{"id":"e167ab9e-f9bf-4e46-bef2-fb62112117af","arxiv_id":"2505.14812","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For perfect complexes with power torsion homology, the paper proves level_R F ≥ dim R - dim R/I + 1 and shows the bound is optimal for Koszul complexes.","lead":"This paper proves a lower bound on the homological level of certain finite free complexes over local rings, improving earlier versions of the New Intersection Theorem. The result is used to compute or bound the level of Koszul complexes and to show the bound is optimal.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 3.1 assumes without support that completing a balanced big Cohen-Macaulay algebra with respect to m yields an m-complete balanced big Cohen-Macaulay algebra; this step is load-bearing because the depth inequalities used later require derived m-completeness, but no cited…","rationale":"After checking the internal logic of Theorem 3.1, the derivation of the claim n ≥ dim R − dim R/I + s, the localization argument showing I ⊂ p, and the application of Proposition 2.10 all appear internally consistent. The single point where the written proof depends on an unsupported assertion is the transition from an arbitrary balanced big Cohen-Macaulay algebra to an m-complete one by completion. This is load-bearing because the m-completeness is not optional: without it, Section 2.4 cannot be used to make F ⊗_R S derived m-complete, and the inequalities (2.6.1) and (2.6.2) are not available. The reader's weakest assumption already pointed toward the dependence on a deep existence theorem, but the sharper concern is that the paper states a specific preservation fact—completion preserves balancedness and yields derived m-completeness—for which no reference is provided and which is not generally automatic for non-Noetherian algebras. If the existence theorem for m-complete balanced big Cohen-Macaulay algebras is available, the proof is repairable by a citation change; otherwise the central bound lacks its foundation. The other issues raised by the reader (the minimal generating set hypothesis in Proposition 3.5 and the height IS correction in Corollary 4.1) are real but do not affect Theorem 3.1 itself. I therefore do not move the verdict and support a conditional acceptance pending verification of the completion step.","tokens_in":9505,"tokens_out":42993,"duration_ms":422195,"concrete_test":"Check whether Andre [2], Bhatt [7], or [24, Theorem 2.7] explicitly proves that every noetherian local ring admits a balanced big Cohen-Macaulay algebra that is m-complete (in the derived sense used in Section 2.4). If such a theorem exists, replace the completion sentence in Section 3 with a direct citation and verify the rest of the proof unchanged. If no such theorem is stated, test the completion claim directly: let R = k[[x]] and S = R[y]_{(x,y)}, which is a balanced big Cohen-Macaulay R-algebra; compute S^ = lim S/x^n S and check whether x is a nonzerodivisor in S^ and whether S^ is derived x-complete in the sense of [24]. If regularity or derived completeness fails in this example, the assertion that completing an arbitrary balanced big Cohen-Macaulay algebra yields an m-complete balanced big Cohen-Macaulay algebra is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the first paragraph of the proof of Theorem 3.1 the paper says: 'we take a balanced big Cohen-Macaulay R-algebra, and we complete it with respect to m, obtaining an m-complete big Cohen-Macaulay R-algebra, which we denote by S.' Every subsequent step needs S to have both properties simultaneously. Balancedness is used to assert depth_R S = dim R and depth_Rp S_p = dim R_p in (3.1.2) and in the final equality depth_R S = dim R. Derived m-completeness is used to apply Section 2.4, so that F ⊗_R S is derived m-complete, and then to apply inequalities (2.6.1) and (2.6.2). The cited literature (Hochster-Huneke, Andre, Bhatt) establishes existence of balanced big Cohen-Macaulay algebras, but the paper cites no theorem saying that their m-adic completion is again balanced big Cohen-Macaulay, nor that it is derived m-complete in the sense of [24]. For non-Noetherian algebras, m-adic completion can change the behavior of regular sequences, and classical m-adic completeness does not automatically imply derived completeness. Thus the construction of S is asserted rather than proved. This is not a failure of Andre's theorem itself, but it is a genuine gap in the written proof: if no m-complete balanced big Cohen-Macaulay algebra exists, the inequalities (2.6.1) and (2.6.2) are unavailable and the central bound does not follow as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem 3.1: for a commutative noetherian local ring R, an ideal I, and a finite free R-complex F with H0(F) nonzero, if Hi(F) is I-power torsion for i≥1 and some minimal generator of H0(F) is I-power torsion, then level_R F ≥ dim R − dim R/I + 1. The proof follows the template of the Improved New Intersection Theorem proofs in [1] and [9], using balanced big Cohen-Macaulay algebras, derived m-completeness, and syzygy-level estimates. The paper then derives applications: bounds for Koszul complexes (Proposition 3.2), a free-rank bound for minimal dg-algebras and Koszul complexes (Proposition 3.5), an equality for Lech-independent sequences (Corollary 3.7), and a short proof of the superheight bound (Corollary 4.1). Examples in Section 4 show that the quantity dim R − dim R/I cannot be replaced by superheight or bigheight in general.","tokens_in":9818,"tokens_out":27286,"duration_ms":284515,"significance":"If the main theorem is correct, it is a genuine strengthening of the height-based bound in [6, Theorem 4.2] and of the length-based bound in [9, Theorem 2.2], and the examples in Section 4 show that the new invariant is sharp and incomparable with bigheight and superheight. The proof is a coherent synthesis of existing machinery rather than a fundamentally new method, but the resulting bound is natural and the applications to Koszul complexes are useful. The paper is clearly written and the central derivation is transparent, which makes it easy to check the main steps.","major_comments":[{"comment":"The proof asserts that completing a balanced big Cohen-Macaulay R-algebra with respect to m yields an m-complete big Cohen-Macaulay R-algebra, and then applies (2.6.1) and (2.6.2) to this S. This step is load-bearing but is not justified by the cited existence theorems. The completion of a non-Noetherian algebra need not preserve balancedness, and the subsequent applications require S to be derived m-complete, not merely classically m-adically complete. Please provide a proof or a precise citation for the existence of an m-complete balanced big Cohen-Macaulay algebra with the derived-completeness property used in (2.6.1)–(2.6.2).","section":"Section 3, proof of Theorem 3.1, first paragraph and equations (2.6.1)–(2.6.2)"},{"comment":"The sentence \"It is easy to see that H1(g⊗k) is an isomorphism\" is false without an additional minimality hypothesis on x. For example, take R = k[[t]], I = (t), and x = (t,0); then the map H1(g⊗k): k^2 → Tor_1^R(R/I,k) ≅ k has rank one and is not an isomorphism. As written, the proof of Proposition 3.5 does not establish the stated bound for an arbitrary generating set x. The statement should require that x is a minimal generating set of I, or the proof must be modified to reduce to a minimal generating subsequence and control the additional Koszul factors.","section":"Section 3.4, Proposition 3.5 and its proof"}],"minor_comments":[{"comment":"The abstract says \"with power torsion homology\" and \"a power torsion minimal generator\"; the word \"I-\" is missing, and this makes the hypotheses ambiguous on first reading.","section":"Abstract and Introduction"},{"comment":"The sentence \"The proof of Theorem 2.2 in [9] shows that I ⊆ p\" is terse; spelling out the supporting argument or citing the exact statement in [9] would improve readability.","section":"Section 3, proof of Theorem 3.1"},{"comment":"The notation \"TorR(R/I, k)\" and \"TorR(R/I, k) ≅ B ⊗_k Λ\" is ambiguous; it should be written as Tor^R or Tor^R_* to indicate the graded Tor algebra.","section":"Section 3.4, Proposition 3.5"},{"comment":"In the diagram for the factorization through K(x2,...,xn; R), the middle row is visually confusing; labeling the Koszul complex and its differential explicitly would make the example easier to follow.","section":"Section 4.3"}],"recommendation":"major_revision","confidential_remarks":"The central theorem appears defensible and the paper is a competent contribution to local algebra, but the proof of Theorem 3.1 contains a genuine gap concerning the existence of an m-complete balanced big Cohen-Macaulay algebra, and Proposition 3.5 is stated too broadly. Both issues are local and fixable, so I recommend major revision rather than rejection. The paper's novelty and scope are appropriate for the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Kekkou proves that for a finite free complex over a local ring, if the positive-degree homology is I-power torsion and a minimal generator of H0 is I-power torsion, then level_R F >= dim R - dim R/I + 1. This genuinely improves the height-based bound in Avramov-Iyengar-Neeman and the length-based bound in Christensen-Ferraro, and the examples show it is sharp for Koszul complexes on systems of parameters. The proof is assembled from existing machinery - syzygy level bounds, local cohomology depth inequalities, and balanced big Cohen-Macaulay algebras - and I did not find a mistake in the main chain of inequalities. The application to Lech-independent sequences (Corollary 3.7) is a clean, nice statement.\n\nNow the soft spots, in proportion.\n\nProposition 3.5 silently requires that x is a minimal generating set of I; as written, H1(g tensor k) need not be an isomorphism. This is a small fix. Corollary 4.1 prints height I in the proof where height(IS) is needed to conclude via superheight; also minor.\n\nThe more substantial issue is the first step in the proof of Theorem 3.1. The paper takes a balanced big Cohen-Macaulay R-algebra and completes it with respect to m, claiming that this gives an m-complete big Cohen-Macaulay algebra S. Every later step needs S to be both balanced big CM and derived m-complete. The cited literature establishes existence of balanced big CM algebras, but I do not see a quoted theorem or argument showing that m-adic completion preserves the balanced property, nor that classical m-completion gives derived m-completeness in the sense of [24]. Since S is not noetherian and not finitely generated, this is not automatic. This is a genuine gap in the written proof. I suspect it is repairable - some constructions of big CM algebras (notably Andre's) do yield m-complete algebras - but as written it is unsupported.\n\nIf that completion step can be justified, the main theorem stands. The reader's two application-side comments are right, and the stress-test note has a real point. I would send this to peer review; a competent referee can sort out the big CM completion step. The paper is a solid contribution, not a new framework, but it is honest, clearly written, and the main inequality is new.","headline":"A solid strengthening of level bounds that deserves a serious referee, but the proof has a load-bearing gap in completing a big Cohen-Macaulay algebra that needs to be justified or fixed.","tokens_in":660,"tokens_out":1669,"would_cite":true,"duration_ms":56794,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-07T15:30:04.520248+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}