{"id":"5367eaeb-93b5-4a37-ac76-9c30a320ce6c","arxiv_id":"2412.01392","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For certain local rings, the authors prove a depth inequality for tensor products under a 'complexity plus one' condition, give a counterexample to a relaxed version, and prove a weaker bound using syzygy numbers.","lead":"This math paper studies when the depth of a tensor product of two modules over a local ring is forced to be as large as a known bound. It extends a known theorem to a broader class of rings and gives a counterexample to an open question from 2015 about weakening a hypothesis.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.2(b) uses R-syzygies where Q-syzygies are needed; as written x cannot be regular on M1, so the self-contained proof of the TE case has a repairable gap.","rationale":"After going through the proof, the main load-bearing point is indeed the TE lifting lemma, not the broad architecture of Theorem 1.2. The induction in part (a) depends on Lemma 4.3, which depends on Lemma 4.2 to keep the deformed ring TE; a failure there would remove the inductive hypothesis. The specific error is mechanical: an R-syzygy of an R-module is an R-module, so it is annihilated by x, while a Q-syzygy can be x-torsion-free. Replacing Ω_R by Ω_Q makes the proof coherent. The base case cx=0 is also terse: it uses Lemma 4.1 without spelling out that M locally free on the punctured spectrum makes Tor^R_i(M,N) finite length, so the depth(Tor)=0 alternative in Lemma 4.1 holds. This is repairable by an explicit sentence. The rest of the TE argument, including Lemma 4.5 and the induction step, appears algebraically sound after these corrections; the complete intersection case is standard, and Example 5.1 correctly answers [14, 3.10]. Because the identified problem is a clear typo with an independent reference supporting the lemma, the appropriate verdict remains conditional: the mathematical claims are likely true, but the paper as written needs a revision before the TE case is self-contained.","tokens_in":18157,"tokens_out":30721,"duration_ms":252310,"concrete_test":"Re-derive Lemma 4.2(b) with Ω^r_Q in place of Ω^r_R and check each step explicitly: (1) high Q-syzygies are totally reflexive because Q is Gorenstein; (2) x is regular on M1 and N1; (3) the Rees isomorphism Ext^{i+1}_Q(M1/xM1,N1) ≅ Ext^i_R(M1/xM1,N1/xN1) applies; (4) ET over R and the change-of-rings sequence force Tor^Q_i(M1/xM1,N1/xN1)=0 for i≫0; (5) tensor the short exact sequences with N1/xN1 and M1 and apply Nakayama to conclude Tor^Q_i(M1,N1)=0 for i≫0. If the corrected proof succeeds, the manuscript needs only a typographical revision. If it cannot be completed, Theorem 1.2(a) would rest entirely on [24, 2.5(3)] rather than on the paper's own argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step in Theorem 1.2(a) is the reduction to a codimension-one deformation Q (Lemma 4.3), which requires the deformed ring Q to inherit the TE property from R. That is exactly Lemma 4.2. In the proof of Lemma 4.2(b), the authors set M1 = Ω^r_R M and N1 = Ω^r_R N and then assert that x is regular on M1 and N1. This is impossible: R = Q/(x) annihilates every R-module, so the action of x on any R-module is zero. The intended construction must be Ω^r_Q, using high Q-syzygies; since Q is Gorenstein these are totally reflexive and x is regular on them. With that substitution, the subsequent Rees isomorphism, the change-of-rings sequence for Tor, and the final Nakayama argument appear to go through. The authors also note that the TE case was independently proved in [24, 2.5(3)], so this is a gap in the paper's self-contained proof rather than evidence against the theorem. A separate wording problem in Lemma 4.3, which passes to the completion while writing 'M is isomorphic to M̂', reinforces that the TE section needs a careful correction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the depth of tensor products of finitely generated modules over local rings. Its main result, Theorem 1.2, states that under either (a) R is a complete local TE ring with infinite residue field, vpd_R(M)<∞, and M is locally free on the punctured spectrum, or (b) R is a local complete intersection and all higher Tor modules of M,N have finite length, the condition depth_R(M⊗_R N) ≥ cx_R(M,N)+1 forces the depth inequality depth_R(M⊗_R N) ≥ depth_R(M)+depth_R(N)−depth_R(R), with equality iff the pair is Tor-independent. Part (b) recovers a result of Celikbas, Sadeghi, and Takahashi; part (a) is new. The paper also constructs a counterexample (Example 5.1) to a question of Celikbas--Sadeghi--Takahashi and proves a weaker lower bound, Theorem 1.3, under the relaxed hypothesis λ(Tor_i^R(M,N))<∞ for i≫0. Section 3 develops lifting results for hulls and approximations, and Section 4 carries out an inductive complexity reduction.","tokens_in":18346,"tokens_out":12565,"duration_ms":105278,"significance":"If the main theorem is correct, the paper provides the first TE-ring analogue of the Huneke--Wiegand/Celikbas--Sadeghi--Takahashi depth bounds, in addition to a clean recovery of the complete intersection case and a definitive negative answer to [14,3.10]. The lifting lemmas in Section 3 are potentially useful and are proved in detail. The proof strategy is coherent and uses standard tools: projective hulls, quasilifting, complexity, and change-of-rings sequences. However, the self-contained proof of the TE case contains a repairable but genuine gap in Lemma 4.2(b), and Lemma 4.3 has an unclarified completion step; these need to be fixed before the new result can be regarded as fully established.","major_comments":[{"comment":"The proof of Lemma 4.2(b) is not self-consistent as written. It sets M_1 = Ω^r_R M and N_1 = Ω^r_R N and then asserts that x is regular on M_1 and N_1. This is impossible: since R = Q/(x), every R-module is annihilated by x, so multiplication by x on any nonzero R-module is zero. The intended construction must use high Q-syzygies, e.g. M_1 = Ω^r_Q M and N_1 = Ω^r_Q N; since Q is Gorenstein, these are totally reflexive and x is regular on them. With that replacement, the subsequent Rees isomorphism, the change-of-rings sequence, and the Nakayama conclusion are plausible. Because Lemma 4.2 is used in Lemma 4.3 to inherit the TE property in the complexity-reduction step, this gap affects the proof of Theorem 1.2(a). The authors' citation of [24,2.5(3)] may cover the TE case externally, but it does not repair the proof given here.","section":"Lemma 4.2(b)"},{"comment":"The proof of Lemma 4.3 contains an unjustified completion step. The sentence 'As M is finitely generated, M ≅ \\hat M' is false unless R is complete; at best one has M ⊗_R \\hat R ≅ \\hat M. The proof then says 'By taking completion, if necessary, we can assume Q is complete,' while the statement of the lemma requires R = Q/(x). If the lemma is intended only for complete R, the statement should say so; if it is intended for arbitrary local R, the proof must explain how the completed construction descends to R, or replace (i) by \\hat R = Q/(x). This is load-bearing because Lemma 4.3 is the complexity-reduction step in Theorem 1.2(a).","section":"Lemma 4.3"}],"minor_comments":[{"comment":"In the last sentence of the proof of Lemma 4.2, 'the implication follows from the first assertion above' should presumably refer to the second assertion, since part (a) is being deduced from the ET statement proved for Q. Please correct this cross-reference.","section":"Lemma 4.2, proof"},{"comment":"The term 'complete TE ring' is not defined. If it means 'complete local TE ring', say so explicitly; if it means 'complete intersection TE ring', note that the complete intersection property already implies TE, so the phrase would be redundant.","section":"Lemma 4.3"},{"comment":"The assertion 'The hypothesis on M implies λ(Ext^i_R(M,N))<∞ for all i≫0' is not immediate and should be justified. It follows from local freeness of M on the punctured spectrum (or from the cited argument in [12,2.4]), but the reader should not have to supply this.","section":"Lemma 4.3, proof"},{"comment":"The final paragraph of the proof treats the complete intersection case very tersely. Since Lemma 4.4 produces a deformation of \\hat R, not of R, the proof should explicitly verify that depth, Tor vanishing, and complexity are preserved under completion, and that the induction is applied to (\\hat M, \\hat N) over \\hat R. This would make the recovery of [14,1.2] fully rigorous.","section":"Theorem 1.2, proof, part (b)"},{"comment":"There are several minor typographical issues, including 'verticle' for 'vertical' in the proof of Proposition 3.5 and 'T ors' in the proof of Lemma 4.4. A careful proofreading pass is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The new TE-ring result is plausible and the main gap is repairable by replacing R-syzygies with Q-syzygies in Lemma 4.2 and by clarifying the completion reduction in Lemma 4.3. I would not reject the paper on the basis of these issues. The authors should also position their contribution carefully relative to [24,2.5(3)], since they themselves note that the TE lifting statement already appears there; the report should make clear exactly which parts of Theorem 1.2(a) are new and which are imported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main news: the paper settles an open question from Celikbas–Sadeghi–Takahashi, gives a TE-ring analogue of their depth bound, and provides a clean counterexample in Example 5.1. Theorem 1.2(a) is genuinely new, and Theorem 1.3's syzygy lower bound is a reasonable weaker substitute. The lifting machinery in Section 3 is useful on its own. Part (b) recovers [14,1.2] but the proof is self-contained and the HJW quasilifting setup is a sensible framework. The stress-test concern is real. Lemma 4.2(b) sets M1 = Ω^r_R M and then asserts x is regular on M1, which is impossible because R is annihilated by x. The intended construction must be Ω^r_Q, using high Q-syzygies. Once you make that substitution, the Rees isomorphism, the change-of-rings sequence, and the Nakayama argument appear to go through. The authors also disclose that [24,2.5(3)] proves the same TE lifting, so the theorem is not in danger but the self-contained proof needs a correction. Lemma 4.3 has a similar wording issue: it writes 'M is isomorphic to M̂' when it really passes to the completion. Both are mechanical and repairable. The citation pattern is honest. No self-citations, the overlap with [24] is disclosed, and Theorem 4.10(b) is explicitly presented as a restatement of [14,3.1]. The results derive from external theorems rather than assuming the target conclusion. This is not a breakthrough but a solid extension within an established research program. The counterexample is simple and convincing, and the TE-ring result gives specialists a new tool. The paper is for commutative algebraists working on Tor vanishing and depth formulas. Recommendation: send it to peer review. A serious referee should ask for the Lemma 4.2 and Lemma 4.3 corrections, but the central mathematics is defensible and worth having.","headline":"Solid, honest paper that answers an open question and proves a TE-ring depth bound; the only real problems are two repairable typos in the self-contained proof.","tokens_in":659,"tokens_out":1627,"would_cite":true,"duration_ms":32099,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D07","13C14","13C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single lower bound on $\\operatorname{depth}_R(M\\otimes_R N)$ forces the classical depth formula on complete intersections and TE rings, with equality detecting Tor-vanishing.","keywords":["tensor products of modules","depth formula","complete intersection rings","TE rings","Tor-vanishing","complexity of modules","Gorenstein dimension","hulls and approximations"],"falsifier":"Re-run Lemma 4.2 with the $Q$-syzygy $\\Omega_Q^r M$ in place of the $R$-syzygy $\\Omega_R^r M$; the printed proof cannot be correct as written because $x$ annihilates every $R$-module. Independently, any example of a non-TE local ring $Q$ with a non-zero-divisor $x$ such that $Q/(x)$ is TE would refute the lemma and therefore Theorem 1.2(a).","tokens_in":17877,"feed_emoji":"📐","tokens_out":13196,"duration_ms":110706,"temperature":0.7,"pith_summary":"The paper aims to turn one numerical inequality about tensor products into structural conclusions: if $\\operatorname{depth}_R(M\\otimes_R N)\\ge \\operatorname{cx}_R(M,N)+1$ for nonzero finitely generated modules over a local complete intersection ring, or over a complete local TE ring with $M$ of finite virtual projective dimension and locally free off the maximal ideal, then the classical depth formula $\\operatorname{depth}_R(M\\otimes_R N)\\ge \\operatorname{depth}_R(M)+\\operatorname{depth}_R(N)-\\operatorname{depth}_R(R)$ holds, with equality exactly when the pair is Tor-independent. A TE ring is one in which eventual vanishing of Tor forces eventual vanishing of Ext; the complete intersection case recovers a theorem from [14]. The paper also answers negatively a question left open in [14]: the finiteness of all $\\lambda(\\operatorname{Tor}_i^R(M,N))$ cannot be relaxed to finiteness only for all sufficiently large $i$. With the relaxed hypothesis, it proves a weaker lower bound in which the two depths are replaced by the syzygy orders $\\operatorname{syz}_R(M)$ and $\\operatorname{syz}_R(N)$. A reader should care because a depth inequality alone can certify Tor-vanishing, and the result extends this detection principle from complete intersections to the broader TE class.","feed_headline":"One depth inequality forces the classical depth formula","feed_subtitle":"For complete intersections and TE rings, depth(M⊗N) ≥ complexity+1; equality detects Tor-vanishing.","key_machinery":"The engine is the quasilifting construction of [18]: for $R=Q/(x)$ with $x$ regular on $Q$, a projective hull of $M$ over $R$ determines an approximation $0\\to\\widetilde X\\to\\widetilde G\\to M\\to0$ over $Q$ in which $\\widetilde G$ is totally reflexive, meaning isomorphic to its double dual with no higher Ext to the ring, and $\\operatorname{pd}_Q(\\widetilde X)=\\operatorname{G-dim}_R(M)$. The proof combines this with a complexity-reduction result from [12], yielding $\\operatorname{cx}_Q(\\widetilde G,N)=\\operatorname{cx}_R(M,N)-1$, and with the change-of-rings long exact sequence and an acyclicity lemma that transfer vanishing and depth information between $R$ and $Q$. The same projective-hull and approximation machinery, iterated, produces the module $L$ in Lemma 5.3 that underlies the syzygy-level bound of Theorem 1.3.","core_discovery":"The central claim, Theorem 1.2 (restated as Theorem 4.10), is that under the depth bound the following are equivalent for either ring class: the pair is Tor-independent; the depth formula holds; the depth inequality $\\operatorname{depth}_R(M)+\\operatorname{depth}_R(N)\\ge \\operatorname{depth}_R(R)+\\operatorname{cx}_R(M,N)$ holds; and the eventual vanishing of Tor or Ext holds. The proof achieves this by induction on complexity. A projective hull $0\\to M\\to X\\to G\\to0$ is lifted along a one-step deformation to a $Q$-level approximation whose complexity is one less, and the TE or complete intersection property is used at the base case where complexity is zero. In the last section, the paper constructs an explicit pair of modules over a two-dimensional complete intersection ring satisfying the depth inequality with complexity zero but with nonzero first Tor, showing that the theorem's length hypothesis cannot be weakened to eventual finiteness; Theorem 1.3 substitutes a bound in terms of syzygy order.","pith_inferences":["If TE rings are closed under localization, a question the paper leaves open, then the locally-free-on-the-punctured-spectrum condition in Theorem 1.2(a) could likely be replaced by the length-finiteness condition used in part (b).","The counterexample's construction, taking a module of projective dimension one and its annihilator, looks portable to other rings, so the negative answer to the relaxed question may hold well beyond two-dimensional complete intersections.","The syzygy-order bound in Theorem 1.3 suggests a hierarchy of weaker depth formulas controlled by how many steps back a module is a syzygy; applying the same argument to higher syzygies could yield interpolating bounds.","Should Lemma 4.2's lifting claim survive a corrected proof, the same induction would apply to modules of finite complete intersection dimension whenever a TE analogue of the deformation reduction exists, removing the virtual projective dimension hypothesis."],"forward_implications":["A depth check alone certifies Tor-vanishing: under the theorem's hypotheses, $\\operatorname{depth}_R(M\\otimes_R N)\\ge\\operatorname{cx}_R(M,N)+1$ implies $\\operatorname{Tor}_i^R(M,N)=0$ for all $i\\ge1$.","The classical depth formula follows from the same inequality, so lower bounds on tensor depth are not just bounds; they force the full depth formula.","The complete intersection case reproduces the known result of [14], while the TE case gives the same conclusion for rings strictly between complete intersections and Gorenstein rings.","The relaxed hypothesis, finite length only for $i\\gg0$, fails: Example 5.1 has $\\operatorname{depth}_R(X\\otimes_R C)=\\operatorname{cx}_R(X,C)+1$ but $\\operatorname{Tor}_1^R(X,C)\\ne0$.","Under the relaxed hypothesis, the substitute bound $\\operatorname{depth}_R(M\\otimes_R N)\\ge\\operatorname{syz}_R(M)+\\operatorname{syz}_R(N)-\\operatorname{depth}_R(R)$ still holds."],"supporting_citations":[{"why":"Gives the complete-intersection depth bound and the open question that the paper recovers and answers.","marker":"[14]"},{"why":"Supplies the quasilifting construction that lifts projective hulls across a deformation.","marker":"[18]"},{"why":"Proves that complete intersections are TE and supplies asymptotic Ext/Tor vanishing facts.","marker":"[6]"},{"why":"Provides the complexity-reduction step used in the induction on cx.","marker":"[12]"},{"why":"Characterizes TE rings as Gorenstein ET rings, a step used in Lemma 4.2.","marker":"[23]"},{"why":"Independently proves the TE lifting case; the paper cites it as confirmation of Lemma 4.2.","marker":"[24]"},{"why":"Gives the depth formula for Tor-independent pairs with finite CI-dimension, used repeatedly in the proofs.","marker":"[1]"},{"why":"Supplies the depth formula for modules of finite complete intersection dimension used in the reduction.","marker":"[22]"}],"fun_headline_variants":["Depth inequality forces classical depth formula","Single depth bound yields Tor-vanishing and depth formula","Counterexample shows depth bound cannot be weakened","Lifting hulls gives new depth formula for TE rings"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Part (a) of the main theorem depends on the assertion that the TE property lifts from a quotient ring $R=Q/(x)$ to the deformation $Q$; the proof of that lifting step, Lemma 4.2, appears to use an $R$-syzygy where a $Q$-syzygy is required, so this lifting lemma is the premise whose failure would leave the TE case unproved.","fun_headline_variants_meta":{"raw":{"variants":["Depth inequality forces classical depth formula","Single depth bound yields Tor-vanishing and depth formula","Counterexample shows depth bound cannot be weakened","Lifting hulls gives new depth formula for TE rings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000449,"raw_usage":{"total_tokens":2218,"prompt_tokens":849,"completion_tokens":1369,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":465,"completion_tokens_details":{"reasoning_tokens":1310}},"tokens_in":465,"tokens_out":1369,"duration_ms":11492,"temperature":1.0,"reasoning_tokens":1310,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:26:06.129586+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run Lemma 4.2 with the $Q$-syzygy $\\Omega_Q^r M$ in place of the $R$-syzygy $\\Omega_R^r M$; the printed proof cannot be correct as written because $x$ annihilates every $R$-module. Independently, any example of a non-TE local ring $Q$ with a non-zero-divisor $x$ such that $Q/(x)$ is TE would refute the lemma and therefore Theorem 1.2(a).","supporting_citations":[{"cited_title":"Celikbas, A","cited_arxiv_id":null,"evidence_quote":"Gives the complete-intersection depth bound and the open question that the paper recovers and answers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the quasilifting construction that lifts projective hulls across a deformation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves that complete intersections are TE and supplies asymptotic Ext/Tor vanishing facts."},{"cited_title":"Celikbas and H","cited_arxiv_id":null,"evidence_quote":"Provides the complexity-reduction step used in the induction on cx."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Characterizes TE rings as Gorenstein ET rings, a step used in Lemma 4.2."},{"cited_title":"Araya and Y","cited_arxiv_id":null,"evidence_quote":"Gives the depth formula for Tor-independent pairs with finite CI-dimension, used repeatedly in the proofs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the depth formula for modules of finite complete intersection dimension used in the reduction."}],"review_version":1}