{"id":"b28d7324-133e-4034-a078-5178bad22b72","arxiv_id":"2502.01133","paper_version":6,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper proves that, under mild hypotheses, a pseudo-Gorenstein ring that is nearly Gorenstein must actually be Gorenstein, and uses this to compare several generalizations of the Gorenstein property. It also gives a way to build quasi-Gorenstein rings from Veronese subalgebras.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the central theorem holds up once the two proof slips in Theorem 3.4 are corrected.","rationale":"The reader's conditional verdict is reasonable: the proof as printed is not fully rigorous, but the defects are localized and repairable. The regular-sequence assumption identified by the reader is the real constraint and is automatic in the main domain application. I see no counterexample or circular step. The abstract's phrase 'mild assumptions' must be read with Theorem 1.2's hypotheses; Remark 4.5 shows they cannot simply be dropped. Thus the verdict should remain conditional/unchanged rather than being strengthened or reversed.","tokens_in":13408,"tokens_out":39853,"duration_ms":474492,"concrete_test":"Re-derive the final paragraph of Theorem 3.4 from Equation (1) without normalizing indeg_R(I)=1: set u=fg_i0, define ψ(1_R)=u^{-1}f, and verify both ψφ=id_I and the degree conclusion fg_i0∈R_0. If these checks pass, Theorem 3.4 and hence Corollary 3.9 stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing objection to the central claim. The theorem's weight rests on Theorem 3.4, where the written proof has two repairable slips: it asserts indeg_R(I)=1 although only indeg_R(M)<∞ is assumed (the argument works with the actual indegrees after taking degree-indeg_R(m_R) components), and it defines the splitting map ψ by ψ(1_R)=fg_i0, which does not satisfy ψφ=id_I; the correct definition is ψ(1_R)=(fg_i0)^{-1}f. With that correction, φ(f)=fg_i0 is a unit, the degree bound deg(fg_i0)<indeg_R(m_R) still forces fg_i0∈R_0, and I indeed has an R(-indeg_R(I))-free summand. The genuinely load-bearing hypothesis is the existence of the length-two R-regular sequence θ1,θ2 in R_{indeg_R(m_R)} inside tr_R(ω_R); this is exactly what the main advertised application (standard graded nearly Gorenstein domain, dim≥2, infinite residue field) supplies, so the central claim does not appear threatened.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies when pseudo-Gorenstein graded rings are Gorenstein. The main result, Theorem 3.7 and Corollary 3.9, says that under a trace-ideal hypothesis—an R-regular sequence of two elements of degree indeg_R(m_R) inside tr_R(omega_R)—a generically Gorenstein ring that is pseudo-Gorenstein is quasi-Gorenstein, and if Cohen–Macaulay it is Gorenstein. The proof proceeds by showing that the canonical module, viewed as a graded ideal, has a free summand (Theorem 3.4). The paper then applies this to compare nearly Gorenstein, almost Gorenstein, and level rings for standard/semi-standard graded domains, and gives a Veronese-subalgebra criterion for quasi-Gorensteinness (Theorem 4.8, Corollary 4.9). Remark 3.10 explicitly notes that the two-element regular sequence condition cannot be weakened to one non-zero divisor, as one-dimensional pseudo-Gorenstein nearly Gorenstein domains show.","tokens_in":13447,"tokens_out":28744,"duration_ms":277526,"significance":"If the main theorem is correct, the paper gives a clean and usable condition under which pseudo-Gorenstein implies Gorenstein, a question that has been studied in several classes of graded rings. The application to nearly Gorenstein standard graded domains of dimension at least two is valuable, as it clarifies the hierarchy among generalizations of Gorensteinness in higher dimension. The author also honestly records the sharpness boundary of the hypothesis in Remark 3.10, which is a useful service to the community. The central argument is plausible and the overall strategy—passing through a free summand of the canonical module—is coherent. However, as written, several load-bearing proofs contain concrete errors that must be repaired before the claims can be accepted.","major_comments":[{"comment":"The proof asserts 'indeg_R(I)=1' from the graded isomorphism M ≅ I⊕N and indeg_R(I)<indeg_R(N). This is not a hypothesis: Theorem 3.4 only assumes indeg_R(M)<∞, and indeg_R(I) need not be 1. The subsequent argument uses f as an R0-basis of I_1 and compares degrees against 1, so the proof does not cover the stated generality as written. The theorem appears repairable by setting d=indeg_R(I), using dim_{R0} I_d=1 (which follows from dim_{R0} M_indeg(M)=1 together with the shift), and replacing the occurrences of 1 by d; nevertheless the current text is invalid.","section":"Theorem 3.4, proof (first paragraph)"},{"comment":"The splitting map ψ is misdefined. Setting ψ(1_R)=fg_i0 gives, for u=fg_i0, ψφ(f)=ψ(u)=u·fg_i0=u^2, not f, so ψφ≠id_I in general. The correct definition is ψ(1_R)=f/(fg_i0), which lies in I because fg_i0 is a unit in R0. This is not cosmetic: the existence of the R(−indeg_R(I))-free summand is exactly the conclusion of the theorem.","section":"Theorem 3.4, proof (final paragraph)"},{"comment":"The proof reuses the letter I for the ideal isomorphic to omega_R, after the statement has already used I for the arbitrary ideal generated by a subset of R1. As a result, the concluding inclusion (n_R^{k−1}I)^{(k)} is asserted for the canonical-model ideal, not for the I of the lemma. To prove the lemma, one must name the canonical ideal, say L, and explicitly use the hypothesis tr_R(L)=tr_R(omega_R)⊇I when applying [30, Theorem 6.1]. Since Lemma 2.20 is the key input for Theorem 4.8, this needs repair.","section":"Lemma 2.20, proof"},{"comment":"The proof states that from omega_R(k)=(omega_R)^{(k)} and a_R∈kZ it follows that a_R(k)=a_R. This is false for the Veronese grading [R^{(k)}]_i=R_{ik}; the correct relation is a_R(k)=a_R/k. The dimension equality used later is still correct because [omega_R(k)]_{-a_R(k)} = [omega_R]_{-a_R}, so the argument survives after this correction, but the displayed equality as written is wrong.","section":"Theorem 4.8, proof"}],"minor_comments":[{"comment":"In the parenthetical description of [∗Hom_R(M,R)]_i, the symbol N appears in the set-builder condition and in the codomain of the maps; N is not defined in this context. The codomain should be R.","section":"Definition 3.2"},{"comment":"The reduction 'Without loss of generality, we may assume that R0 is infinite (see Remark 2.18)' is asserted without details. Please indicate why the hypotheses—torsion-freeness of the lowest-degree canonical module element, dimension one of that component, and the radical condition on [tr_R(omega_R)]_{indeg_R(m_R)}—are preserved under the base change R→R⊗_{R0}k.","section":"Theorem 4.2, proof"}],"recommendation":"major_revision","confidential_remarks":"The proof relies essentially on [26, Remark 2.13] and [30, Theorem 6.1], both of which are works by the author or coauthored by the author; [26] is an arXiv preprint that postdates the first version of this manuscript. The editor may wish to have those results checked independently, since the proof of Lemma 2.20 applies [30, Theorem 6.1] in a way that is currently obscured by notation. The central theorem seems credible once the Theorem 3.4 slips are fixed, but the number of local proof errors in the present version is sufficient to require a careful revision rather than acceptance as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem is real and worth knowing. For a generically Gorenstein graded ring, pseudo-Gorenstein plus nearly Gorenstein forces quasi-Gorenstein, provided tr_R(ω_R) contains two R-regular elements in degree indeg_R(m_R). For standard graded domains of dimension at least two over an infinite field the condition is automatic, so you get a clean statement: pseudo-Gorenstein + nearly Gorenstein + standard graded domain, dim ≥ 2, implies Gorenstein. That is a genuinely useful criterion. The hierarchy corollaries are good: under the same hypotheses a type-2 ring is level, and an almost Gorenstein standard graded domain is either Gorenstein or of minimal multiplicity. The non-Cohen–Macaulay version and the Veronese construction of quasi-Gorenstein rings (Theorem 1.4) are new and extend the author's earlier work.\n\nNow the soft spots, in proportion.\n\nThe abstract overstates the result. It claims every positively graded domain that is pseudo-Gorenstein and nearly Gorenstein is Gorenstein. That is false, and the paper contains its own counterexample: Remark 4.5 gives a two-dimensional normal affine semigroup ring (Q[x,xy,x^2y^3,x^3y^5]) that is pseudo-Gorenstein, nearly Gorenstein, and not Gorenstein. Theorem 1.2 in the introduction states the hypotheses correctly; the abstract needs either 'standard graded' or the regular-sequence condition.\n\nTheorem 3.4 has two proof slips, both repairable. The proof asserts indeg_R(I)=1, which is not a hypothesis; the argument works with the actual initial degree. And the splitting map is defined by ψ(1_R)=fg_i0, but then ψφ multiplies by (fg_i0)^2; the correct definition is ψ(1_R)=f/(fg_i0), which lies in I because fg_i0 is a unit. With that, the free-summand conclusion goes through. The later theorems build on 3.4, so the chain is not fully rigorous as written, but the fixes are mechanical.\n\nMinor: in Theorem 4.8 the a-invariant of the k-th Veronese is a_R/k, not a_R. The conclusion is unaffected because the dimension of the relevant graded piece is what matters.\n\nThe dependence on the author's own [26] and [30] is worth a referee's glance, but it is not circular: those are independent results about fiber products and a linear variant of nearly Gorenstein, not restatements of the target theorem. Remark 3.10 honestly records that one non-zero-divisor does not suffice.\n\nWho this is for: commutative algebraists working on Gorenstein-type properties, trace ideals, and h-vectors; combinatorial people get the standard graded domain case as a clean corollary. It deserves a serious referee. The main theorem is new, the argument is recognizable and sound once the two slips are corrected, and the literature handling is honest. Ask for the abstract fix, the ψ correction, and the a-invariant correction; all are small.","headline":"Solid paper with a true main theorem: pseudo-Gorenstein plus nearly Gorenstein forces Gorenstein under a mild regular-sequence condition; the abstract overstates it and Theorem 3.4 has two mechanical proof slips, but the paper deserves refereeing.","tokens_in":14162,"tokens_out":14456,"would_cite":true,"duration_ms":125816,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13H10","13A02","05E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Pseudo-Gorenstein rings become Gorenstein once the trace ideal of the canonical module contains a two-element regular sequence in the smallest positive degree; for standard graded nearly Gorenstein domains of dimension at least two, this…","keywords":["Pseudo-Gorenstein ring","Nearly Gorenstein ring","Almost Gorenstein ring","Level ring","Trace ideal","Canonical module","Quasi-Gorenstein ring","Veronese subalgebra"],"falsifier":"Find a standard graded nearly Gorenstein domain of dimension at least two whose last h-vector entry is 1 (so the ring is pseudo-Gorenstein), in which two degree-one elements of the trace ideal form a regular sequence, but which is not Gorenstein; the theorem says no such ring exists. A concrete place to look is a normal affine semigroup ring such as $R=\\mathbb{Q}[x,xy,x^2y^3,x^3y^5]$ from Remark 4.5, where computing the trace ideal and checking the degree-one regular-sequence condition would test the boundary of the hypothesis.","tokens_in":13013,"feed_emoji":"","tokens_out":12824,"duration_ms":109020,"temperature":0.7,"pith_summary":"This paper investigates when the weak Gorenstein-like condition called pseudo-Gorensteinness actually forces a graded ring to be Gorenstein. Its main theorem says that if the ring is generically Gorenstein, its canonical module contains a torsion-free element of lowest degree, and the trace ideal of the canonical module contains a two-element regular sequence lying in the smallest positive degree of the ring, then pseudo-Gorensteinness implies Gorensteinness. For standard graded nearly Gorenstein domains of dimension at least two these hypotheses are automatically satisfied, so in that class every pseudo-Gorenstein ring is Gorenstein. The paper applies this to sort out how nearly Gorenstein, almost Gorenstein, and level rings relate in higher-dimensional graded domains, and it constructs quasi-Gorenstein rings as Veronese subalgebras of certain non-Cohen-Macaulay rings.","feed_headline":"A trace-ideal condition turns pseudo-Gorenstein rings Gorenstein","feed_subtitle":"Two regular-sequence elements in the trace ideal's smallest degree are enough; without them, counterexamples appear.","key_machinery":"The machinery is the trace ideal $\\operatorname{tr}_R(\\omega_R)$, the ideal built from all graded module maps $\\omega_R\\to R$, together with a two-element regular sequence — a pair of elements each of which is a non-zero-divisor modulo the previous one — living in the smallest positive degree of $R$. The key step is Lemma 3.1: with two homogeneous non-zero-divisors $f_1,f_2$ in an ideal $I$ with $f_2\\notin Rf_1$, a regular sequence $\\theta_1,\\theta_2$ in $R_{\\operatorname{indeg}_R(\\mathfrak m_R)}$ forces one of the fractions $\\theta_i f_2/f_1$ to lie outside $R$, meaning the trace ideal contains an element that cannot come from inside $R$. Feeding this into the identity $\\operatorname{tr}_R(I)=I\\cdot I^{-1}$ for ideals containing a non-zero divisor, Theorem 3.4 shows that the canonical module's ideal summand has an $R$-free summand; then the trace ideal is the whole ring, so the ring is quasi-Gorenstein. In the Cohen-Macaulay setting, pseudo-Gorensteinness is read off the h-vector as $h_{s(R)}=1$, which is what connects the trace computation to the Gorenstein conclusion.","core_discovery":"The central discovery is a sufficient condition for pseudo-Gorenstein rings to be Gorenstein, stated as Theorem 1.2. Let $R$ be a Cohen-Macaulay generically Gorenstein graded ring whose canonical module $\\omega_R$ contains a torsion-free homogeneous element of degree $-a_R$, and suppose $\\operatorname{tr}_R(\\omega_R)$ contains an $R$-regular sequence $\\theta_1,\\theta_2$ in $R_{\\operatorname{indeg}_R(\\mathfrak m_R)}$. If $R$ is pseudo-Gorenstein, meaning $\\dim_{R_0}([\\omega_R]_{-a_R})=1$, then $R$ is Gorenstein. The proof actually establishes a stronger non-Cohen-Macaulay statement, Theorem 3.7: under the same hypotheses, $\\dim_{R_0}([\\omega_R]_{-a_R})=1$ forces $R$ to be quasi-Gorenstein, i.e. the canonical module is a shifted copy of $R$. In the Cohen-Macaulay case quasi-Gorenstein is Gorenstein. The paper's applications show that in standard graded nearly Gorenstein domains of dimension at least two, the hypotheses are automatic, and consequently pseudo-Gorensteinness alone becomes a Gorenstein certificate.","pith_inferences":["An implicit combinatorial consequence of Corollary 4.7 is that for Ehrhart rings of lattice polytopes, which are standard graded domains, a nearly Gorenstein Ehrhart ring whose h-vector ends in 1 must be Gorenstein; this can be tested on families of polytopes with known non-symmetric h-vectors.","The proof suggests that the real engine is low-degree abundance of the trace ideal: two regular-sequence elements in the minimal degree already make the trace ideal the whole ring. A natural test is whether the two-element regular sequence can be relaxed to a weaker condition in dimension at least two, since Remark 3.10 shows dimension one is genuinely exceptional.","The Veronese construction offers a recipe for producing quasi-Gorenstein rings: any standard graded domain whose canonical module has a one-dimensional bottom piece and whose trace ideal contains the maximal ideal yields quasi-Gorenstein Veronese subalgebras without requiring the original ring to be Cohen-Macaulay; applying this to concrete semigroup rings could produce new quasi-Gorenstein but no"],"forward_implications":["In any standard graded nearly Gorenstein domain of dimension at least two, pseudo-Gorensteinness is equivalent to Gorensteinness; in particular such a domain with last h-vector entry 1 must be Gorenstein.","In a nearly Gorenstein graded domain of dimension at least two whose Cohen-Macaulay type is 2, the canonical module is generated in a single degree, so the ring is level (Corollary 4.6(2)).","An almost Gorenstein standard graded domain over an algebraically closed field of characteristic zero is either Gorenstein or has minimal multiplicity, and in either case it is level (Corollary 4.7(3)).","For a Noetherian standard graded domain whose trace ideal contains the maximal ideal and whose lowest nonzero canonical-module piece is one-dimensional, a Veronese subalgebra of index $k$ with depth at least two is quasi-Gorenstein; if that subalgebra is Cohen-Macaulay, it is Gorenstein (Corollary 4.9).","Under the weaker radical condition $\\sqrt{[\\operatorname{tr}_R(\\omega_R)]_{\\operatorname{indeg}_R(\\mathfrak m_R)}R}\\supseteq\\mathfrak m_R$, a Cohen-Macaulay ring of dimension at least two that is pseudo-Gorenstein is Gorenstein, and type 2 forces levelness (Corollary 4.4)."],"supporting_citations":[{"why":"supplies the definition of pseudo-Gorenstein rings as those whose canonical module has a one-dimensional lowest graded piece.","marker":"[10]"},{"why":"introduces nearly Gorenstein rings through the trace ideal and provides the identity $\\operatorname{tr}_R(I)=I\\cdot I^{-1}$ used throughout the proof.","marker":"[20]"},{"why":"supplies the canonical-trace facts that a ring is quasi-Gorenstein exactly when the trace of the canonical module is the whole ring and that generic Gorensteinness embeds the canonical module as an ideal.","marker":"[26]"},{"why":"defines canonical modules and records the Veronese submodule relation $\\omega_{R^{(k)}}\\cong(\\omega_R)^{(k)}$ used in the Veronese applications.","marker":"[15]"},{"why":"provides the trace containment for Veronese subalgebras and the existence of a non-zero divisor in degree one for semi-standard graded rings.","marker":"[30]"},{"why":"contributes the theorem that almost Gorenstein standard graded domains are pseudo-Gorenstein or have minimal multiplicity, used in Corollary 4.7(3).","marker":"[21]"},{"why":"is the earlier comparison result that Corollary 3.5 generalizes, showing the new theorem's scope.","marker":"[29]"},{"why":"defines level rings and supplies the h-vector facts that identify pseudo-Gorensteinness with $h_{s(R)}=1$.","marker":"[38]"}],"fun_headline_variants":["Two trace-ideal elements turn pseudo-Gorenstein into Gorenstein","Pseudo-Gorenstein domains become Gorenstein via trace-ideal regular sequence","Trace ideal's two elements suffice for pseudo-Gorenstein to be Gorenstein","Pseudo-Gorenstein rings trace-ideal condition yields Gorenstein","A regular sequence in trace ideal makes pseudo-Gorenstein Gorenstein"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on finding two elements inside the trace ideal that both lie in the smallest positive degree of the ring and that remain non-zero-divisors after dividing out by one another; if only a single such element is available, the conclusion can fail in dimension one.","fun_headline_variants_meta":{"raw":{"variants":["Two trace-ideal elements turn pseudo-Gorenstein into Gorenstein","Pseudo-Gorenstein domains become Gorenstein via trace-ideal regular sequence","Trace ideal's two elements suffice for pseudo-Gorenstein to be Gorenstein","Pseudo-Gorenstein rings trace-ideal condition yields Gorenstein","A regular sequence in trace ideal makes pseudo-Gorenstein Gorenstein"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000911,"raw_usage":{"total_tokens":3888,"prompt_tokens":895,"completion_tokens":2993,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":511,"completion_tokens_details":{"reasoning_tokens":2893}},"tokens_in":511,"tokens_out":2993,"duration_ms":20776,"temperature":1.0,"reasoning_tokens":2893,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T16:29:57.979155+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a standard graded nearly Gorenstein domain of dimension at least two whose last h-vector entry is 1 (so the ring is pseudo-Gorenstein), in which two degree-one elements of the trace ideal form a regular sequence, but which is not Gorenstein; the theorem says no such ring exists. A concrete place to look is a normal affine semigroup ring such as $R=\\mathbb{Q}[x,xy,x^2y^3,x^3y^5]$ from Remark 4.5, where computing the trace ideal and checking the degree-one regular-sequence condition would test the boundary of the hypothesis.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the definition of pseudo-Gorenstein rings as those whose canonical module has a one-dimensional lowest graded piece."},{"cited_title":"Herzog, T","cited_arxiv_id":null,"evidence_quote":"introduces nearly Gorenstein rings through the trace ideal and provides the identity $\\operatorname{tr}_R(I)=I\\cdot I^{-1}$ used throughout the proof."},{"cited_title":"Goto and K","cited_arxiv_id":null,"evidence_quote":"defines canonical modules and records the Veronese submodule relation $\\omega_{R^{(k)}}\\cong(\\omega_R)^{(k)}$ used in the Veronese applications."},{"cited_title":"Higashitani,Almost Gorenstein homogeneous rings and theirh-vectors, Journal of Algebra456(2016), 190–206.↑2, 4","cited_arxiv_id":null,"evidence_quote":"contributes the theorem that almost Gorenstein standard graded domains are pseudo-Gorenstein or have minimal multiplicity, used in Corollary 4.7(3)."},{"cited_title":"Miyashita,Comparing generalized Gorenstein properties in semi-standard graded rings, Journal of Algebra647(2024), 823–843.↑2, 7, 10","cited_arxiv_id":null,"evidence_quote":"is the earlier comparison result that Corollary 3.5 generalizes, showing the new theorem's scope."},{"cited_title":"41, Springer Science & Business Media, 2007.↑1, 3, 4, 10, 11 (S","cited_arxiv_id":null,"evidence_quote":"defines level rings and supplies the h-vector facts that identify pseudo-Gorensteinness with $h_{s(R)}=1$."}],"review_version":1}