{"id":"7e3c8831-bfaf-4c13-b77e-179fad29ddd2","arxiv_id":"2608.10049","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Virtually Gorenstein Artin algebras are weakly Gorenstein, and several long-standing homological conjectures hold for them.","lead":"This paper proves that, for every virtually Gorenstein Artin algebra, the semi-Gorenstein-projective modules are exactly the Gorenstein projective ones. As a result, several classical homological conjectures hold for this broad class of algebras.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central equality ⊥∞Λ=GP rests on Theorem 3.7(2), whose proof is omitted ('similar'); until that dual Gorenstein-projective-dimension criterion is written out for infinitely generated modules, the main theorem is unverified.","rationale":"The most load-bearing assertion in the paper is exactly the missing proof of Theorem 3.7(2), which is also the reader's weakest assumption. The main theorem, Corollary 1.7, and the claimed resolutions of classical conjectures for this class all pass through this criterion. The text's 'proof of (2) is similar' is an explicit omission, and the surrounding lemmas do not transparently cover the infinitely generated case that the statement demands. I therefore cannot regard the central claim as fully established. However, this is a proof-completeness concern, not a demonstrated counterexample; the proved parts, especially Theorem 2.8 and the detailed argument for Theorem 3.7(1), lend genuine plausibility and are not contradicted by anything in the manuscript. The reader's verdict of CONDITIONAL is appropriate: the paper should be accepted only after the full dual argument, including a non-finitely-generated Gorenstein-projective construction, is supplied. No change to the reader's recommended verdict is needed; the concern reinforces it.","tokens_in":10304,"tokens_out":16718,"duration_ms":166793,"concrete_test":"Write out the n=0 case of Theorem 3.7(2) for an arbitrary M∈Mod-Λ: starting from Ext^i(M,Λ)=0 for all i≥1, construct a complete projective resolution for M, explicitly locating where each projective module and each acyclicity after Hom(-,Λ) comes from, and check whether the construction uses M∈mod-Λ at any point (e.g. via Lemma 3.2 or Prop 2.9). If no such construction can be given without a finite-generation hypothesis, the proof of 3.7(2) is incomplete and the main theorem reduces to the finitely generated statement; equivalently, one should find or rule out an infinitely generated M in ⊥∞Λ\\GP over a virtually Gorenstein algebra.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single load-bearing step is Theorem 3.7(2): it asserts Gpd_Λ(M)=inf{n | Ext^{n+l}_Λ(M,Λ)=0 for all l≥1} for every M∈Mod-Λ. Theorem 3.10 then obtains ⊥∞Λ=GP by taking n=0. The proof in the text is one sentence: 'We only prove (1); the proof of (2) is similar.' This is not a routine symmetry: Theorem 3.6(2), the injective-side analogue, relies on Lemma 3.4, which says D(Λ^op)^⊥∞⊆GI. The projective-side containment needed for 3.7(2), namely ^⊥∞Λ⊆GP for arbitrary (not necessarily finitely generated) modules, is not proved anywhere. Lemma 3.2, the closest supporting input, is explicitly stated for M∈mod-Λ and produces Q,C in add(filt(X)); Proposition 2.9 (⊥X=GP) is also f.g.-module context. Passing to all of Mod-Λ requires an argument that every module with Ext^i(M,Λ)=0 for i≥1 admits a complete projective resolution, e.g. via a GP-precover/embedding and an inductive splice; no such argument is present. If the omitted dual proof needs finiteness (e.g. uses Lemma 3.2 on M or on its syzygies), then 3.7(2) holds only for finitely generated modules and Theorem 3.10's main equality fails in Mod-Λ. Secondary but related: Corollary 3.8 is not a direct consequence of 3.7(2); from the hypotheses one only gets Gpd(M)=0, i.e. M∈GP, and the additional implication Hom(M,Λ)=0 ⇒ M=0 is not supplied.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies virtually Gorenstein Artin algebras and claims that for such algebras every semi-Gorenstein-projective module (i.e., every module M with Ext^i(M,Λ)=0 for all i≥1) is Gorenstein projective. This is stated as Theorem 3.10, from which the paper derives weak Gorensteinness, the Strong Nakayama Conjecture, the Auslander-Gorenstein Conjecture, Tachikawa's First Conjecture, and related statements. The central technical tool is Theorem 3.7, which gives a criterion for projective and Gorenstein-projective dimensions in terms of vanishing of Ext against GP or against Λ, respectively. The paper also contains a characterization of modules of virtually finite injective dimension (Theorem 2.8) and applications to the Gorenstein Symmetry Conjecture.","tokens_in":10666,"tokens_out":9480,"duration_ms":95850,"significance":"If the main result is correct, it resolves a natural case of Ringel and Zhang's Question 1.4 and shows that several classical homological conjectures hold for virtually Gorenstein Artin algebras. The paper also proposes new dimension criteria that could be useful beyond this class. However, the decisive arguments are delegated: Theorem 3.6 is said to follow by adapting a lemma of Beligiannis, and Theorem 3.7(2) is dismissed with 'the proof is similar'. Since the main theorem and its corollaries rest on these unproved assertions, the current manuscript does not yet provide a verifiable proof of its central claim.","major_comments":[{"comment":"The proof of the Gorenstein projective dimension criterion is omitted; the text states only 'We only prove (1); the proof of (2) is similar.' This equality is load-bearing: Theorem 3.10 and Corollary 3.8 use precisely the case n=0 of (2), i.e., the assertion that ⊥∞Λ ⊆ GP for all modules in Mod-Λ. This is not a routine dual of (1): proving it requires constructing a complete projective resolution for an arbitrary (not necessarily finitely generated) module M with Ext^i(M,Λ)=0 for all i≥1, or invoking a previously established theorem of the same strength. The author should supply the full argument, or state and prove the precise known result being adapted, with all finiteness conditions on the modules involved made explicit.","section":"§3, Theorem 3.7(2)"},{"comment":"Both equalities in Theorem 3.6 are delegated: (1) is said to follow by adapting [8, Lemma 5.1], and (2) is dismissed as 'similar'. These criteria for injective and Gorenstein injective dimension are used in the proof of Theorem 3.7(1) (via the duality argument) and in Theorem 3.14. Since the proof of Theorem 3.6 is not written out, the reader cannot check whether the adaptation works for infinitely generated modules, which is essential for the claimed scope of the paper. Please provide the complete proof or a precise statement of the quoted lemma with the necessary modifications.","section":"§3, Theorem 3.6"},{"comment":"The proof that Ext^i(M,Λ)=0 for all i≥0 forces M=0 is not supplied; the text says only that it is 'a direct consequence of Theorem 3.7'. From Theorem 3.7(2) one obtains Gpd(M)=0, i.e., M is Gorenstein projective. The additional step is to show that a Gorenstein projective module with Hom(M,Λ)=0 must be zero. This can be proved by embedding M into a projective module P via a complete resolution and noting that the cokernel is Gorenstein projective, hence Ext^1(coker,Λ)=0, so Hom(P,Λ)→Hom(M,Λ) is surjective; if Hom(M,Λ)=0 then every map P→Λ vanishes on M, and since projective modules are separated by maps to Λ, M=0. This argument (or an equivalent one) should be written out explicitly.","section":"§3, Corollary 3.8"}],"minor_comments":[{"comment":"The notation ⊥∞Λ is used for ⊥∞{Λ}, but the convention at the beginning defines ⊥∞C for a class C; writing ⊥∞{Λ} or defining the singleton case would avoid ambiguity.","section":"§1, Definition 1.2"},{"comment":"There is a duplicated word 'from from' in the sentence 'Hence we see from from the exact sequence (♣) that im(hi)∈ ⊥∞(X⊥).'","section":"§2, proof of Theorem 2.8"},{"comment":"The phrase 'dimension shfting' appears twice and should be 'dimension shifting'. In the same proof, 'Ext ≥i Λ (V,Λ)=0' should presumably be 'Ext^≥i_Λ(V,M)=0', since the vanishing is against M, not against Λ.","section":"§3, proof of Lemma 3.3"},{"comment":"The sentence 'Since the set S finite' should read 'Since S is finite'.","section":"§3, proof of Lemma 3.3"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is interesting, but the paper's key technical results (Theorems 3.6 and 3.7(2)) are asserted without proof. I would advise the editor to require the author to provide complete proofs of these statements before the manuscript can be accepted; otherwise the central claim remains unverified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper claims a clean, significant result: over a virtually Gorenstein Artin algebra, every semi-Gorenstein-projective module is Gorenstein projective, which answers Ringel-Zhang's Question 1.4 for this class and yields SNC, AGC, and GSC as corollaries. The new input is an Ext-criterion for Gorenstein projective dimension, Theorem 3.7(2), and the argument is a reasonable extension of known cotorsion-pair and duality techniques. That part is genuinely new and the consequences are natural, not forced.\n\nThe trouble is exactly where your reader put the finger. Theorem 3.7(2) is the engine, and its proof is one sentence: 'We only prove (1); the proof of (2) is similar.' That is not a routine symmetry. The projective-side containment, that Ext^{i}(M,Λ)=0 for all i≥1 forces M to be Gorenstein projective for arbitrary (not necessarily finitely generated) modules, is the heart of the paper, and no supporting lemma in the text establishes it. Lemma 3.2 is finitely generated only; Lemma 3.4 gets you into GI on the injective side, but the dual step for GP is not written. A referee can probably reconstruct the argument, but that is not the same as having the proof.\n\nTwo other soft spots, in proportion. Corollary 3.8 is not a direct consequence of Theorem 3.7(2): from the hypotheses you get Gpd(M)=0, i.e. M∈GP, and you still need to argue Hom(M,Λ)=0 plus GP implies M=0. That implication is not generally true for arbitrary modules over arbitrary Artin algebras, so it needs a real proof in the virtually Gorenstein setting. Also, Lemma 3.3 has a typo in its proof: the displayed vanishing should be Ext^{≥i}(V,M)=0, not Ext^{≥i}(V,Λ)=0. Minor, but it sits in a key argument.\n\nOn balance, the paper is honest about being a short note, the citations are appropriate, and the strategy is credible. It is not ready for acceptance as written, but it is exactly the kind of paper that should get a serious referee rather than a desk rejection. My recommendation: send it to review with a clear instruction that the authors must supply a complete proof of Theorem 3.7(2) and the missing step in Corollary 3.8.","headline":"The main theorem is probably true and worth publishing, but the paper as submitted rests on an omitted proof of the crucial Ext-criterion, so a referee should demand a full write-up before acceptance.","tokens_in":11249,"tokens_out":14000,"would_cite":false,"duration_ms":140271,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G10","18G25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Over virtually Gorenstein Artin algebras, every semi-Gorenstein-projective module is Gorenstein projective.","keywords":["virtually Gorenstein Artin algebras","weakly Gorenstein Artin algebras","Gorenstein projective modules","semi-Gorenstein-projective modules","Auslander-Gorenstein Conjecture","Strong Nakayama Conjecture","Auslander-Reiten Conjecture","cotorsion theory"],"falsifier":"Construct a virtually Gorenstein Artin algebra $\\Lambda$ and a module $M$ with $\\operatorname{Ext}^i_\\Lambda(M,\\Lambda)=0$ for all $i\\ge1$ such that $M$ is not Gorenstein projective; this directly contradicts $\\perp^\\infty\\Lambda=\\mathcal{GP}$. Equivalently, exhibit a module for which the formula in Theorem 3.7(2) gives a value strictly below the true Gorenstein projective dimension.","tokens_in":10054,"feed_emoji":"","tokens_out":11130,"duration_ms":97397,"temperature":0.7,"pith_summary":"This paper proves that over any virtually Gorenstein Artin algebra, the classes of semi-Gorenstein-projective and Gorenstein projective modules coincide: a module $M$ with $\\operatorname{Ext}^i_\\Lambda(M,\\Lambda)=0$ for all $i\\ge1$ is Gorenstein projective ($\\perp^\\infty\\Lambda=\\mathcal{GP}$). The proof derives this from a new dimension formula, computing the Gorenstein projective dimension of $M$ as $\\inf\\{n\\mid \\operatorname{Ext}^{n+l}_\\Lambda(M,\\Lambda)=0\\text{ for all }l\\ge1\\}$. If correct, the result settles a 2020 open question for this class and makes every virtually Gorenstein Artin algebra weakly Gorenstein. It also yields the Strong Nakayama Conjecture, the Auslander-Gorenstein Conjecture, and Tachikawa's First Conjecture for these algebras, along with the Auslander-Reiten Conjecture for algebras all of whose Gorenstein projective modules are projective.","feed_headline":"Virtually Gorenstein algebras are weakly Gorenstein","feed_subtitle":"It settles a 2020 open question and carries several classical homological conjectures.","key_machinery":"The load-bearing device is the cotorsion theory $(\\perp\\mathcal{GI},\\mathcal{GI})$ -- a pair of module classes orthogonal under $\\operatorname{Ext}^1$ -- which, for virtually Gorenstein algebras, is part of a hereditary cotorsion triple $(\\mathcal{GP},\\perp\\mathcal{GI},\\mathcal{GI})$. The paper builds a set $X$ of kernels and images taken from injective resolutions of the indecomposable injective modules and from the minimal injective resolution of $\\Lambda$, and proves that $\\perp\\mathcal{GI}=\\perp(X^\\perp)$ and that $\\perp X=\\mathcal{GP}$. The dimension formulas of Theorems 3.6 and 3.7 then convert vanishing of Ext against $\\Lambda$ into a bound on Gorenstein projective dimension, and the equality $\\perp^\\infty\\Lambda=\\mathcal{GP}$ follows.","core_discovery":"The central claim is Theorem 3.10: if $\\Lambda$ is a virtually Gorenstein Artin algebra, then $\\perp^\\infty\\Lambda=\\mathcal{GP}$. In words, a right module $M$ over $\\Lambda$ has $\\operatorname{Ext}^i_\\Lambda(M,\\Lambda)=0$ for every $i\\ge1$ if and only if $M$ is Gorenstein projective. Since the opposite algebra of a virtually Gorenstein algebra is again virtually Gorenstein, the statement holds on both sides, so $\\Lambda$ is weakly Gorenstein. The proof passes through a characterization of modules of virtually finite injective dimension and through new formulas (Theorems 3.6 and 3.7) that compute injective, projective, and Gorenstein-projective dimensions by vanishing of Ext against a single test module: the algebra $\\Lambda$ itself or the character module $D(\\Lambda^{\\mathrm{op}})$.","pith_inferences":["The proof identifies Ext-vanishing against the ring itself as the only test needed for Gorenstein projectivity over virtually Gorenstein algebras; one could test whether the same is true with other test modules $C$ in place of $\\Lambda$, which would produce a family of weak Gorenstein conditions relative to $C$.","The paper does not consider whether the equality $\\perp^\\infty\\Lambda=\\mathcal{GP}$ is preserved under common algebra constructions such as one-point extensions, tensor products, or iterated tilts; checking this would show how far the theorem extends beyond the examples discussed.","The new projective-dimension criterion may feed into the finitistic dimension problem: over virtually Gorenstein algebras, bounding the relevant Ext-vanishing index for finitely generated modules of finite projective dimension would bound finitistic dimension directly."],"forward_implications":["Every virtually Gorenstein Artin algebra is weakly Gorenstein: the containment $\\mathcal{GP}\\subseteq\\perp^\\infty\\Lambda$ becomes an equality, so semi-Gorenstein-projective modules over these algebras are exactly the Gorenstein projective ones.","The Strong Nakayama Conjecture holds for all modules over virtually Gorenstein Artin algebras: if $\\operatorname{Ext}^n_\\Lambda(M,\\Lambda)=0$ for all $n\\ge0$, then $M=0$.","The Generalized Nakayama Conjecture, the Nakayama Conjecture, the Auslander-Gorenstein Conjecture, and Tachikawa's First Conjecture follow for virtually Gorenstein Artin algebras.","For an Artin algebra in which the Gorenstein projective modules are precisely the projective modules, the Auslander-Reiten Conjecture holds for all modules, and projective dimension is given by the Ext-vanishing formula.","The Gorenstein Symmetry Conjecture is reproved for virtually Gorenstein Artin algebras by identifying five numerical invariants of the algebra."],"supporting_citations":[{"why":"Introduces semi-Gorenstein-projective and weakly Gorenstein modules and poses the question that the paper answers for virtually Gorenstein algebras.","marker":"[29]"},{"why":"Supplies the definition of virtually Gorenstein algebras used throughout.","marker":"[10]"},{"why":"Provides the cotorsion triple and duality properties of virtually Gorenstein algebras used in the main lemmas and dimension formulas.","marker":"[7]"},{"why":"Gives the hereditary complete cotorsion theory $(\\perp\\mathcal{GI},\\mathcal{GI})$ that Proposition 2.1 and Lemma 3.4 rely on.","marker":"[31]"},{"why":"Provides the filtration and Ext-Tor duality lemmas used to characterize $\\perp\\mathcal{GI}$ and to transfer Ext-vanishing across the character duality.","marker":"[20]"},{"why":"Supplies Lemma 2.5, identifying finitely presented modules in $\\perp(X^\\perp)$ with $\\operatorname{add}(\\operatorname{filt}(X))$, used in Lemma 3.2.","marker":"[27]"},{"why":"Supplies the continuous-chain Ext-vanishing lemma used in Theorem 2.8 to establish hereditariness of the cotorsion theory.","marker":"[25]"},{"why":"Provides the cotorsion theory and cover/envelope basics, including completeness of cotorsion theories generated by a set, used throughout Section 2.","marker":"[18]"},{"why":"Underlies the equality of Gorenstein injective dimension and injective dimension used in Theorem 3.14.","marker":"[21]"},{"why":"Gives existence of projective envelopes over Artin algebras, used in Proposition 2.9 to prove $\\perp X=\\mathcal{GP}$.","marker":"[1]"}],"fun_headline_variants":["Virtually Gorenstein means weakly Gorenstein","Semi-Gorenstein-projective equals Gorenstein projective here","One test module gives all dimensions in this class","2020 open question settled for virtually Gorenstein","Classical conjectures follow from virtually Gorenstein"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the unproved half of Theorem 3.7, which says that over these algebras, the Gorenstein projective dimension of every module $M$ is the least $n$ such that $\\operatorname{Ext}^{n+l}_\\Lambda(M,\\Lambda)=0$ for all $l\\ge1$; the text points out only that the proof is similar to the dual statement, and if this formula fails then the equality $\\perp^\\infty\\Lambda=\\mathcal{GP}$ need not follow.","fun_headline_variants_meta":{"raw":{"variants":["Virtually Gorenstein means weakly Gorenstein","Semi-Gorenstein-projective equals Gorenstein projective here","One test module gives all dimensions in this class","2020 open question settled for virtually Gorenstein","Classical conjectures follow from virtually Gorenstein"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000804,"raw_usage":{"total_tokens":3479,"prompt_tokens":836,"completion_tokens":2643,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":452,"completion_tokens_details":{"reasoning_tokens":2564}},"tokens_in":452,"tokens_out":2643,"duration_ms":21153,"temperature":1.0,"reasoning_tokens":2564,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:18:53.557762+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a virtually Gorenstein Artin algebra $\\Lambda$ and a module $M$ with $\\operatorname{Ext}^i_\\Lambda(M,\\Lambda)=0$ for all $i\\ge1$ such that $M$ is not Gorenstein projective; this directly contradicts $\\perp^\\infty\\Lambda=\\mathcal{GP}$. Equivalently, exhibit a module for which the formula in Theorem 3.7(2) gives a value strictly below the true Gorenstein projective dimension.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces semi-Gorenstein-projective and weakly Gorenstein modules and poses the question that the paper answers for virtually Gorenstein algebras."},{"cited_title":"Beligiannis, I","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of virtually Gorenstein algebras used throughout."},{"cited_title":"Beligiannis, Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras,J","cited_arxiv_id":null,"evidence_quote":"Provides the cotorsion triple and duality properties of virtually Gorenstein algebras used in the main lemmas and dimension formulas."},{"cited_title":"ˇSaroch, J","cited_arxiv_id":null,"evidence_quote":"Gives the hereditary complete cotorsion theory $(\\perp\\mathcal{GI},\\mathcal{GI})$ that Proposition 2.1 and Lemma 3.4 rely on."},{"cited_title":"G ¨obel, J","cited_arxiv_id":null,"evidence_quote":"Provides the filtration and Ext-Tor duality lemmas used to characterize $\\perp\\mathcal{GI}$ and to transfer Ext-vanishing across the character duality."},{"cited_title":"Moradifar, J","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 2.5, identifying finitely presented modules in $\\perp(X^\\perp)$ with $\\operatorname{add}(\\operatorname{filt}(X))$, used in Lemma 3.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the continuous-chain Ext-vanishing lemma used in Theorem 2.8 to establish hereditariness of the cotorsion theory."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the cotorsion theory and cover/envelope basics, including completeness of cotorsion theories generated by a set, used throughout Section 2."},{"cited_title":"Holm, Rings with finite gorenstein injective dimension,Proc","cited_arxiv_id":null,"evidence_quote":"Underlies the equality of Gorenstein injective dimension and injective dimension used in Theorem 3.14."},{"cited_title":"Asensio Mayor, J","cited_arxiv_id":null,"evidence_quote":"Gives existence of projective envelopes over Artin algebras, used in Proposition 2.9 to prove $\\perp X=\\mathcal{GP}$."}],"review_version":1}