On nearly Gorenstein affine semigroups

Francesco Strazzanti; Raheleh Jafari; Santiago Zarzuela Armengou

REVIEW 3 major objections 5 minor 3 cited by

On nearly Gorenstein affine semigroups

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that a nearly Gorenstein affine semigroup ring of embedding dimension d+3 that is not Gorenstein has Cohen-Macaulay type t with d ≤ t ≤ 3, and that both bounds are sharp.

desk verdict The codimension-three type bound is a genuine advance and the trace description is useful, but the proof of Lemma 3.8(4) has a gap that currently blocks Theorem 3.9. read the letter →

arxiv 2411.12081 v1 pith:TM7ERSPO submitted 2024-11-18 math.AC

Raheleh Jafari , Francesco Strazzanti , Santiago Zarzuela Armengou This is my paper · ORCID

classification math.AC MSC 20M2505E4013H10

keywords nearlyGorensteinaffinesemigroupCohen-MacaulaytypetraceidealcanonicalmoduleApérysetsimplicialquasi-Frobeniuselement

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

An affine semigroup is a finitely generated set of lattice points closed under addition, and its semigroup ring is the algebra of monomials with exponents in that set. This paper asks how large the Cohen-Macaulay type of such a ring can be when the ring is nearly Gorenstein, a property that relaxes the Gorenstein condition by allowing the trace of the canonical module to be the whole ring or just its maximal ideal. Working with simplicial semigroups, the authors describe the canonical module and its trace entirely in terms of the maximal elements of the Apéry set, and they characterize the nearly Gorenstein condition by an arithmetic condition on the semigroup generators. The main theorem gives that a nearly Gorenstein, non-Gorenstein semigroup ring of embedding dimension d+3 has type between d and 3, with both bounds sharp and every intermediate value attained. This matters because it settles a natural question about the size of the type in codimension three and extends results previously known only for numerical semigroups.

What carries the argument

The load-bearing object is the Apéry set $\operatorname{Ap}(S,E)$ of the semigroup with respect to its $d$ extremal generators, ordered by the relation $b \preceq_S a$ when $a-b$ lies in $S$. Its maximal elements correspond one-to-one with quasi-Frobenius elements, and their number is the Cohen-Macaulay type. The trace ideal $\operatorname{tr}(S)$ is described as the set of $b \in S$ for which, for some maximal element $m_i$, the shifts $b+m_i-m_j$ all lie in $S$; requiring every generator of $S$ to lie in this set is the arithmetic form of near-Gorensteinness. With exactly three non-extremal generators, the equality $T(a_1)+a_1 = m + \lambda_1 a_{d+1} + \lambda_2 a_{d+2} + \lambda_3 a_{d+3}$ forces $m$ to fall into one of the classes $M_i^1$, $M_i^2$, $M_{i,j}$, and Lemma 3.8 constrains how many classes can be inhabited, yielding the type bound.

What would settle it

A single explicit example of a simplicial affine semigroup with the paper's standing hypotheses whose ring is nearly Gorenstein and not Gorenstein, has embedding dimension d+3, and has four maximal Apéry elements would invalidate Theorem 3.9, since the type would be four.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 3.9: if $K[S]$ is nearly Gorenstein and not Gorenstein, and $S$ is minimally generated by $d+3$ vectors (so the embedding dimension is $d+3$), then the Cohen-Macaulay type satisfies $d \leq \operatorname{type}(S) \leq 3$. The proof shows that the trace ideal $\operatorname{tr}(S)$ consists of elements $b$ for which $b$ plus one fixed maximal Apéry element stays in $S$ after subtracting any other maximal element, and that near-Gorensteinness is equivalent to each semigroup generator having this property. The type of $K[S]$ equals the number of maximal elements of $\operatorname{Ap}(S,E)$ with respect to the natural order, so the authors count these maximal elements by analyzing, for each choice of the semigroup generators, the ways that $T(a_1)+a_1$ can be written as $m$ plus a combination of the three non-extremal generators. A structured case analysis (Lemmas 3.1–3.8) shows that at most three maximal elements can exist, and the paper's examples exhibit type 2 and type 3 cases, including a dimension-three example with type 3.

Load-bearing premise

The entire counting argument depends on identifying the Cohen-Macaulay type of $K[S]$ with the number of maximal Apéry-set elements, a fact taken from the literature; if that identification fails for simplicial affine semigroups with the stated normalization, the bounds would not be about the actual type.

Editorial extensions

If this is right

Since $d \leq \operatorname{type}(S) \leq 3$, a nearly Gorenstein non-Gorenstein semigroup ring of embedding dimension $d+3$ can only exist in dimension $d \leq 3$; in higher dimensions such a ring must be Gorenstein.
The sharpness examples mean that, for each $d \leq 3$, every integer $t$ with $d \leq t \leq 3$ occurs as the type of some nearly Gorenstein non-Gorenstein semigroup ring of embedding dimension $d+3$.
Near-Gorensteinness becomes a finite combinatorial check: for each generator, it suffices to verify the finitely many containment conditions $b+m_i-m_j \in S$ over the finite set of maximal Apéry elements.
The theorem recovers, for $d=1$, the known fact that nearly Gorenstein numerical semigroup rings of embedding dimension 4 have type at most 3, and supplies the first extension to arbitrary dimension with the same codimension.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same decomposition machinery could in principle be run for more than three extra generators, yielding a bound on the type that depends only on the number of non-extremal generators; the paper does not claim such a result.
One could scan small simplicial affine semigroups using the finite trace-ideal criterion to see whether type 3 cases are rare and whether the paper's examples are structurally typical.
The trace-ideal description may transfer to other monomial algebras with a discrete Apéry-like set, such as Ehrhart rings of lattice polytopes, where near-Gorensteinness has been studied; this is an inference beyond the paper.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper proves the right theorem and gives a usable new tool, but one load-bearing lemma is not fully proved in the text I have.

The genuinely new result is Theorem 3.9: a nearly Gorenstein simplicial affine semigroup of embedding dimension d+3 has type at most 3 (and at least d when not Gorenstein). The d=1 case was known by Moscariello–Strazzanti; the generalization to all dimensions and the sharpness examples are new. The lower bound type≥d (Cor 2.4) is also new and has a clean proof. The trace ideal characterization in Proposition 1.5, expressing tr(S) via maximal Apéry elements, is a useful combinatorial tool and I expect it to be reused.

The main body of Section 3 is a detailed case analysis. I checked the early lemmas (3.1–3.7) and they are fine; the counting argument in Theorem 3.9 is clever. The problem is Lemma 3.8(4). The text I have contains a corrupted comparison symbol where the argument concludes c is not ≤_S certain maximal elements and then c ≤_S T(a2). More seriously, the final paragraph uses an equation

(λ_j + 1 + μ_j)a_{d+j} = l_i a_{d+i} + l_k a_{d+k} + a_2

with l_i, l_k never defined and no derivation from (3.15)–(3.17). That identity is then used to contradict the slope ordering (3.14). Without it, the exclusion of the mixed cases in Lemma 3.8(4) is not proved. There's also a "μ ≤ g_{i,k} − 1" earlier that appears from nowhere to get (3.13). These are internal holes, not disagreements with prior results. They might be fixable with more care, and the surrounding structure suggests the theorem is true, but as it stands the proof is incomplete.

On the broader picture: no sign of circularity, no fitted parameters. The examples check out by hand, and the authors say they used Macaulay2/GAP, though they don't ship scripts.

Who's this for? People working on nearly Gorenstein rings, affine semigroup rings, and type bounds. It's a specialist result, not a breakthrough for a general audience. But it settles an open case.

My recommendation: send to peer review, but the referee must ask for a repaired Lemma 3.8(4) with all symbols defined. If the authors can fill the gap, the paper is solid. If not, the main theorem is only a conjecture.

Referee Report

3 major / 5 minor

Summary. The paper studies simplicial affine semigroup rings K[S] and their canonical module and trace ideal. The main structural result is Proposition 1.5, which describes the trace ideal tr(S) in terms of the maximal elements of the Apéry set Ap(S,E) with respect to the partial order ≼_S. From this, the authors derive a characterization of near-Gorensteinness (Corollary 1.6), a lower bound type(S) ≥ d for non-Gorenstein nearly Gorenstein semigroups (Corollary 2.4), and, for embedding dimension d+3, an upper bound type(S) ≤ 3 (Theorem 3.9). The paper also gives examples showing that the bounds are sharp and that every integer between them is attained when d is small.

Significance. If the main theorem is correct, it is a meaningful contribution to the study of nearly Gorenstein affine semigroup rings: it generalizes known type bounds for numerical semigroup rings to simplicial affine semigroups of codimension three, and it provides a combinatorial description of the trace ideal that can be used in further work. The paper is careful in stating its hypotheses (simplicial, fully embedded, smallest generator on each ray) and uses standard tools such as the Goto–Suzuki–Watanabe canonical module, Apéry sets, and trace ideal criteria. The sharpness examples are computable and the proofs are detailed. The main reservation is that the proof of Lemma 3.8(4), which is essential for Theorem 3.9, contains several unexplained or corrupted steps; until those are repaired, the claimed bound is not fully established.

major comments (3)

[§3, Lemma 3.8(4), proof of exclusion of M_{j,k}] In the paragraph beginning 'If M_{j,k} is not empty', the proof uses the corrupted symbol '/notprecedesoreqlS' in the assertions 'a1+a2 = c+ad+i /notprecedesoreqlS T(a1)+a1' and 'c /notprecedesoreqlS m for m ∈ {n_i, m_{i,k}, m_{j,k}}'. These comparisons are essential for concluding that c is only ≼_S to T(a2)=n_j, and hence for deriving the contradiction with (3.17). The intended symbol is presumably 'not ≼_S', but as printed the inference cannot be checked. The proof of Lemma 3.8(4) is incomplete unless these comparisons are stated with correct notation and justified from the definitions.
[§3, Lemma 3.8(4), final paragraph] The equation '(λ_j + 1 + μ_j)a_{d+j} = l_i a_{d+i} + l_k a_{d+k} + a_2' is introduced without defining the integers l_i and l_k, and without deriving it from the preceding identities (3.15)–(3.17). This equation is then used to conclude that in the slope ordering (3.14) one has t = j, and together with (3.17) to reach a contradiction. Since this step is the one that rules out the case where both M_i^2 and M_j^2 are nonempty, the exclusion of M_{i,k} and M_{j,k} in that case is not actually proved. The authors need to supply the missing definition or derivation.
[§3, Lemma 3.8(4), around (3.13)] The inequality 'μ ≤ g_{i,k} − 1' appears without derivation, right before equation (3.13). The proof states 'Thus, μ ≤ g_{i,k} − 1 < g_{i,k} + h_k' after bounding λ_i and λ_j, but the bound on μ does not follow from the displayed equations alone, since μ was introduced earlier as the coefficient of a_{d+k} in the expression T(a_1) = λ_i a_{d+i} + λ_j a_{d+j} + μ a_{d+k}, and the relation between μ and g_{i,k} is not explained. Equation (3.13) is used to deduce d=2 and to set up the slope ordering (3.14), so this gap is load-bearing.

minor comments (5)

[Throughout] The word 'slop' is used instead of 'slope' in several places, e.g., in Lemma 3.5 and Lemma 3.8; this should be corrected for readability.
[Section 1, citation [15]] The reference to [15, Definition 3.1] contains the typo 'Defenition'; please correct.
[Theorem 3.9, Case 1] In the case analysis, the sentence 'If M_j^2 is also empty' appears immediately after assuming that M_i^2 and M_j^2 are nonempty, which is contradictory as written. The intended case distinction should be rephrased, likely referring to M_k^2 or M^1_k.
[Lemma 3.8(4) statement] The inclusion 'M_k^2 ∪ M_k^1 ⊂ M_i^2 ∪ M_j^2' uses a strict subset symbol; if equality is possible, the notation should be clarified.
[Abstract and Introduction] The abstract states a result for 'codimension at most three', while Theorem 3.9 is stated for embedding dimension d+3, i.e., codimension exactly three. This is consistent because the codimension-two case was already known, but the wording could be made uniform.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the type bound is derived by combinatorial case analysis from an independently stated trace description; the cited type-counting bridge is prior work with stated assumptions, not a fitted input.

full rationale

The paper's derivation chain is self-contained rather than circular. Theorem 1.1 proves that K[ω_S] is the canonical module by combining standard external facts ([7, Theorem 3.8], [19, Theorem 1.1]) with a new generation argument; it does not define canonical module in terms of ω_S. Proposition 1.5 then derives the trace ideal formula from this canonical module description and the standard trace formula [11, Lemma 1.1], and Corollary 1.6 is a direct translation, not an assumed equivalence. The type bounds in Section 3 count |max≼_S Ap(S,E)| and use [15, Proposition 3.3] to identify this count with the Cohen-Macaulay type. This citation is from the first author's prior work, but it is a parameter-free theorem whose stated assumptions (Cohen-Macaulayness) do not include the target bound type(S) ≤ 3 or any nearly Gorenstein conclusion; it is therefore independent support and does not make the derivation circular. Theorem 3.9 is an internal combinatorial case analysis of the sets M_i^1, M_i^2, M_{i,j}; no fitted parameter is renamed as a prediction, no equation is shown to reduce to itself, and no uniqueness theorem is imported to declare a choice forced. The sharpness examples are separate computations. The reviewer's noted proof gaps in Lemma 3.8(4)—the corrupted comparison symbol, the undefined l_i,l_k, and the underived equation before (3.17)—are genuine correctness/completeness concerns, but they are not instances of circularity: they do not show that any asserted conclusion is equivalent by construction to its input. Accordingly, the appropriate finding is no significant circularity, score 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard commutative algebra and on the cited results [7], [11], [15], [18], [19]. No free parameters or invented entities are introduced; the M^i_j partition of maximal Apéry elements is internal proof machinery, not a new postulate.

assumptions (7)

domain assumption S is a simplicial affine semigroup, fully embedded in N^d, minimally generated by a1,...,a_{d+r} with a1,...,ad as extremal rays; each a_i is the smallest generator on its ray.
Standing hypothesis stated at the start of Section 1. The minimality of a_i on each ray is needed for Lemma 2.1 (uniqueness of T(a_i)), which underpins Section 3.
domain assumption K[S] is Cohen-Macaulay.
Nearly Gorenstein is only defined for Cohen-Macaulay rings (Definition 1.3). The canonical module, trace ideal, and type are used throughout under this assumption.
domain assumption The canonical module of K[S] is K[C_S] with C_S = -(∩_{i=1}^d C_i), as in Goto-Suzuki-Watanabe [7, Theorem 3.8].
Invoked in the proof of Theorem 1.1 to show K[ω_S] is the canonical module.
domain assumption The number of quasi-Frobenius elements of S equals the Cohen-Macaulay type of K[S] ([15, Proposition 3.3]).
Bridges combinatorial counting of max Apéry elements to the ring invariant type(S). This is the weakest load-bearing assumption, since the main theorem's bound on |max Ap| is only meaningful as a type bound via this bridge.
domain assumption K[S] is Cohen-Macaulay if and only if for all w1,w2 in Ap(S,E), w1-w2 in group(a1,...,ad) implies w1=w2 (and the equivalent formulation in Proposition 1.8(3)), from Rosales-García-Sánchez [19, Theorem 1.5 and Corollary 1.6].
Used in many proofs (e.g., Proposition 2.3, Lemma 2.9, Lemma 3.3) to force equality of Apéry elements from an expression comparison.
domain assumption For a Cohen-Macaulay positively graded K-algebra, a prime p is not in the Gorenstein locus if and only if tr(ω_R) ⊆ p ([11, Lemma 2.1]).
Provides the definition of nearly Gorenstein and is used in Proposition 1.12 to characterize Gorenstein on the punctured spectrum.
domain assumption For numerical semigroup rings of embedding dimension 4, nearly Gorenstein implies type at most 3 ([18, Theorem 2.4]).
Used in Theorem 3.9 to handle the base case d=1.

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Pith. "Pith review of On nearly Gorenstein affine semigroups." pith.science (2026). https://pith.science/paper/TM7ERSPO

@misc{pith2026241112081,
  author       = {Pith},
  title        = {Pith review of: On nearly Gorenstein affine semigroups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TM7ERSPO}},
  note         = {Machine review of arXiv:2411.12081}
}

abstract

We describe the canonical module of a simplicial affine semigroup ring $\mathbb{K}[S]$ and its trace ideal. As a consequence, we characterize when $\mathbb{K}[S]$ is nearly Gorenstein in terms of arithmetic properties of the semigroup $S$. Then, we find some bounds for the Cohen-Macaulay type of $\mathbb{K}[S]$ when it is nearly Gorenstein. In particular, if it has codimension at most three, we prove that the Cohen-Macaulay type is at most three and this bound is sharp.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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Nearly Gorenstein and almost symmetric properties in shifted numerical semigroups
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In shifted numerical semigroups, being nearly Gorenstein or almost symmetric eventually repeats with period r_k, via a corrected pseudo-Frobenius bijection.
When do pseudo-Gorenstein rings become Gorenstein?
math.AC 2025-02 conditional novelty 6.0 of 10

A pseudo-Gorenstein graded ring becomes Gorenstein when the trace ideal of its canonical module contains a length-two regular sequence in the initial degree, with applications to nearly and almost Gorenstein rings.
The canonical trace of Stanley-Reisner rings that are Gorenstein on the punctured spectrum
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Nearly Gorenstein Stanley-Reisner rings of dimension at least three are Gorenstein, and canonical traces of punctured-Gorenstein Stanley-Reisner rings are exactly the ring, the maximal ideal, or its square.

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Works this paper leans on

21 extracted references · 21 canonical work pages · cited by 3 Pith papers

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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