{"id":"9169df39-e6c4-4d21-85bb-8eee003986b6","arxiv_id":"2608.08228","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors construct commutative local artin algebras that are G-regular and weakly Gorenstein but admit infinitely generated non-projective Gorenstein projective modules, refuting Chen's Problems A, B, and C.","lead":"A new class of rings, called strongly G-regular rings, is introduced and characterized in terms of derived categories and subcategory finiteness. The paper constructs explicit counterexamples showing that three open problems of Chen on Gorenstein projective modules all have negative answers.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The counterexample's G-regularity and (tr) rest entirely on an imported classification from [2, Example 2.5(a)]; if that citation fails for p=q=2, the negative answers to Chen's problems do not follow.","rationale":"I read the manuscript in good faith and traced the proof of Theorem 5.7 carefully. The construction of the module M=Im c is self-contained: the definitions of a, b, g, and h in Proposition 5.6 check out, including the delicate index manipulation involving the bijections λ and μ. The argument that M is Gorenstein projective and not projective is internally sound. The paper's main conclusion, however, is not just the existence of such M; it is that R is simultaneously G-regular and weakly Gorenstein (i.e., satisfies (tr)). Those two properties are not proved in the text but are quoted from [2, Example 2.5(a)] and Remark 3.3(2). If that cited classification is accurate, the paper settles all three of Chen's problems and is a substantial advance. If not, the counterexample merely shows that strong G-regularity is strictly stronger than G-regularity plus (tr) in this family, and the open problems remain open. Since the paper does not re-derive the cited result and I cannot verify it independently, the most load-bearing concern is exactly this external dependency. The reader's weakest_assumption identified the same citation, so I agree. My proposed check is to reproduce [2, Example 2.5(a)] in the minimal nontrivial case p=q=2; if the classification holds for that case, the concern is resolved. I see no other internal objection that would change the conditional verdict.","tokens_in":462,"tokens_out":10819,"duration_ms":266136,"concrete_test":"Obtain [2] and reproduce Example 2.5(a) for the case p=q=2, i.e., A=k[x,y]/(x,y)^2 and B=k[s,t]/(s,t)^2. Specifically, independently classify all finitely generated R-modules M with Ext^>0_R(M,R)=0 and all with Tor^R_>0(M,E_R(k))=0; since I^3=0 this is a finite linear algebra problem. If any nonfree module satisfies either condition, recompute Theorem 5.7's G-regularity and (tr) claims. Also check the hypotheses in [2] (e.g., p,q≥2 vs >2, characteristics, field assumptions) against Theorem 5.7's use.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction in Theorem 5.7 explicitly builds an infinitely generated Gorenstein projective module M=Im c that is not projective, establishing that R is not strongly G-regular. However, the paper's conclusion that R is G-regular and satisfies (tr) — and hence the negative answers to Chen's Problems A, B, and C — is not proved in the text. It is imported verbatim from [2, Example 2.5(a)] via Remark 3.3(2). If that cited classification is incorrect, applies only under stronger hypotheses (e.g., p,q>2, algebraically closed k, or extra conditions on A and B), or is misquoted, then R may fail to be G-regular or weakly Gorenstein. In that case the existence of a non-projective Gorenstein projective module would only show that strong G-regularity fails, not that a G-regular/weakly Gorenstein ring can fail it; the answers to Chen's problems would remain open. The paper gives no independent verification of the cited example, and I cannot access [2] to confirm its hypotheses and conclusion match the present setting.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces and studies strongly G-regular rings, i.e., commutative noetherian rings over which every Gorenstein projective module, not necessarily finitely generated, is projective. It establishes a classification of thick subcategories over locally quasi-dominant rings (Theorem 2.11 and Corollary 2.13), characterizes strong G-regularity in terms of the derived category, quasi-dominance, and the singular/Gorenstein loci (Theorem 3.8 and Corollary 3.11), and proves a higher-dimensional analogue of the Beligiannis–Krause characterization via covariantly/contravariantly finite thick subcategories (Theorem 4.5). The central new result is Theorem 5.7: for any field k there is a commutative local finite-dimensional k-algebra R that is weakly Gorenstein and G-regular (CM-free) but not strongly G-regular; an explicit infinitely generated Gorenstein projective module is constructed. Corollary 5.9 then gives negative answers to Chen's Problems A, B, and C. The only point of fragility is that the G-regularity and the condition (tr) of the constructed ring are imported verbatim from Atkins–Vraciu [2, Example 2.5(a)], and no independent verification or precise statement of that example is provided.","tokens_in":25853,"tokens_out":9878,"duration_ms":90496,"significance":"If the cited result is correct, the paper resolves a cluster of open problems due to Chen and Ringel–Zhang in the negative, and it provides a substantial new theory connecting strong G-regularity with quasi-dominance and thick subcategory finiteness. The construction of a totally acyclic complex in Proposition 5.6 is explicit and the verification of total acyclicity in Theorem 5.7 is detailed and appears correct; there is no post-hoc data selection. The paper also contains useful new families of dominant and strongly G-regular rings. The main caveat is that the decisive G-regularity and (tr) properties of the counterexample are not proved in the text; they rest on a single citation. This makes the central claim conditionally sound rather than fully self-contained.","major_comments":[{"comment":"The final sentence of Theorem 5.7, 'By [2, Example 2.5(a)] and Remark 3.3(2), R is G-regular and satisfies (tr)', is the sole justification for the properties that make R a counterexample to Chen's Problems A, B, and C in Corollary 5.9. The cited example is not stated, and the paper does not verify that its hypotheses hold for an arbitrary field k and for all p,q ⩾ 2. If the cited example requires additional assumptions (for instance, an algebraically closed field, or p,q > 2), or if it is misquoted, then the ring R may fail to be G-regular or to satisfy (tr), and the negative answers to Chen's problems would not follow. Since this is a load-bearing step, please either state [2, Example 2.5(a)] precisely and confirm that the present setting satisfies its hypotheses, or give a self-contained proof of the G-regularity and (tr) of R.","section":"§5, Theorem 5.7"},{"comment":"The paper concludes that the constructed rings are weakly Gorenstein in the sense of Ringel–Zhang by combining (tr) with the equivalence asserted in Remark 3.3(2). This is an additional advertised property in Theorem 1.4 and the abstract, but the connection between (tr) and weak Gorensteinness for artin algebras is only referenced indirectly. Please provide a precise reference to [45] for the equivalence and, if needed, a one-sentence explanation showing that the artinian local algebra R satisfies the definition of weak Gorensteinness directly from (tr).","section":"§5, Remark 5.8(2) and Corollary 5.9"}],"minor_comments":[{"comment":"The statement 'The ring R is local with maximal ideal I=mR+nR' is used without proof; it follows from the standard fact that the tensor product over a field of two local k-algebras is local, but the authors should either justify it or supply a citation.","section":"§5, Theorem 5.7"},{"comment":"The isomorphism m⊗^L_R R/xR ≅ m⊗_R R/xR is used without comment; it holds because x is R-regular, but a brief justification would improve clarity.","section":"§2, Proposition 2.7(2)"},{"comment":"The displayed statement of Theorem 1.1 contains a garbled typesetting of the correspondence; the final version should ensure that the bijection between thick subcategories and generalization-closed subsets is rendered correctly.","section":"Introduction, Theorem 1.1"},{"comment":"The phrase 'strongly G-regular' in the sense of Atkins–Vraciu is defined using Tor-condition; the paper should explicitly note that for artinian local rings this condition is equivalent to the conjunction of G-regularity and the condition (tr), and cite the relevant theorem of [45] for the equivalence with weak Gorensteinness.","section":"§3, Remark 3.3(2)"},{"comment":"The notation 'dim k(0 :R m)' in the proof should be typeset as 'dim_k(0:_R m)' for readability; this is a purely typographical issue.","section":"§5, Corollary 5.2"}],"recommendation":"major_revision","confidential_remarks":"The main risk in this manuscript is the unverified reliance on [2, Example 2.5(a)] for the G-regularity and (tr) of the counterexample ring. If the editors can obtain a referee report that checks the applicability of that cited example, the concern would be resolved. The rest of the paper appears careful and explicit, and the construction in Theorem 5.7 is a valuable contribution regardless. The requested revision is local: either quote and verify the cited example, or provide a self-contained proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The theorem that will be remembered is Theorem 5.7: for any field k and artinian local k-algebras A, B with square-zero maximal ideals of k-dimension at least 2, the tensor product R = A ⊗_k B is G-regular and satisfies (tr) but is not strongly G-regular. The proof that R is not strongly G-regular is explicit and self-contained: they build an infinitely generated Gorenstein projective module M = Im c that is not projective. The acyclicity argument in Proposition 5.6 is clever and checkable.\n\nWhat is genuinely new: the definition of strong G-regularity for all Gorenstein projective modules, the characterization in terms of thick subcategories and quasi-dominance (Theorem 3.8), and the extension of Takahashi's subcategory classification to non-CM rings (Corollary 2.13). These sections are careful, detailed, and the paper is honest about which results are imported.\n\nThe soft spot: the claim that R is G-regular and satisfies (tr) is not proved in the text; it is imported from [2, Example 2.5(a)] via Remark 3.3(2). If that citation is wrong, applies only under stronger hypotheses, or is misquoted, the negative answers to Chen's Problems A, B, and C would not follow from this construction—though the existence of a non-projective Gorenstein projective module over these tensor product rings would remain. This is a genuine fragility, but not a confirmed flaw: the citation is to a peer-reviewed paper and the stated hypotheses (dim_k m, dim_k n >= 2, m^2 = n^2 = 0) match the setting. Given the stakes, a referee should ask the authors to either reproduce the relevant argument or state the exact hypotheses of [2, Example 2.5(a)] so that the reader can verify it without hunting down the source.\n\nThe rest of the paper, Sections 2–4, is solid and the classification theorems are substantial. The paper is written for specialists in Gorenstein homological algebra; a nonspecialist can extract the main example from Section 5 without reading the earlier sections.\n\nI would send this to a serious referee. The main construction is explicit, the structural results are important, and the citation issue is a reasonable request for clarification, not a reason to reject.","headline":"Settles Chen's problems in the negative via an explicit tensor-product construction; the main counterexample is self-contained except for one cited classification that should be verified.","tokens_in":26424,"tokens_out":2717,"would_cite":true,"duration_ms":26309,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13C60","13D09","13H10","16E65"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any field k, a commutative local artin algebra can be G-regular yet possess a non-projective Gorenstein projective module.","keywords":["strongly G-regular ring","Gorenstein projective module","quasi-dominant ring","CM-free algebra","weakly Gorenstein algebra","thick subcategory","totally acyclic complex","artin algebra"],"falsifier":"For $k = \\mathbb{F}_2$ and $A = B = k[x,y]/(x^2, xy, y^2)$, compute explicitly with the definition in Theorem 5.7 whether $M = \\operatorname{Im} c$ is Gorenstein projective and non-free. If $M$ is free, or if a finitely generated non-free Gorenstein projective module exists over $R = A \\otimes_k B$, the counterexample fails.","tokens_in":25440,"feed_emoji":"","tokens_out":15288,"duration_ms":119720,"temperature":0.7,"pith_summary":"This paper introduces strongly G-regular rings, defined by requiring every (including infinitely generated) Gorenstein projective module to be projective, and studies how this condition relates to quasi-dominance and to the homological finiteness of thick subcategories. Its main positive results characterize strong G-regularity in terms of quasi-dominant rings and, in the Cohen–Macaulay case, in terms of covariant or contravariant finiteness of a natural thick subcategory. Its main negative result constructs, over any field $k$, a commutative local artin algebra $R$ that is weakly Gorenstein and G-regular (CM-free) but not strongly G-regular; the explicit infinitely generated Gorenstein projective module built from a totally acyclic complex refutes the three open problems (A), (B), and (C) of the artin algebra literature. This matters because those problems asked whether the finiteness of Gorenstein projective modules would force them to decompose into finitely generated pieces, and the answer turns out to be no even in the commutative local artin case.","feed_headline":"A local artin algebra can be CM-free yet not strongly G-regular","feed_subtitle":"An explicit tensor-product ring carries a non-projective Gorenstein module, refuting the three problems.","key_machinery":"The argument runs through two main mechanisms. On the structural side, the paper uses the thick subcategory $\\operatorname{thick}_{D^b(R)}\\{R, D\\}$ generated by the ring and its dualizing complex: strong G-regularity is equivalent to this thick subcategory being the whole bounded derived category, and quasi-dominance is the local statement that the residue field can be built from $\\{R, D, X\\}$ for every $X$ of infinite projective dimension; the classification of such thick subcategories then flows from the correspondence between thick subcategories of $D^b(R)$ and of $\\operatorname{mod} R$, restricted through the category $C(R)$. On the constructive side, Proposition 5.6 builds a one-periodic complex $X = (\\cdots \\to F \\to F \\to \\cdots)$ with differential $c = \\begin{pmatrix} a & b \\\\ b & -a \\end{pmatrix}$ on $F = G \\oplus G$; given endomorphisms $a, b$ of a free module $G$ with $a^2 = b^2 = 0$, $ab = ba$, and additive splittings $g$ and $h$ satisfying $ag + ga = \\operatorname{id}_G$ and $b^*h + hb^* = \\operatorname{id}$, the complex is totally acyclic, so $M = \\operatorname{Im} c$ is Gorenstein projective. In Theorem 5.7, the tensor product $R = A \\otimes_k B$ realizes such $a, b$ on a free module with basis indexed by $\\mathbb{N}^2$ using bijections from $\\mathbb{N} \\times \\{1,\\dots,p\\}$ and $\\mathbb{N} \\times \\{1,\\dots,q\\}$ to $\\mathbb{N}$; the two tensor factors ensure $ab = ba$, while the two acyclicity conditions are checked separately on $B$-bases and $A$-bases.","core_discovery":"The paper's central claim is that strong G-regularity is strictly stronger than G-regularity, even among commutative local artin algebras that are weakly Gorenstein. For any field $k$, take artinian local $k$-algebras $A$ and $B$ with square-zero maximal ideals of $k$-dimension at least 2 and form $R = A \\otimes_k B$; the paper shows that $R$ is G-regular and satisfies (tr) (equivalently, is weakly Gorenstein), but the module $M = \\operatorname{Im} c$, where $c$ is the one-periodic differential on $F = G \\oplus G$ constructed in Section 5, is Gorenstein projective and not projective. Consequently the three problems (A), (B), and (C) all have negative answers: over the same ring $R$, CM-freeness does not force every Gorenstein projective module to be a direct sum of finitely generated ones, nor to be projective, and CM-finiteness does not imply virtual Gorensteinness. The paper also proves a classification theorem for thick subcategories over locally quasi-dominant rings and characterizes strong G-regularity by the equality of the singular and non-Gorenstein loci together with local quasi-dominance.","pith_inferences":["The explicit construction of a totally acyclic complex from commuting square-zero endomorphisms with additive splittings could be adapted to other tensor-product settings, potentially yielding non-projective Gorenstein projective modules over graded or noncommutative algebras.","The counterexample is G-regular and weakly Gorenstein but not strongly G-regular; a known theorem for artin algebras then implies it is not virtually Gorenstein, so weak Gorensteinness does not imply virtual Gorensteinness for commutative artin algebras.","The classification theorem suggests that quasi-dominance may be the right unifying hypothesis for subcategory classification beyond Cohen–Macaulay rings; one could test whether the correspondence extends to categories larger than $C(R)$ or to rings without dualizing complexes.","The non-projective Gorenstein projective module $M = \\operatorname{Im} c$ is a one-periodic image, giving a concrete infinitely generated Gorenstein projective module over an artin algebra; computing its endomorphism ring or deciding whether it decomposes into finitely generated modules would be a natural next step."],"forward_implications":["Problems (A), (B), and (C) all have negative answers: CM-freeness does not force every Gorenstein projective module to be a direct sum of finitely generated ones, nor to be projective, and CM-finiteness does not imply virtual Gorensteinness.","Strong G-regularity is strictly stronger than G-regularity even in the artinian local setting: there is a CM-free ring with a non-projective Gorenstein projective module.","Under mild assumptions, strong G-regularity is a local property and is characterized by local quasi-dominance together with equality of the singular and non-Gorenstein loci.","For a henselian Cohen–Macaulay non-Gorenstein local ring, strong G-regularity is equivalent to the covariant (equivalently contravariant) finiteness of the thick subcategory generated by the ring and its canonical module.","New families of strongly G-regular rings are obtained: quotients of regular rings by powers of complete-intersection ideals and certain products of parameter ideals are locally dominant and strongly G-regular when the number of generators exceeds one."],"supporting_citations":[{"why":"Supplies the definition of quasi-dominant rings and the subcategory classification theorems that this paper extends to the non-Cohen–Macaulay case.","marker":"[50]"},{"why":"Provides the classification of the tensor product rings used in the counterexample, asserting G-regularity and (tr) for the constructed ring.","marker":"[2]"},{"why":"Gives the equivalence between strong G-regularity and the equality of the thick subcategory generated by the ring and its dualizing complex with the whole derived category.","marker":"[35]"},{"why":"Formulates problems (A), (B), and (C) and the CM-free notion that the paper answers negatively.","marker":"[18]"},{"why":"Relates CM-finiteness, virtual Gorensteinness, and the decomposition of Gorenstein projective modules into finitely generated summands for artin algebras.","marker":"[10]"},{"why":"Characterizes virtual Gorenstein algebras via covariant/contravariant finiteness of a thick subcategory, used in the artin algebra analogue.","marker":"[11]"},{"why":"Provides the bijection between thick subcategories of the bounded derived category and of the module category used in the classification theorem.","marker":"[40]"},{"why":"Introduces G-regular rings and supplies the localization/quotient examples that separate G-regularity from strong G-regularity.","marker":"[48]"}],"fun_headline_variants":["Strong G-regularity strictly stronger than G-regularity","Weakly Gorenstein but not strongly G-regular: a counterexample","Three Chen problems refuted by explicit ring","Strong G-regularity fails for weakly Gorenstein local algebras"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The counterexample inherits from a cited classification the claim that the tensor product ring $A \\otimes_k B$ (with both maximal ideals of dimension at least two and square zero) is G-regular and satisfies (tr); if that classification fails, the example no longer separates the two notions.","fun_headline_variants_meta":{"raw":{"variants":["Strong G-regularity strictly stronger than G-regularity","Weakly Gorenstein but not strongly G-regular: a counterexample","Three Chen problems refuted by explicit ring","Strong G-regularity fails for weakly Gorenstein local algebras"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000645,"raw_usage":{"total_tokens":2956,"prompt_tokens":927,"completion_tokens":2029,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":1958}},"tokens_in":543,"tokens_out":2029,"duration_ms":14697,"temperature":1.0,"reasoning_tokens":1958,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:14:35.687332+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $k = \\mathbb{F}_2$ and $A = B = k[x,y]/(x^2, xy, y^2)$, compute explicitly with the definition in Theorem 5.7 whether $M = \\operatorname{Im} c$ is Gorenstein projective and non-free. If $M$ is free, or if a finitely generated non-free Gorenstein projective module exists over $R = A \\otimes_k B$, the counterexample fails.","supporting_citations":[{"cited_title":"Takahashi , Dominant local rings and subcategory classification, Int","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of quasi-dominant rings and the subcategory classification theorems that this paper extends to the non-Cohen–Macaulay case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classification of the tensor product rings used in the counterexample, asserting G-regularity and (tr) for the constructed ring."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the equivalence between strong G-regularity and the equality of the thick subcategory generated by the ring and its dualizing complex with the whole derived category."},{"cited_title":"Beligiannis , On algebras of finite Cohen-Macaulay type, Adv","cited_arxiv_id":null,"evidence_quote":"Relates CM-finiteness, virtual Gorensteinness, and the decomposition of Gorenstein projective modules into finitely generated summands for artin algebras."},{"cited_title":"Beligiannis; H","cited_arxiv_id":null,"evidence_quote":"Characterizes virtual Gorenstein algebras via covariant/contravariant finiteness of a thick subcategory, used in the artin algebra analogue."},{"cited_title":"Krause; G","cited_arxiv_id":null,"evidence_quote":"Provides the bijection between thick subcategories of the bounded derived category and of the module category used in the classification theorem."},{"cited_title":"Takahashi , On G-regular local rings, Comm","cited_arxiv_id":null,"evidence_quote":"Introduces G-regular rings and supplies the localization/quotient examples that separate G-regularity from strong G-regularity."}],"review_version":1}