Nearly Gorenstein Families

• Nearly Gorenstein families are rings (or algebras) that exhibit only mild deviations from Gorensteinness, measured by the canonical trace ideal and combinatorial invariants.
• They encompass diverse structures including commutative local rings, noncommutative algebras, Stanley–Reisner rings, Ehrhart rings, and affine semigroup rings.
• These families bridge the gap between Gorenstein and Cohen–Macaulay rings, offering insights for singularity classification and combinatorial optimization.

A nearly Gorenstein family consists of commutative or noncommutative rings for which Gorensteinness fails only in a “mild” or controlled way, as quantified by the canonical trace ideal and precise module-theoretic or combinatorial invariants. This property appears in several algebraic and combinatorial frameworks, including commutative local and graded rings, noncommutative finite-dimensional algebras, Stanley–Reisner and Ehrhart rings, cyclic quotient singularities, and toric or affine semigroup rings. The unifying principle is that nearly Gorenstein rings interpolate between Gorenstein and more general Cohen–Macaulay rings, with the trace of the canonical module capturing the deviation from Gorensteinness.

1. Canonical Trace and the Nearly Gorenstein Definition

Let $R$ be a (graded) Cohen–Macaulay ring with canonical module $\omega_R$. The trace ideal $$\operatorname{tr}(\omega_R) = \sum_{\phi \in \operatorname{Hom}_R(\omega_R, R)} \phi(\omega_R)$$ is the key structural object. The ring $R$ is Gorenstein if and only if $\operatorname{tr}(\omega_R) = R$.

The nearly Gorenstein property is defined as: $\operatorname{tr}(\omega_R) \supseteq \mathfrak{m}$ where $\mathfrak{m}$ is the (graded) maximal ideal. Thus, $R$ is nearly Gorenstein if the non-Gorenstein locus is isolated at the closed point: every linear form lies in the trace of the canonical module, but $\operatorname{tr}(\omega_R)$ may be strictly smaller than $R$ (Hall et al., 2023). In the noncommutative context, left nearly Gorenstein algebras are those where the categories of Gorenstein projective and stable modules coincide (Marczinzik, 2017).

2. Algebraic Families: Constructions and Characterizations

2.1 Families of Commutative Rings via Rees Algebras

The construction $$R(I)_{a,b} = \bigoplus_{n \geq 0} I^n t^n / (t^2 + a t + b)$$, for $a, b \in R$, unifies Nagata's idealization ($a = b = 0$) and amalgamated duplication ($a=0$, $b=-1$). The Gorenstein and almost Gorenstein properties in this family depend only on the data $(R, I)$, not on $a,b$.

• $R(I)_{a,b}$ is Gorenstein if and only if $R$ is Cohen–Macaulay and $I$ is a canonical ideal (Barucci et al., 2015).
• $R(I)_{a,b}$ is almost Gorenstein (in dimension 1) if and only if the data $(R, I)$ satisfy explicit canonical module and reduction conditions.

2.2 Noncommutative Nearly Gorenstein Families

A finite-dimensional algebra $A$ over a field is left nearly Gorenstein if the subcategory of Gorenstein projective modules equals the subcategory of stable modules (those $M$ with $\operatorname{Ext}^i_A(M, A) = 0$ for all $i \geq 1$). For such algebras, the Gorenstein dimension coincides with the Gorenstein projective dimension of the regular bimodule $A$: $\mathrm{Gdim}(A) = \mathrm{Gpd}_{A^e}(A)$ where $A^e = A^{\mathrm{op}} \otimes_K A$ is the enveloping algebra (Marczinzik, 2017). All Gorenstein and all representation-finite algebras are left nearly Gorenstein.

2.3 Stanley–Reisner and Ehrhart Ring Constructions

In combinatorial commutative algebra, nearly Gorensteinness is characterized via the canonical trace for Stanley–Reisner rings $R = K[\Delta]$ and Ehrhart rings $A(P)$.

For Stanley–Reisner rings, $R$ is nearly Gorenstein if and only if $\mathfrak{m} \subseteq \operatorname{tr}(\omega_R)$. In dimension at least $3$, nearly Gorenstein Stanley–Reisner rings are Gorenstein; the non-Gorenstein nearly Gorenstein rings only appear in dimension $0$ (disjoint unions of vertices) or $1$ (paths) (Miyashita et al., 17 Dec 2024).

Ehrhart rings $A(P)$ of lattice polytopes are nearly Gorenstein when the Minkowski sum decomposition

$P = \lfloor a_P P \rfloor + \{P\}$

holds, where $a_P$ is the codegree and $\lfloor a_P P \rfloor$, $\{P\}$ are explicit polytopes determined by the facet data of $P$ (Hall et al., 2023, Miyashita, 8 Jul 2024). This combinatorial criterion connects the nearly Gorenstein property to integer decomposition properties and polytope structure.

3. Arithmetic and Combinatorial Criteria for Nearly Gorenstein Rings

For simplicial affine semigroup rings $K[S]$ with extremal generators $E = \{a_1, \ldots, a_d\}$, the key objects are the Apéry set $$\operatorname{Ap}(S, E) = \{x \in S : x - e \notin S \text{ for } e \in E\}$$ and its maximal elements. The canonical module and its trace have explicit descriptions:

• $\omega_{K[S]} \cong K[w_S]$, $w_S = -\max \operatorname{Ap}(S, E) + S$
• $$\operatorname{tr}(\omega_{K[S]}) = K[w_S + w_S^{-1}]$$

$K[S]$ is nearly Gorenstein if every generator $a_i$ satisfies $a_i \in w_S + w_S^{-1}$, or equivalently, for every $a_i$, there exists $m \in \max \operatorname{Ap}(S, E)$ such that $a_i + m - m_j \in S$ for all $j$ (Jafari et al., 18 Nov 2024). This gives a purely arithmetic criterion in terms of Apéry sets and maximal Apéry elements.

Combinatorially, for Ehrhart rings of polytopes with integer decomposition property (IDP), nearly Gorensteinness is characterized by the Minkowski sum formula involving the codegree and certain face polytopes (Hall et al., 2023).

4. Infinite Families and Classification Theorems

Nearly Gorenstein property admits rich infinite families in various settings:

• Cyclic Quotient Singularities: For $R^G = k[[x_1,\ldots,x_d]]^G$ with $G$ cyclic and small, nearly Gorensteinness is characterized by a combinatorial congruence condition on the group weights. All 2-dimensional cyclic quotient singularities are nearly Gorenstein; in higher dimension, explicit weight patterns (e.g., all-ones, almost all-ones) give infinite families (Caminata et al., 2020).
• Veronese Subrings and Dilates: If a standard graded ring $R$ (resp. a polytope $P$) satisfies an appropriate “linear” trace condition, then for all large $k$, the Veronese subring $R^{(k)}$ (resp. the Ehrhart ring of $kP$) is nearly Gorenstein. In the polytope case, this relies on the decomposition $P = [P] + \{P\}$ (Miyashita, 8 Jul 2024).
• Hibi Rings and Polytopes: Nearly Gorensteinness for Hibi rings corresponds to the poset being a disjoint union of pure connected components whose ranks differ by at most one; for $(0,1)$-polytopes, a product decomposition criterion applies with codegree gaps $\leq 1$ (Hall et al., 2023).
• Stable-Set Polytopes of h-Perfect Graphs: The Ehrhart ring is nearly Gorenstein if and only if each component is Gorenstein and their clique numbers differ by at most one (Miyazaki, 2022).

5. Relations with Gorenstein, Pseudo-Gorenstein, and Almost Gorenstein Properties

Nearly Gorensteinness interacts with several generalizations of Gorensteinness:

• Every Gorenstein ring is nearly Gorenstein, but the converse fails unless structural assumptions are imposed (e.g., for Stanley–Reisner rings in dimension $\ge 3$ or for pseudo-Gorenstein nearly Gorenstein domains under mild grade constraints, nearly Gorensteinness implies Gorensteinness (Miyashita et al., 17 Dec 2024, Miyashita, 3 Feb 2025)).
• Level and pseudo-Gorenstein properties, together with nearly Gorensteinness, typically force actual Gorensteinness in dimension at least 2 (Miyashita, 3 Feb 2025).
• Almost Gorenstein rings are nearly Gorenstein under additional conditions, with corresponding relationships involving trace ideals, Cohen–Macaulay type, and canonical module structure (Barucci et al., 2015).

The following table summarizes selected implications in the graded and toric settings:

Property Combination	Implies	Paper
Nearly Gorenstein + pseudo-Gorenstein + grade $\geq2$	Gorenstein	(Miyashita, 3 Feb 2025)
Level + pseudo-Gorenstein	Gorenstein	(Miyashita, 3 Feb 2025)
Almost Gorenstein, $s(R)\ge 2$	pseudo-Gorenstein	(Miyashita, 3 Feb 2025)
Nearly Gorenstein Stanley–Reisner, $\dim \ge 3$	Gorenstein	(Miyashita et al., 17 Dec 2024)

6. Cohen–Macaulay Type and Structural Bounds

The Cohen–Macaulay type of nearly Gorenstein rings is sharply bounded in low codimension:

• For nearly Gorenstein simplicial affine semigroup rings $K[S]$ of codimension $\le 3$, the type is at most $3$ (and in fact between $d$ and $3$ where $d$ is the dimension) (Jafari et al., 18 Nov 2024).
• In higher codimension, such bounds fail without further assumptions.

The type calculation relies explicitly on the structure of the Apéry set and quasi-Frobenius elements, with equality of type and the number of maximal Apéry elements.

7. Applications, Open Questions, and Further Directions

Nearly Gorenstein families have relevance in singularity theory and combinatorial commutative algebra due to their proximity to Gorenstein structures. They regularly arise in:

• The study of boundary cases in the hierarchy: Gorenstein → level/pseudo-Gorenstein/almost Gorenstein → nearly Gorenstein → Cohen–Macaulay.
• Classification of singularities: Cyclic quotient singularities, Stanley–Reisner rings, and Ehrhart rings provide explicit families with controlled canonical trace.
• Ring-theoretic and combinatorial optimization: Graph polytopes, matroid base polytopes, and edge polytopes possess precise nearly Gorenstein criteria.

Prominent open problems include:

• Whether the Gorenstein dimension equals the Gorenstein projective dimension of the regular bimodule for all finite-dimensional algebras (Marczinzik, 2017).
• Explicit classifications of nearly Gorenstein rings in broader classes beyond current affine semigroup, Ehrhart, or Stanley–Reisner frameworks.
• The behavior of the nearly Gorenstein property under Segre and tensor products, and its implications for multi-graded and noncommutative algebras (Miyazaki, 2022).

The concept of nearly Gorenstein families thus provides a robust, unifying language for characterizing rings with tightly controlled departures from Gorensteinness, with classification results guided by canonical trace computations, combinatorial geometry, and module-theoretic splits.