{"id":"a2388ddf-0aa7-4fc1-88a8-ecfe190be784","arxiv_id":"2411.12081","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Nearly Gorenstein affine semigroup rings of codimension three have Cohen-Macaulay type at most three (and at least the dimension when not Gorenstein), with both bounds sharp.","lead":"This paper proves that any nearly Gorenstein affine semigroup ring whose generators exceed the ambient dimension by exactly three has Cohen-Macaulay type no larger than three, unless it is Gorenstein. It also gives a concrete description of the canonical module and its trace ideal in terms of Apéry sets, which is what makes the bound possible.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Lemma 3.8(4), essential for Theorem 3.9, contains a corrupted comparison and an underived equation; until that case analysis is completed, type≤3 is not fully established.","rationale":"The reader's weakest_assumption isolates the dependence on [15, Proposition 3.3] that identifies type(S) with |max_{≼_S} Ap(S,E)|; that is a legitimate external dependency, though nothing in the paper suggests it fails. The reader's rationale also flags Lemma 3.8(4) as containing a corrupted symbol and an unjustified equation, and that is exactly the concern I regard as most load-bearing: it is an internal gap in the proof of the main theorem. The theorem may well be true—the examples in Section 3 are consistent, and the counting strategy is plausible—but as written the case analysis in Lemma 3.8(4) does not close. I therefore agree with the CONDITIONAL verdict: the paper should be accepted only after the passage is clarified and preferably supported by a computational or machine-checkable verification. My recommendation is UNCHANGED because the reader already reached CONDITIONAL and my concern does not move the verdict; it reinforces the same condition.","tokens_in":23840,"tokens_out":7545,"duration_ms":71472,"concrete_test":"Provide a complete, line-by-line proof of Lemma 3.8(4) with the corrupted comparison symbol replaced and the missing equation (λ_j + 1 + μ_j)a_{d+j} = l_i a_{d+i} + l_k a_{d+k} + a_2 derived from (3.15)–(3.17) using Lemma 2.11(3). In parallel, independently test the statement for small cases: use Normaliz/Macaulay2 to enumerate simplicial affine semigroups with d = 2, 3, r = 3 and generator coordinates bounded by, say, 10, filter for Cohen–Macaulay and nearly Gorenstein, compute the type from a minimal free resolution, and verify both that type ≤ 3 and that the structural claims of Lemma 3.8(4) hold in every enumerated instance.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central bound type(S) ≤ 3 in Theorem 3.9 depends on the exclusion of configurations in Lemma 3.8(4). In the proof of that lemma, the string '/notprecedesoreqlS' replaces a comparison symbol in two sentences used to conclude that c is not ≤_S certain maximal elements; without knowing whether the intended symbol is 'not ≼_S' or something else, those inferences cannot be checked. More seriously, the final paragraph claims 'By Lemma 2.11(3), we have (λ_j + 1 + μ_j)a_{d+j} = l_i a_{d+i} + l_k a_{d+k} + a_2' but the quantities l_i, l_k are not defined in the text and the equation is not derived from the preceding identities (3.15)–(3.17). This equation is then used to contradict the slope ordering (3.14), so the exclusion of M_{i,k} and M_{j,k} in the case where both M^2_i and M^2_j are nonempty is not actually proved. Earlier in the same lemma, the inequality 'μ ≤ g_{i,k} − 1' also appears without derivation; it is needed to obtain equation (3.13). These gaps are internal to the proof, not disagreements with prior consensus: if Lemma 3.8(4) cannot be repaired, the case analysis in Theorem 3.9 has a hole and the claimed sharp bound is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":24040,"tokens_out":3855,"duration_ms":35125,"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":[{"comment":"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.","section":"§3, Lemma 3.8(4), proof of exclusion of M_{j,k}"},{"comment":"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.","section":"§3, Lemma 3.8(4), final paragraph"},{"comment":"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.","section":"§3, Lemma 3.8(4), around (3.13)"}],"minor_comments":[{"comment":"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":"Throughout"},{"comment":"The reference to [15, Definition 3.1] contains the typo 'Defenition'; please correct.","section":"Section 1, citation [15]"},{"comment":"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.","section":"Theorem 3.9, Case 1"},{"comment":"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.","section":"Lemma 3.8(4) statement"},{"comment":"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.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of math.AC and the main theorem, if established, is a good contribution. However, the proof of Lemma 3.8(4) has three interconnected gaps: a corrupted comparison symbol, an undefined equation with l_i and l_k, and an unjustified inequality. These are not cosmetic; they block the exclusion of cases that Theorem 3.9 relies on. The issues appear repairable within the manuscript's framework, so I recommend major revision rather than rejection. The authors should be asked to rewrite the proof of Lemma 3.8(4) completely, preferably with the corrupted symbol restored and the missing derivations supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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\n\n(λ_j + 1 + μ_j)a_{d+j} = l_i a_{d+i} + l_k a_{d+k} + a_2\n\nwith 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.\n\nOn 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.\n\nWho'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.\n\nMy 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.","headline":"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.","tokens_in":24684,"tokens_out":3081,"would_cite":true,"duration_ms":28879,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20M25","05E40","13H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["nearly Gorenstein","affine semigroup","Cohen-Macaulay type","trace ideal","canonical module","Apéry set","simplicial affine semigroup","quasi-Frobenius element"],"falsifier":"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.","tokens_in":23562,"feed_emoji":"🧮","tokens_out":11170,"duration_ms":95812,"temperature":0.7,"pith_summary":"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.","feed_headline":"In codimension 3, nearly Gorenstein rings have type at most 3","feed_subtitle":"A combinatorial Apéry-set count pins the Cohen-Macaulay type between the dimension and 3, with all values attained.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the identification of the Cohen-Macaulay type with the number of maximal Apéry elements, which is the bridge between the combinatorial count and the ring invariant.","marker":"[15]"},{"why":"Provides the description of the canonical module of an affine semigroup ring that the paper uses to prove $K[\\omega_S]$ is canonical.","marker":"[7]"},{"why":"Introduces the trace of the canonical module and the nearly Gorenstein property, and gives the criterion that the trace contains the maximal ideal.","marker":"[11]"},{"why":"Establishes the numerical semigroup case ($d=1$) where nearly Gorenstein rings of embedding dimension 4 have type at most 3, serving as base case and source of sharpness examples.","marker":"[18]"},{"why":"Gives the Cohen-Macaulay criterion for simplicial affine semigroups that is used throughout to translate ring properties into Apéry-set conditions.","marker":"[19]"}],"fun_headline_variants":["Codim-3 nearly Gorenstein semigroups have type at most 3","Type bounded by 3 near Gorenstein in codim 3","Sharp type bound: 3 for codim-3 near Gorenstein","Counting Apéry elements caps type at 3 in codim 3","Nearly Gorenstein: CM type ≤3 when codim ≤3"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Codim-3 nearly Gorenstein semigroups have type at most 3","Type bounded by 3 near Gorenstein in codim 3","Sharp type bound: 3 for codim-3 near Gorenstein","Counting Apéry elements caps type at 3 in codim 3","Nearly Gorenstein: CM type ≤3 when codim ≤3"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000251,"raw_usage":{"total_tokens":1538,"prompt_tokens":904,"completion_tokens":634,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":520,"completion_tokens_details":{"reasoning_tokens":533}},"tokens_in":520,"tokens_out":634,"duration_ms":6630,"temperature":1.0,"reasoning_tokens":533,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:56:27.647768+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Jafari and M","cited_arxiv_id":null,"evidence_quote":"Supplies the identification of the Cohen-Macaulay type with the number of maximal Apéry elements, which is the bridge between the combinatorial count and the ring invariant."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the description of the canonical module of an affine semigroup ring that the paper uses to prove $K[\\omega_S]$ is canonical."},{"cited_title":"Herzog, T","cited_arxiv_id":null,"evidence_quote":"Introduces the trace of the canonical module and the nearly Gorenstein property, and gives the criterion that the trace contains the maximal ideal."},{"cited_title":"Moscariello and F","cited_arxiv_id":null,"evidence_quote":"Establishes the numerical semigroup case ($d=1$) where nearly Gorenstein rings of embedding dimension 4 have type at most 3, serving as base case and source of sharpness examples."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Cohen-Macaulay criterion for simplicial affine semigroups that is used throughout to translate ring properties into Apéry-set conditions."}],"review_version":1}