{"id":"e80dc33e-77f6-46e8-8507-2abe9864d2d2","arxiv_id":"2501.12109","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For resolving subcategories satisfying a new condition (A), the level of a complex is always at least its resolution dimension plus its lowest nonzero cohomology degree plus one.","lead":"This paper proves a general lower bound on how many steps it takes to build a complex from a chosen class of modules. The bound recovers and extends several known results in homological algebra.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.12 depends on Lemma 3.6, stated without proof, to make the splitting argument go through; until that lemma is proved in the stated generality, the central lower bound is not fully verified.","rationale":"The reader's weakest-assumption identification is exactly the gap I find most load-bearing. Theorem 3.12 is a clean generalization whose proof structure is otherwise coherent: the ghost-map composition is built from X itself, so the Ghost Lemma is applied correctly, and the only real bridge between the constructed complex C(-i) and the projective resolution P is Lemma 3.6. The paper explicitly states that this lemma is omitted, and it is used exactly where the argument must convert a quasi-isomorphism of complexes into an isomorphism of Ext groups on cokernels. Without a proof of Lemma 3.6 in the full abelian-category setting, the central inequality rests on an unverified assertion. I do not claim the lemma is false: in fact a proof via the mapping cone and the closure properties of a resolving subcategory is plausible. But the manuscript does not supply it, and the module-based citation does not obviously cover the stated generality. Thus the correct disposition is the reader's CONDITIONAL verdict, with no adjustment needed.","tokens_in":21015,"tokens_out":17280,"duration_ms":194048,"concrete_test":"Write out a complete proof of Lemma 3.6 in the stated generality, following the truncation/cone strategy: for a quasi-isomorphism f:P->Q in C^-(X) and each integer v, construct an exact triangle or exact sequence relating C^vP, C^vQ, and C^v(cone(f)), and use M in X^perp plus the closure properties of X to show the induced Ext^i(-,M) maps are isomorphisms. Verify explicitly that every step uses only that A has enough projective objects and that X contains proj A and is closed under extensions and kernels of epimorphisms. If any step requires extra hypotheses (e.g., P and Q bounded below, or C^vP of finite X-resolution dimension), check whether those hypotheses hold for the projective resolution P and the complex X appearing in Theorem 3.12. If the proof goes through, the concern is resolved; if it requires extra assumptions, the theorem must be modified or the use justified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.12 hinges on Lemma 3.6. In the final paragraph, the argument needs Ext^1_A(C^{-g+1}X, X^{-g}) -> Ext^1_A(C^{-g+1}P, X^{-g}) to be an isomorphism so that Lemma 3.4 can be applied and the bottom row splits; this is precisely the content of Lemma 3.6 with v = -g+1 and M = X^{-g}. If that splitting fails, the contradiction g <= g-1 evaporates, and the Ghost Lemma only shows that the nonzero map psi is a composition of g+i ghosts. Lemma 3.6 is stated for arbitrary abelian categories, all M in X^perp, all positive i, and all integers v, but the proof is omitted with 'we omit the proof as the same argument is valid' (Section 3, before Lemma 3.6). The module case in [6, Lemma 3.2] may use properties of module categories that do not automatically transfer to a general abelian category with only enough projectives, such as enough injectives or freely constructed Ext arguments. Moreover, the lemma quantifies over all integers v, and the comparison must hold even when C^vP and C^vQ are not known to have finite X-resolution dimension; this needs a separate verification. Since the lemma is load-bearing and its proof is absent, the central claim is not yet fully established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a condition (A) on a resolving subcategory X of an abelian category with enough projectives and proves that for every nonzero object M in the bounded derived category, level^X_{D^b(A)} M is at least X-resol.dim M + inf M + 1 (Theorem 3.12). The proof is based on an approximation of M by a complex built from objects in X∩X^\\perp and on a ghost-map argument. The author then applies this theorem to projective, Gorenstein projective/injective, C-projective/injective modules with respect to a semidualizing module, and to contravariantly finite resolving subcategories, recovering results of Altmann–Grifo–Montaño–Sanders–Vu and of Awadalla–Marley and extending them to abelian categories.","tokens_in":21305,"tokens_out":28774,"duration_ms":299608,"significance":"If the proof is correct, the main theorem offers a clean, general lower bound for levels in derived categories, unifying several known results. The condition (A) is a natural hypothesis, and the applications to semidualizing modules and contravariantly finite resolving subcategories are valuable. The paper is clearly structured and the ghost-map strategy is elegant. The central reservation is Lemma 3.6, which is load-bearing but stated without proof; once that lemma is properly established, the paper would be a solid contribution to the subject.","major_comments":[{"comment":"Lemma 3.6 is stated without proof, with only the remark that the same argument as in [6, Lemma 3.2] is valid. This lemma is used essentially in the final paragraph of the proof of Theorem 3.12, where the quasi-isomorphism P→X is used to identify Ext^1_A(C^{-g+1}X, X^{-g}) with Ext^1_A(C^{-g+1}P, X^{-g}). Since the lemma is asserted for arbitrary abelian categories, all integers v, and all M in X^\\perp, the transfer from the module-theoretic setting is not automatic. A full proof, or a precise reference that covers the stated generality, must be supplied. The argument needs to explain why the cokernel exact sequences of the mapping cone induce isomorphisms at the same Ext-degree for every v, and the statement for B^v—though not used in the proof—also needs justification.","section":"Section 3, Lemma 3.6"},{"comment":"Corollary 6.4 asserts the lower bound for every contravariantly finite resolving subcategory X of mod Λ over an artin algebra, citing Proposition 6.3. However, Proposition 6.3 has the additional hypothesis that X is contained in Sub(X^\\perp). The corollary does not explain why this hypothesis is automatically satisfied in the artin algebra setting, nor does it cite a specific result that verifies it. If the hypothesis is indeed automatic, the proof should say so explicitly; otherwise, the claimed application is not fully established by the preceding proposition.","section":"Section 6, Corollary 6.4"}],"minor_comments":[{"comment":"There are typographical errors such as 'resolutuion' in Lemma 3.2 and Proposition 3.7; these should be corrected.","section":"Sections 3 and 6"},{"comment":"The statement says 'left R-modules' whereas the convention in Section 1 defines Mod R as right R-modules; the handedness should be made consistent.","section":"Corollary 4.4"},{"comment":"The proof begins by assuming -g < i. It would be clearer to state explicitly that when -g ≥ i the desired inequality is trivial, since level is at least 1 for nonzero objects, before proceeding with the contradiction argument.","section":"Theorem 3.12, proof"},{"comment":"The proof uses [14, Lemma 2.4(1)] as a black box. Including the statement of that lemma would make the paper more self-contained and would help the reader verify that it applies in the asserted generality of abelian categories with enough projectives.","section":"Proposition 3.7"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the main theorem is attractive. The main issue is Lemma 3.6: it is load-bearing for Theorem 3.12 and is stated without proof. If the author can supply a correct proof in the stated generality, I would support acceptance; if the lemma is false, the theorem would need to be restricted to settings where the lemma holds. The applications are well chosen and the recoveries of existing results are properly credited."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take: this is a genuinely new generalization, not a repackaging. Theorem 3.12 gives a lower bound for level^X in D^b(A) in terms of X-resolution dimension plus inf M plus 1, under a resolvent condition (A). It recovers the Altmann–Grifo–Montaño–Sanders–Vu and Awadalla–Marley theorems as special cases and adds new corollaries for C-projective, Add C, add C, and contravariantly finite resolving subcategories. The ghost-map proof is coherent; I found no circularity and no fitted parameters.\n\nThe one real soft spot is Lemma 3.6, stated without proof. The stress-test is right: in the final paragraph of Theorem 3.12, the splitting argument needs Ext^1(C^{-g+1}X, X^{-g}) to be isomorphic to Ext^1(C^{-g+1}P, X^{-g}), and that is exactly Lemma 3.6. The lemma quantifies over all v and all M in X^⊥, and the module proof in [6] may rely on module-specific facts. I suspect the lemma is true and the generalization is routine, but 'we omit the proof' is not enough for a load-bearing step in an arbitrary abelian category. This is a fixable gap, not a fatal one, but the main theorem is conditional until it is proved.\n\nLess concerning: Proposition 3.7 invokes [14, Lemma 2.4] as a black box, and Section 5 leans on [10]. Both are published, but a referee should check that the cited statements cover the full generality used.\n\nBottom line: the framework is useful and the applications are real. Homological algebraists working on levels, Gorenstein dimensions, and resolving subcategories will want to read it, and the paper deserves a serious referee. My recommendation: send it to peer review, with a report asking for a proof of Lemma 3.6 (or a precise reference that covers the abelian-category version). I would be comfortable with conditional acceptance after that.","headline":"A genuinely new and useful generalization of level lower bounds, but the proof currently hinges on an unproved load-bearing lemma.","tokens_in":21837,"tokens_out":6025,"would_cite":false,"duration_ms":66247,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13C60","13D09","18G80"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a universal lower bound on the level of a complex in the bounded derived category of any abelian category, expressed through its resolution dimension and the infimum of its cohomology.","keywords":["level","derived category","resolving subcategory","ghost lemma","resolution dimension","Gorenstein projective","semidualizing module"],"falsifier":"Try to construct a resolving subcategory X satisfying condition (A) and a bounded complex M for which level^X M is strictly less than X-resol.dim M + inf M + 1; for example, search over finite-dimensional algebras and their contravariantly finite resolving subcategories. Alternatively, directly test Lemma 3.6: take a quasi-isomorphism between two right-bounded complexes of X-objects that are not homotopy equivalent, and an M in X^⊥, and compute Ext^i(C^v Q, M) and Ext^i(C^v P, M) for some i and v; any difference would falsify the lemma and hence the theorem as proved.","tokens_in":20810,"feed_emoji":"🧮","tokens_out":10575,"duration_ms":97610,"temperature":0.7,"pith_summary":"This paper proves a single general inequality that controls the 'level' of a complex — the number of mapping-cone steps needed to assemble it from a chosen subcategory — in the bounded derived category of any abelian category. For any resolving subcategory X satisfying a short-exact-sequence condition (A), every nonzero complex M satisfies level^X M ≥ X-resol.dim M + inf M + 1. The level measures how hard M is to construct from X, while X-resol.dim measures how well M can be approximated by bounded complexes of objects of X; the inequality says the construction cost must exceed the approximation length by at least the cohomological range. The theorem recovers the known lower bounds for projective and Gorenstein projective levels and extends the same bound to Gorenstein injective, semidualizing-module, and contravariantly finite resolving subcategories.","feed_headline":"Levels of complexes obey one universal bound","feed_subtitle":"The proof recovers known bounds and extends them to Gorenstein and semidualizing subcategories.","key_machinery":"The central object is the ghost map — a morphism in the derived category that induces zero on all Ext groups with objects of X — together with the Ghost Lemma, which limits how many ghost maps can compose before they vanish when the source has bounded level. The proof also relies on condition (A) to construct, by descending induction, a bounded representative of M whose low-degree terms lie in X∩X^⊥; a quasi-isomorphism-invariance lemma (Lemma 3.6) and a splitting lemma (Lemma 3.4) then carry the contradiction. The named identity is the inequality level^X_{D^b(A)} M ≥ X-resol.dim M + inf M + 1, which packages the whole argument.","core_discovery":"The central discovery is Theorem 3.12: let A be an abelian category with enough projective objects, and X a resolving subcategory satisfying condition (A) — for every X in X, there is a short exact sequence 0 → X → Y → X' → 0 with Y ∈ X∩X^⊥ and X' ∈ X. Then for every nonzero object M in D^b(A), the X-level of M is bounded below by X-resol.dim M + inf M + 1. The proof runs by contradiction: if the level u is finite, then M admits a bounded complex whose low-degree terms lie in X∩X^⊥, and composing the hard-truncation maps yields a composite of exactly X-resol.dim M + inf M ghost maps. The Ghost Lemma forces the level of M to be strictly larger than that number, and a splitting argument rules out the possibility that this composite is zero. The argument yields the inequality as a universal statement, independent of the particular abelian category or resolving subcategory, provided condition (A) holds.","pith_inferences":["The theorem suggests that the level of a complex is never smaller than the length of the shortest 'resolution gap' plus one; this can be read as a quantitative version of the intuition that constructing a complex from a subcategory is at least as hard as resolving it.","Condition (A) is satisfied in many natural settings (projectives, Gorenstein projectives, semidualizing-module classes, contravariantly finite resolving subcategories with the sub-Sub condition); the theorem indicates these are all instances of a single structural phenomenon rather than separate results.","If Lemma 3.6 were proven to fail in some category, the theorem would still hold for all instances where the lemma holds; a natural test is to search for a resolving subcategory where the quasi-isomorphism invariance of Ext into X^⊥ fails, which would carve out the exact boundary of the theorem's applicability.","The inequality may be sharp: examples where equality holds would give a lower bound that is exactly the obstruction to building the complex from X in fewer steps; the paper does not address sharpness, but the form of the bound suggests that resolution dimension plus infimum is the exact cost in many module categories."],"forward_implications":["For X = Proj R over a ring R, the theorem reduces to level^Proj M ≥ pd_R M + inf M + 1, recovering the Altmann–Grifo–Montaño–Sanders–Vu bound.","For X = GProj R or GInj R, it recovers and unifies the Awadalla–Marley bounds for Gorenstein projective and injective dimensions.","For X = Add C or add C with C a semidualizing module over a commutative noetherian ring, it gives new lower bounds for C-projective levels.","For any contravariantly finite resolving subcategory X of mod Λ over an artin algebra Λ, the theorem yields a lower bound for X-levels.","The dual theorem (Theorem 3.13) gives the corresponding lower bound for coresolving subcategories with condition (A*) in terms of coresolution dimension and sup M."],"supporting_citations":[{"why":"Introduces the notion of level in a triangulated category, the central invariant studied here.","marker":"[5]"},{"why":"Supplies the ghost lemma, the splitting lemma, and the Gorenstein projective lower bound that the paper generalizes.","marker":"[6]"},{"why":"Provides the projective-level lower bound recovered as Corollary 4.4(1).","marker":"[3]"},{"why":"Gives the resolution-dimension characterization used in Lemmas 3.1 and 3.2.","marker":"[12]"},{"why":"Motivates condition (A) and the approximation construction of Lemma 3.9.","marker":"[13]"},{"why":"Is the source of the ghost lemma cited in the proof of Theorem 3.12.","marker":"[20]"},{"why":"Proves that G_C Proj and tref_C(R) are resolving and satisfy condition (A), enabling Section 5.","marker":"[22]"},{"why":"Supplies Wakamatsu's lemma used in Proposition 6.3 for contravariantly finite subcategories.","marker":"[4]"}],"fun_headline_variants":["Universal floor for levels from resolution dimension","One inequality sets level bound via resolution dimension","Resolution dimension + infimum lower-bounds all levels","Derived levels have a resolution-dimension bound","Universal bound: level ≥ resolved dimension + infimum"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument hinges on an unproved claim about extension groups being preserved when a complex is replaced by a quasi-isomorphic one from the chosen subcategory; if that claim turns out to be false, the proof's central contradiction no longer works.","fun_headline_variants_meta":{"raw":{"variants":["Universal floor for levels from resolution dimension","One inequality sets level bound via resolution dimension","Resolution dimension + infimum lower-bounds all levels","Derived levels have a resolution-dimension bound","Universal bound: level ≥ resolved dimension + infimum"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001428,"raw_usage":{"total_tokens":5716,"prompt_tokens":856,"completion_tokens":4860,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":4790}},"tokens_in":472,"tokens_out":4860,"duration_ms":39019,"temperature":1.0,"reasoning_tokens":4790,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:31:14.862616+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Try to construct a resolving subcategory X satisfying condition (A) and a bounded complex M for which level^X M is strictly less than X-resol.dim M + inf M + 1; for example, search over finite-dimensional algebras and their contravariantly finite resolving subcategories. Alternatively, directly test Lemma 3.6: take a quasi-isomorphism between two right-bounded complexes of X-objects that are not homotopy equivalent, and an M in X^⊥, and compute Ext^i(C^v Q, M) and Ext^i(C^v P, M) for some i and v; any difference would falsify the lemma and hence the theorem as proved.","supporting_citations":[{"cited_title":"Christensen; A","cited_arxiv_id":null,"evidence_quote":"Gives the resolution-dimension characterization used in Lemmas 3.1 and 3.2."},{"cited_title":"Christensen; S","cited_arxiv_id":null,"evidence_quote":"Motivates condition (A) and the approximation construction of Lemma 3.9."},{"cited_title":"Rouquier , Dimensions of triangulated categories, J","cited_arxiv_id":null,"evidence_quote":"Is the source of the ghost lemma cited in the proof of Theorem 3.12."},{"cited_title":"Altmann; E","cited_arxiv_id":null,"evidence_quote":"Provides the projective-level lower bound recovered as Corollary 4.4(1)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the notion of level in a triangulated category, the central invariant studied here."},{"cited_title":"A w adalla; T","cited_arxiv_id":null,"evidence_quote":"Supplies the ghost lemma, the splitting lemma, and the Gorenstein projective lower bound that the paper generalizes."},{"cited_title":"White , Gorenstein projective dimension with respect to a semidua lizing module, J","cited_arxiv_id":null,"evidence_quote":"Proves that G_C Proj and tref_C(R) are resolving and satisfy condition (A), enabling Section 5."},{"cited_title":"Auslander; I","cited_arxiv_id":null,"evidence_quote":"Supplies Wakamatsu's lemma used in Proposition 6.3 for contravariantly finite subcategories."}],"review_version":1}