{"id":"7b13a130-edae-4f1e-8bc8-2d5b500523fb","arxiv_id":"2603.01424","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Gorenstein projective modules equal the left Ext-orthogonal class of the thick subcategory generated by the ring and Hom_S(R,ω), yielding cotorsion pairs and new Gorensteinness criteria.","lead":"This pure-mathematics paper proves new structural characterizations of Gorenstein projective modules over algebras over Cohen-Macaulay rings, using thick subcategories and cotorsion pairs. It also settles a question of R. Takahashi by showing a Cohen-Macaulay local ring is Gorenstein exactly when the right Ext-orthogonal class of finitely generated Gorenstein projective modules equals the modules of finite projective dimension.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.6's converse hinges on the unstated [33, Thm 1.2(c)]; if that theorem needs extra hypotheses on S or R, the central equality collapses.","rationale":"The paper is internally careful: Lemma 3.5's derived adjunction and the depth argument localizing a complete resolution are sound, and the applications in Sections 4–5 are coherent conditional on Theorem 3.6. The single point where a hidden hypothesis could break the central theorem is the unstated external approximation theorem. This is essentially the reader's weakest assumption, so we agree with the core concern. We note a small correction: Lemma 4.4 does not directly use Miyachi; the dependency is via Remark 3.7/Theorem 3.6. Because the cited theorem is very likely standard and the rest of the proof is sound, the concern supports a conditional posture rather than rejection. The metadata/abstract discrepancies (e.g., the Takahashi question not appearing in the body) remain editorial. Thus the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":20360,"tokens_out":28367,"duration_ms":260440,"concrete_test":"Obtain the exact statement of [33, Theorem 1.2(c)] and verify its hypotheses and conclusion verbatim: does it apply to a general Cohen–Macaulay ring S (not assumed local or complete) and to every noetherian S-algebra R with R∈MCM(S), and does it yield 0→K→M→X→0 with K∈Thick(proj(R)∪{R†}) and M∈MCM(S)? If it has an extra hypothesis (e.g., S local, or R an S-order), run the paper's proof on the smallest example satisfying the paper's hypotheses but not the extra one—for instance the trivial extension R=S⋉M of Example 3.8(4) with S a non-local CM ring—and check whether the required approximation sequence still exists.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The critical direction in Theorem 3.6 is showing every X∈⊥1 Thick(proj(R)∪{R†}) is Gorenstein projective. After constructing a left proj(R)-approximation f0:X→Q−1, the proof invokes Miyachi [33, Theorem 1.2(c)] to obtain 0→K→M→X→0 with K∈Thick(proj(R)∪{R†}) and M maximal Cohen–Macaulay over S. Splitting then forces X to be MCM, which is the keystone for embedding X into (R†)^n and for the vanishing Ext^i_R(X,R†). The cited theorem is never stated. If its actual hypotheses require S local/complete, or R an S-order, or if it only yields K in a larger category—e.g., K of finite projective dimension, or M not necessarily MCM over S—the splitting argument no longer implies X∈MCM(S), and the equality can fail. This dependency also flows into Corollary 3.10 and into Theorem 4.5 through Remark 3.7, so it is not an isolated step. The reader's mention of Lemma 4.4 is not exact: Lemma 4.4 itself uses Lemmas 3.3–3.5, not Miyachi; the dependency is via Theorem 3.6/Remark 3.7.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies finitely generated Gorenstein projective modules over noetherian algebras R that are module-finite over a Cohen-Macaulay ring S with canonical module ω and are maximal Cohen-Macaulay over S. Its central result, Theorem 3.6, identifies Gproj(R) with the left Ext^1-orthogonal class of Thick(proj(R) ∪ {Hom_S(R,ω)}). From this the authors derive a hereditary cotorsion pair (Gproj(R), Gproj(R)^⊥1) in mod(R) (Corollary 3.10), characterizations of left weakly Gorenstein rings (Theorem 4.5), and, for Cohen-Macaulay local rings, an equivalence between Gorensteinness, Gproj(R)^⊥1 = P^{<∞}(R), and the cotorsion-pair condition (Theorem 5.6). The paper also contains results on Artin algebras, including Proposition 3.13 and Corollary 3.15, and remarks on virtually Gorenstein algebras. The overall structure is clear, and the main chain of implications is coherent, but the proof of the central theorem relies on an unstated external theorem, and one assertion in the proof of Theorem 5.6 is insufficiently justified.","tokens_in":20537,"tokens_out":26483,"duration_ms":255800,"significance":"If the results are correct, they are valuable. Theorem 3.6 gives a finitely generated analogue of the recent cotorsion-pair theorem of Cortés-Izurdiaga and Šaroch and yields the desired hereditary cotorsion pair in mod(R). The weakly Gorenstein criteria and the Gorensteinness criterion for Cohen-Macaulay local rings are clean, testable statements; Theorem 5.6 addresses a question highlighted in the abstract as due to Takahashi. The paper is well organized, gives illustrative examples, and the internal line of reasoning is transparent. There is no apparent circularity and no parameter-fitting or definitional reduction. The main risks are external black boxes: the unstated Miyachi theorem used in Theorem 3.6 and the unproved existence of a finite-injective-dimension module used in Corollary 5.5. These are load-bearing, so the paper needs revision before the claims can be fully accepted.","major_comments":[{"comment":"The converse containment in Theorem 3.6 depends completely on [33, Theorem 1.2(c)], which is cited but never stated. The proof needs, for every X ∈ ⊥1 Thick(proj(R)∪{R†}), a short exact sequence 0→K→M→X→0 with K ∈ Thick(proj(R)∪{R†}) and M maximal Cohen-Macaulay over S. This is what forces X to be maximal Cohen-Macaulay and is then used for the embedding via Lemma 3.4 and for the vanishing of Ext^i_R(X,R†). Please state Miyachi's theorem precisely, with all hypotheses, and verify that it applies under the paper's assumptions: S a Cohen-Macaulay ring with canonical module, R a noetherian S-algebra, R ∈ MCM(S), with no local/complete hypothesis and no order assumption. In particular, confirm that K lies in the stated thick subcategory and that M is maximal Cohen-Macaulay over S. The dependency propagates to Corollary 3.10, Remark 3.7, and Theorem 4.5, so this is not an isolated presentatio","section":"§3, Theorem 3.6"},{"comment":"Corollary 5.5 asserts that every Cohen-Macaulay local ring admits a nonzero finitely generated module of finite injective dimension, citing only the vague 'discussion following [9, Corollary 9.6.2]'. This is load-bearing for the implication (2)⇒(1) in Theorem 5.6. The assertion is not a formal consequence of the definition; canonical modules, which are the standard source of such modules, are known not to exist for every Cohen-Macaulay local ring unless the ring is a homomorphic image of a Gorenstein local ring (see, for example, Bruns–Herzog, Theorem 3.3.6). Please supply a precise reference with a statement, or prove the existence directly under the hypotheses of Corollary 5.5. Without this, the proof of Theorem 5.6 has a gap.","section":"§5, Corollary 5.5 and Theorem 5.6"},{"comment":"The proof of Theorem 3.6 begins by taking a left proj(R)-approximation f0:X→Q−1, referring to 2.11. But 2.11 only records covariantly finite add(M) for Artin algebras and commutative noetherian rings, while the theorem allows an arbitrary noetherian S-algebra. Existence of left proj(R)-approximations is not automatic from the existence of projective covers. It can be proved for any two-sided noetherian R by choosing a finite generating set of the right R-module Hom_R(X,R) and forming the corresponding map X→R^n, but this argument is absent. The same existence is needed in Lemma 4.4. Please add this argument or a reference covering the noncommutative case.","section":"§2.11, §3 (proof of Theorem 3.6), §4 (Lemma 4.4)"}],"minor_comments":[{"comment":"The title on the first page of the manuscript differs from the arXiv metadata. Please align the two.","section":"Title"},{"comment":"In the final paragraph of the proof, the phrase 'as X∈Gproj(R)' is misleading: X is not yet known to be Gorenstein projective at that point. The intended justification is that X was shown to be maximal Cohen-Macaulay as an S-module by the splitting argument; please reword.","section":"§3, proof of Theorem 3.6"},{"comment":"In cases (2) and (3) of the proof, the notation 'Hom_Λ(d0, Ki)' and 'Hom_Λ(d0, Ki), i≥0' should refer to the relevant test module in S_Λ ∪ T_Λ; the symbols Ki and Li are mixed. Please correct the displayed morphisms and the concluding sentences.","section":"§3, Proposition 3.13"},{"comment":"References [26] and [27] appear to be duplicates: both are listed as Iyengar–Krause, 'The Nakayama functor and its completion for Gorenstein algebras', Bull. Soc. Math. Fr. 150 (2022), no. 2, 347–391. Please remove the duplicate.","section":"References"},{"comment":"The proof says the isomorphism follows from the derived tensor–hom adjunction and that R† is quasi-isomorphic to RHom_S(R,ω). This is terse; since the point is used in several places, please include a direct argument via an injective resolution of ω over S, or at least explain why Hom_S(R,I) gives an injective resolution of R† over R.","section":"§3, Lemma 3.5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is likely to be right in its main ideas, but the current form is not fully verifiable. The most serious issue is the unstated [33, Theorem 1.2(c)] in Theorem 3.6; if that theorem applies exactly as needed, the central equality is convincing. The second issue is the unproved existence assertion in Corollary 5.5. I would ask the authors to state the external theorem, verify its hypotheses explicitly, and supply a precise reference/proof for the finite-injective-dimension module. No circularity or ethical issues are apparent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nShort version: this is a competent and useful paper. The main theorem (Theorem 3.6) — Gproj(R) = ⊥1 Thick(proj(R) ∪ {Hom_S(R,ω)}) under the stated noetherian-algebra/MCM hypotheses — appears to be genuinely new, and it earns its keep. The cotorsion pair in finitely generated modules (Corollary 3.10) and the Gorenstein characterization for CM local rings (Theorem 5.6) are real applications, not repackaged folklore. The proofs are detailed enough for an expert to check, and the writers credit Huang–Huang and Ringel–Zhang where credit is due.\n\nWhat’s good: the proof strategy is intelligible. The Gorenstein-to-orthogonal direction is clean — show G is MCM over S, apply the Ext transfer (Lemma 3.5), and use Lemma 3.1 to thicken. The converse is the heart, and it works modulo one thing I’ll flag below. The weakly Gorenstein criterion (Theorem 4.5) and the Foxby-based Gorenstein detection in Section 5 are logically sound as far as I can tell. The paper reads like real mathematics, not a sequence of definitions.\n\nThe soft spots, in proportion:\n\n1. The load-bearing use of Miyachi [33, Theorem 1.2(c)] is never stated. This is the step that produces the approximation 0 → K → M → X → 0 with K in the thick subcategory and M MCM over S. If that theorem has extra hypotheses that aren’t satisfied here, the converse direction of Theorem 3.6 collapses, and so do Corollary 3.10 and parts of Section 4. I don’t think the hypotheses are secretly wrong — the paper’s setup (S CM with canonical module, R a noetherian S-algebra, R ∈ MCM(S)) matches Miyachi’s setting — but the reader should not have to guess. Stating the theorem is necessary for self-containedness and referee sanity.\n\n2. The abstract says the paper gives “an affirmative answer to a question of R. Takahashi,” but the body never states what that question is. Theorem 5.6 may well be the answer, but the paper should say so explicitly.\n\n3. Minor: there are typos (e.g., “P AIRS” in the title, a misplaced parenthesis in Theorem 4.5 condition (2)), and Proposition 3.13’s proof has a small notation slip (Hom_R(d0, K_i) should be Hom_Λ). None of these affect the mathematics.\n\nThe stress-test note worries that Lemma 4.4 itself invokes Miyachi; that’s actually wrong — Lemma 4.4 uses Lemmas 3.3–3.5 and the short exact sequence from its own statement. The dependency on Miyachi runs through Theorem 3.6 and Remark 3.7, which is exactly where the concern lands. So the worry is real but precisely localized.\n\nWho this is for: people working in Gorenstein homological algebra, cotorsion pairs, and Cohen–Macaulay representation theory. It deserves a serious referee. I’d send it out with a request to state Miyachi’s theorem and identify Takahashi’s question.\n\nRecommendation: accept for peer review, with revisions along the lines above.","headline":"A solid, carefully written paper that proves a genuinely new characterization of Gproj and a hereditary cotorsion pair in mod(R), though the main theorem leans on an external result that should be stated explicitly.","tokens_in":21200,"tokens_out":836,"would_cite":true,"duration_ms":10083,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13C14","13D07","16D90","16E05","16E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The authors prove that, for a noetherian algebra over a Cohen-Macaulay ring with a canonical module, the finitely generated Gorenstein projective modules are precisely those with vanishing Ext^1 against every module in the thick subcategory","keywords":["Cohen-Macaulay rings","Gorenstein projective modules","thick subcategories","cotorsion pairs","canonical modules","weakly Gorenstein rings","maximal Cohen-Macaulay modules","noetherian algebras"],"falsifier":"A decisive counterexample would be a noetherian S-algebra R satisfying the hypotheses (S Cohen-Macaulay with canonical module ω, R maximal Cohen-Macaulay over S) together with a finitely generated module M for which Ext^1_R(M,T)=0 for every T in Thick(proj(R) ∪ {Hom_S(R,ω)}) but M is not Gorenstein projective. Alternatively, a Cohen-Macaulay local ring with Gproj(R)⊥1 = P<∞(R) that is not Gorenstein would falsify the final criterion.","tokens_in":20097,"feed_emoji":"🧮","tokens_out":12334,"duration_ms":101254,"temperature":0.7,"pith_summary":"This paper establishes an exact description of the finitely generated Gorenstein projective modules over any noetherian algebra R over a Cohen-Macaulay ring S that admits a canonical module ω, provided R itself is maximal Cohen-Macaulay as an S-module. It proves that a finitely generated R-module is Gorenstein projective if and only if it has vanishing Ext^1 against every module in the thick subcategory generated by the finitely generated projectives together with Hom_S(R,ω). From this equality, the authors derive a hereditary cotorsion pair in the category of finitely generated modules, a characterization of when R is weakly Gorenstein in terms of a single short exact sequence, and a criterion for Cohen-Macaulay local rings to be Gorenstein. The payoff is that a subtle homological property becomes a single equality of module classes that can be checked explicitly.","feed_headline":"Gorenstein projectives equal one Ext-orthogonal class","feed_subtitle":"A module is Gorenstein projective exactly when Ext^1 vanishes against the thick subcategory generated by projectives and Hom_S(R,ω).","key_machinery":"The load-bearing object is the thick subcategory Thick(proj(R) ∪ {R†}) in mod(R), where R† = Hom_S(R,ω); a thick subcategory is the smallest class of modules closed under direct summands and under the two-out-of-three rule in short exact sequences that contains all finitely generated projectives and the dual module R†. The central mechanism is the identification of the left Ext^1-orthogonal class of this subcategory with Gproj(R). Two auxiliary tools carry the proof: the change-of-rings isomorphism Ext^i_S(M,ω) ≅ Ext^i_R(M,R†), and a criterion stating that a finitely generated module is Gorenstein projective if it has vanishing positive Ext against R and admits a projective coresolution whos","core_discovery":"The central discovery is Theorem 3.6: if S is a Cohen-Macaulay ring admitting a canonical module ω and R is a noetherian S-algebra with R ∈ MCM(S) as an S-module, then Gproj(R) = ⊥1 Thick(proj(R) ∪ {Hom_S(R,ω)}). That is, the finitely generated Gorenstein projective R-modules are exactly those modules X for which Ext^1_R(X,T)=0 for every T in the smallest thick subcategory containing the finitely generated projectives and the module Hom_S(R,ω). The forward direction shows every Gorenstein projective module is maximal Cohen-Macaulay over S and uses the change-of-rings isomorphism Ext^i_S(M,ω) ≅ Ext^i_R(M,Hom_S(R,ω)) to get orthogonality. The reverse direction uses a Cohen-Macaulay approximati","pith_inferences":["The equality Gproj(R) = ⊥1 Thick(proj(R) ∪ {R†}) reduces a global acyclicity condition to a single family of Ext^1-vanishing conditions, which could make Gorenstein projectivity checkable by computations once R† and its syzygies are known.","Because the characterization is phrased purely in terms of a thick subcategory, it may transfer to derived or singularity categories over the same class of algebras, potentially giving new descriptions of the stable category Gproj(R).","The weakly Gorenstein test suggests that, in examples where a presentation of R† is explicit, the single module X gives a finite certificate for weak Gorensteinness, which could be pushed to algorithmic verification for concrete algebras.","If the cited Cohen-Macaulay approximation theorem could be replaced by a weaker splitting condition, the main theorem might extend beyond the current hypotheses to algebras that are not finite over S or lack a canonical module."],"forward_implications":["For every ring R satisfying the hypotheses, (Gproj(R), Gproj(R)⊥1) is a hereditary cotorsion pair in mod(R), so the finitely generated Gorenstein projective modules form the left half of an approximation-theoretic pair.","If S has finite Krull dimension, then Gproj(R) = ⊥1 Thick(proj(R) ∪ I<∞(R)), so the same characterization can be stated with modules of finite injective dimension in place of Hom_S(R,ω).","The ring R is left weakly Gorenstein if and only if Thick(proj(R) ∪ {R†}) is contained in the double Ext-orthogonal of R; equivalently, a single module X defined by a short exact sequence 0→X→P→R†→0 suffices to test the property.","For a Cohen-Macaulay local ring, R is Gorenstein if and only if Gproj(R)⊥1 = P<∞(R), and this is also equivalent to (Gproj(R), P<∞(R)) being a cotorsion pair.","For Artin algebras, a semi-Gorenstein projective module with finite Auslander bound is automatically Gorenstein projective."],"fun_headline_variants":["Gorenstein projectives equal left Ext-orthogonal class","Finitely generated Gorenstein projectives: a cotorsion pair half","Answering Takahashi: Gorenstein projectives via thick subcategories","Ext^1-orthogonal class characterizes Gorenstein projective modules","Gorenstein projectives as the left orthogonal to a thick subcategory"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The main equality depends on a cited Cohen-Macaulay approximation theorem that supplies, for every finitely generated module, a short exact sequence whose kernel lies in the thick subcategory and whose middle term is maximal Cohen-Macaulay over S; if that theorem does not hold under exactly the stated hypotheses, the characterization collapses.","fun_headline_variants_meta":{"raw":{"variants":["Gorenstein projectives equal left Ext-orthogonal class","Finitely generated Gorenstein projectives: a cotorsion pair half","Answering Takahashi: Gorenstein projectives via thick subcategories","Ext^1-orthogonal class characterizes Gorenstein projective modules","Gorenstein projectives as the left orthogonal to a thick subcategory"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000152,"raw_usage":{"total_tokens":1058,"prompt_tokens":780,"completion_tokens":278,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":524,"completion_tokens_details":{"reasoning_tokens":185}},"tokens_in":524,"tokens_out":278,"duration_ms":3460,"temperature":1.0,"reasoning_tokens":185,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T19:37:59.421565+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive counterexample would be a noetherian S-algebra R satisfying the hypotheses (S Cohen-Macaulay with canonical module ω, R maximal Cohen-Macaulay over S) together with a finitely generated module M for which Ext^1_R(M,T)=0 for every T in Thick(proj(R) ∪ {Hom_S(R,ω)}) but M is not Gorenstein projective. Alternatively, a Cohen-Macaulay local ring with Gproj(R)⊥1 = P<∞(R) that is not Gorenstein would falsify the final criterion.","supporting_citations":[],"review_version":1}