018, Löfwall et al., 2012, Dubey et al., 15 Jul 2025, Eto et al., 2021, Singh et al., 2023, Herzog et al., 2020, Watanabe, 2018, D'Anna et al., 2018).