{"id":"101ad56a-5bf2-4044-9825-d84952405cbe","arxiv_id":"2505.00441","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Cohen-Macaulay local rings, the left and right depth formula conditions are characterized by the Uniform Auslander Condition and a newly introduced Uniform Buchweitz Condition, respectively.","lead":"This paper introduces two conditions, called (ldep) and (rdep), that control the depth of tensor products of modules over Cohen-Macaulay local rings. It proves that these conditions are equivalent to known uniform bounds on Ext vanishing, and derives a formula for the degree of the last nonzero Tor group.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.4's (3)⇒(4) applies Lemma 3.9 to f_M⊗f_N although Lemma 3.9 concerns f_M⊗N; the implicit replacement f_N≈N is false in general.","rationale":"The reader's flagged step is exactly the load-bearing weakness. The proof as written has a mismatch between Lemma 3.9 and the modules it is applied to, and the natural fix (f_N≅N⊕free) is false in general; hence the inference cannot be accepted without additional argument. The main theorem's equivalence (1)–(6) depends on this transfer, so a conditional verdict is appropriate. I do not see a separate equally serious concern: the other apparent misreferences (condition (5) for (7), Proposition 3.9 for Lemma 3.9) are typos consistent with an unpolished proof, but they do not by themselves falsify the theorem. No machine-checked proof or reproducible code is provided. Theorem 1.2 and Section 6 are less affected, but Theorem 1.1 is central. Thus no change to the reader's conditional verdict is needed.","tokens_in":28438,"tokens_out":23786,"duration_ms":256033,"concrete_test":"Re-derive the inference from Lemma 3.9 by explicitly matching variables: applying Lemma 3.9 to the modules f_M and e_N yields Tor_{1≤i≤d}(f_M/x f_M, e_N)=0, not the desired vanishing. To test the implicit replacement, compute in R=k[[t^3,t^4,t^5]] with N=R and x=t^3: verify that Ω^1(R/xR)=m and that m is not free over R while R is free; this disproves e_N≅N⊕free in general. If no amended argument is supplied, the proof has a genuine gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Load-bearing gap: in Theorem 4.4, step (3)⇒(4) (proof, pp. 15–16), condition (3) is applied to A=M/xM and B=N/xN, giving q(f_M,e_N)=0 and f_M⊗e_N MCM, where f_M=Ω^d(M/xM) and e_N=Ω^d(N/xN). The text then says 'Lemma 3.9 gives Tor_{1≤i≤d}(M,N)=0 and M⊗N is MCM.' But Lemma 3.9 as stated is asymmetric: its condition (3) is f_M⊗N MCM for the original pair (M,N), not f_M⊗f_N MCM. To invoke it one would need to replace e_N by N in the MCM factor, which requires a stable isomorphism e_N≅N up to free summands. This is not proved and is generally false: for a 1-dimensional CM domain R and N=R, with x a nonzero divisor, e_N=Ω^1(k)=m, while R is free and m is not. Since (3) is otherwise connected to (4) only through this transfer, the equivalence of (1)–(6) in Theorem 1.1 is not established by the written proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies depth inequalities for derived tensor products over commutative Noetherian local rings. It introduces left- and right-hand depth formulas (ldep) and (rdep), together with derived variants, and proves that for Cohen-Macaulay rings derived (ldep) is equivalent to the uniform Auslander condition with bound equal to the dimension; it introduces a dual uniform Buchweitz condition for (rdep), proves transfer properties under regular sequences and completion, gives examples showing failure of localization, and extends a formula of Jorgensen for q_R(M,N). The main theorems are Theorem 1.1 (an eight-condition equivalence for (ldep)), Theorem 1.2 (an analogous statement for (rdep)), and Theorem 1.3 (a local-to-global formula for q_R(M,N)).","tokens_in":28748,"tokens_out":57431,"duration_ms":582938,"significance":"The paper is well organized and generally carefully written, with detailed proofs and several illuminating examples. The proposed equivalence between derived (ldep) and the uniform Auslander condition is a strong and natural structural result if it holds, and the introduction of (UBC) as a dual notion for (rdep) is a useful contribution. The extension of Jorgensen's formula is a genuine added value. However, the proof of the central equivalence contains a gap in the (3) implies (4) step of Theorem 4.4, so the main result is not yet fully established as written.","major_comments":[{"comment":"The proof asserts that once f_M tensor^L_R e_N is MCM and q_R(f_M,e_N)=0, Lemma 3.9 gives Tor^R_{1 <= i <= d}(M,N)=0 and M tensor_R N MCM. This does not follow from the stated Lemma 3.9: its condition (3) is the MCM property of f_M tensor_R N for the original pair (M,N), not of f_M tensor_R e_N. Passing from e_N to N requires a stable isomorphism Omega^d_R(N/xN) congruent to N plus a free module, which is false in general. For example, with R = k[[t^2,t^3]], d = 1, N = m, and x = t^2, one has N/xN isomorphic to k^2, so e_N = Omega^1_R(N/xN) is isomorphic to m direct sum m; if this were stably isomorphic to N = m, ranks would force m direct sum m direct sum R^a isomorphic to m direct sum R^{a+1}, while the minimal number of generators would give 4+a = a+3, a contradiction. Since this step is the only argument linking condition (3) to condition (4), the equivalence of (1)-(6) in Theorem 1.1 is not established as written.","section":"Theorem 4.4, proof of (3) implies (4) (pp. 15-16)"},{"comment":"The displayed identity b_R(M,N^vee) = b_R(A,N^vee) + d has the wrong sign: for M = Omega^d_R(A), dimension shifting gives Ext^i_R(M,N^vee) isomorphic to Ext^{i+d}_R(A,N^vee) for i > 0, so the shift is by -d, not +d. The intended conclusion b_R(M,N^vee) = 0 can nevertheless be recovered directly from condition (5), because M is MCM and hence codepth_R(M) = 0; the erroneous formula should be removed or corrected.","section":"Theorem 4.4, proof of (5) implies (3) (p. 16)"}],"minor_comments":[{"comment":"In the (4) implies (7) part, the text says 'the condition of (3) forces q_R(M,N^vee)=0 and M tensor_R N is MCM'; it should refer to condition (4), and the module obtained as MCM is M tensor_R N^vee, from which Proposition 3.8 then gives b_R(M,N)=0.","section":"Theorem 4.4, proof of (4) implies (7)"},{"comment":"The paragraph labeled 'Next we show (7) implies (4)' invokes condition (5) before (5) has been proved; since the authors later prove (7) implies (5), the proof can be reordered, but as written the paragraph is logically premature and should be relabeled or moved.","section":"Theorem 4.4, proof structure"},{"comment":"The phrase 'derived (rdep) holds on SpecR for the module M' should read 'derived (rdep) holds for the local rings R_p' or similar; the hypotheses are ring conditions, not module conditions, and the current wording is confusing.","section":"Section 6, Lemmas 6.1-6.2 and Theorem 6.3"},{"comment":"The statement that 'neither R nor A satisfies derived (ldep) by Corollary 4.6' is inaccurate for R, since R is Artinian and Corollary 4.6 requires dim(R) > 0; the conclusion for A, and then for B, suffices for the argument.","section":"Example 5.13"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this is a genuinely useful paper, but the proof of Theorem 1.1 has a load-bearing gap at (3)⇒(4). The reader's stress-test is right. In the proof, after showing f_M⊗f_N is MCM (where f_M = Ω^d(M/xM) and f_N = Ω^d(N/xN)), the text says \"Lemma 3.9 gives Tor_{1≤i≤d}(M,N)=0 and M⊗N is MCM.\" But Lemma 3.9 as stated concerns f_M⊗N, not f_M⊗f_N. You would need f_N to be stably isomorphic to N, which is false in general — e.g., for a 1-dimensional CM domain and N=R, f_N is the canonical module's syzygy, essentially the maximal ideal, not free. The gap is not cosmetic: it is the step that connects condition (3) to the Tor-vanishing and MCM conclusions, so the equivalence of (1)–(6) is not established as written.\n\nWhat the paper does well: it splits the depth formula into (ldep) and (rdep), proves that derived (ldep) is equivalent to UAC with b_R = dim(R), introduces a plausible dual condition UBC, extends Jorgensen's formula for q_R(M,N), and gives the first examples showing these conditions need not localize. The writing is clear and the surrounding lemmas are mostly well argued. The q_R formula in Section 6 and the non-localization examples are likely to survive independent of the gap.\n\nWhere I stand: the central idea is good and the paper deserves a serious referee. The gap is real and must be fixed before the main theorem can be trusted, but it does not look obviously fatal — the authors may be able to supply an extra argument or adjust condition (3). I would send this to review, with the explicit instruction that the proof of (3)⇒(4) be repaired. If it is fixable, this is a strong paper. If the authors cannot fix it, the equivalence claim should be weakened.\n\nFor a reading group, this is actually a good paper to work through — the definitions are clean and the gap provides a useful exercise in checking how Lemma 3.9 is being used. I would cite it once the gap is resolved, but not before.","headline":"A useful paper on the depth formula with a real proof gap in the main theorem: (3)⇒(4) of Theorem 4.4 applies Lemma 3.9 to f_M⊗f_N although the lemma requires f_M⊗N.","tokens_in":29231,"tokens_out":1834,"would_cite":false,"duration_ms":19432,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13C15","13D07"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the derived left-hand depth inequality over a Cohen-Macaulay local ring is equivalent to the ring having its dimension as a uniform Auslander bound.","keywords":["depth","tensor product","maximal Cohen-Macaulay","uniform Auslander condition","uniform Buchweitz condition","derived category","Tor","Ext"],"falsifier":"Find a Cohen-Macaulay local ring $R$ of dimension $d>0$ with two maximal Cohen-Macaulay modules $M,N$ satisfying $q_R(M,N)=1$: nonzero Tor in degree 1 and vanishing beyond. Theorem 1.1 predicts that such a ring cannot satisfy derived (ldep), because condition (4) would force $q_R(M,N)=0$. So exhibiting such a ring that does satisfy derived (ldep)—or merely checking whether a ring known to satisfy condition (3) admits such a pair—would settle the equivalence; equivalently, for any candidate ring, computing the Auslander bound $b_R$ and checking whether $b_R=d$ gives a direct numerical test.","tokens_in":28289,"feed_emoji":"🧮","tokens_out":8749,"duration_ms":81464,"temperature":0.7,"pith_summary":"This paper splits the classical depth formula for tensor products over a local ring into two inequalities, called (ldep) and (rdep), and asks when each can hold. Its main theorem is that, over a Cohen-Macaulay local ring $R$ of dimension $d$, the derived left inequality—$\\operatorname{depth}_R(M\\otimes^{\\mathbf{L}}_R N)+\\operatorname{depth}(R)\\ge \\operatorname{depth}_R(M)+\\operatorname{depth}_R(N)$ whenever the derived tensor product has bounded homology—is equivalent to $d$ being the ring's uniform Auslander bound, a strong vanishing condition on Ext. In other words, one global number on Ext-vanishing completely controls whether the left half of the depth formula holds, and the same circle of conditions implies that maximal Cohen-Macaulay modules are Tor-independent whenever their Tor is finite. A dual theorem ties the right inequality to a new 'uniform Buchweitz condition.' The upshot is that depth conditions on tensor products detect representation-theoretic finiteness previously studied in other terms, and they yield a formula for the top nonvanishing Tor degree $q_R(M,N)$ as a supremum of local depth discrepancies.","feed_headline":"One Auslander bound controls tensor-product depth","feed_subtitle":"For Cohen-Macaulay rings, the derived depth inequality is equivalent to a uniform bound on Ext-vanishing.","key_machinery":"The load-bearing objects are the derived tensor product $M\\otimes^{\\mathbf{L}}_R N$, whose depth the paper defines through Ext into the residue field and local cohomology, and the $d$-th syzygy construction $\\Omega^d_R(-)$, normalized so that an MCM complex is one whose local cohomology is concentrated in degree $\\dim(R)$. The paper repeatedly uses $f_M := \\Omega^d_R(M/xM)$ for a maximal regular sequence $x$: it is an MCM module supported on the punctured spectrum, and Lemma 3.9 converts the statement that $f_M \\otimes N$ is MCM into Tor-vanishing between $M$ and $N$. The argument for Theorem 1.1 also passes to the completion to obtain a canonical module and applies local duality, reducing the equivalence to the Uniform Auslander Condition. For the right-hand condition, the new Uniform Buchweitz Condition (UBC) plays the dual role, saying every finite Ext-vanishing $b_R(M,N)$ is at least $\\operatorname{codepth}_R(M)$; the proof reduces (rdep) to constant-rank modules on the punctured spectrum and then to control of nonfree loci via pushforwards along multiplication by a regular element.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1.1: for a Cohen-Macaulay local ring $R$ of dimension $d$, the following are equivalent: derived (ldep); derived (ldep) for modules; the condition that $d$-th syzygies of finite-length modules have maximal Cohen-Macaulay (MCM) derived tensor products whenever their Tor is finite; the condition that MCM modules with finite Tor are Tor-independent with MCM tensor product; the bound $\\operatorname{codepth}_R(M):=\\operatorname{depth}(R)-\\operatorname{depth}_R(M)$ on every finite Ext-vanishing $b_R(M,N)$; and the Uniform Auslander Condition with bound $d$. If this theorem is right, the left half of the depth formula is not a mild hypothesis—it is exactly the statement that the ring's Auslander bound equals its dimension. For Gorenstein rings of positive dimension the paper derives that (ldep) implies (rdep), so the full depth formula follows from the left inequality alone. The companion theorem for (rdep) shows that (1) implies (2), that (2), (3), and (4) are equivalent, that (4) implies (5) implies (6), that (3) implies (7) implies (8), that (2) implies (1) when $d>0$, and that (6) and (8) are equivalent when a canonical module exists. The paper also proves Theorem 1.3: if derived (dep) holds at every localization, then $q_R(M,N)=\\sup\\{\\operatorname{depth}(R_p)-\\operatorname{depth}_{R_p}(M_p)-\\operatorname{depth}_{R_p}(N_p)\\mid p\\in\\operatorname{Supp}(M)\\cap\\operatorname{Supp}(N)\\}$.","pith_inferences":["If Theorem 1.1 is correct, computational searches for rings satisfying (ldep) can be replaced by a single computation of the Auslander bound $b_R$; any ring with $b_R>\\dim(R)$ automatically provides an explicit pair of modules for which the depth formula's left inequality fails.","The internal gap in the proof of (3) implies (4)—Lemma 3.9 is stated for $f_M \\otimes N$ yet applied to $f_M \\otimes f_N$—suggests a stable-isomorphism argument is needed; if no such argument exists, condition (3) may be strictly weaker than derived (ldep), which would be a precise place to look for a counterexample.","Since every Artinian ring satisfies derived (rdep) for modules, the right-hand condition only becomes restrictive in positive dimension; this suggests studying derived (rdep) for complexes, where the paper shows the Artinian obstruction is real, as the genuinely uniform version.","The paper's $q_R$ formula, combined with the observation that derived (dep) localizes for modules of finite complete intersection dimension but not in general, raises the question of exactly which module classes the equality can cover; each such class would give a new extension of Jorgensen's formula."],"forward_implications":["On a Cohen-Macaulay ring of dimension $d$, if the derived left depth inequality holds, then every pair of maximal Cohen-Macaulay modules with finite Tor is Tor-independent and has maximal Cohen-Macaulay tensor product.","For Gorenstein rings of positive dimension, (ldep) implies (rdep), so the full classical depth formula holds whenever its left half does.","The (ldep) and (rdep) conditions ascend and descend along completion and modding out by a regular sequence, but they need not localize; Example 5.15 exhibits a complete Cohen-Macaulay ring satisfying them globally whose localization at a prime fails them.","When derived (dep) holds at every localization, the top nonvanishing Tor degree $q_R(M,N)$ is exactly the supremum of $\\operatorname{depth}(R_p)-\\operatorname{depth}_{R_p}(M_p)-\\operatorname{depth}_{R_p}(N_p)$ over the common support, generalizing Jorgensen's formula to modules beyond complete intersection dimension."],"supporting_citations":[{"why":"Provides the completion lemma and the earlier converse result that makes the Gorenstein case of the depth formula characterize AB-rings.","marker":"[KLOT23]"},{"why":"Supplies ascent and descent of the Auslander condition under completion and the implication that UAC yields the (tr) condition.","marker":"[CH10]"},{"why":"Gives the derived (dep) result for AB-rings, used to conclude (ldep) implies (rdep) in the Gorenstein case.","marker":"[CJ15]"},{"why":"Establishes the depth lemma for complexes and the depth formula for modules of finite complete intersection dimension.","marker":"[Iye99]"},{"why":"Defines maximal Cohen-Macaulay complexes and provides the local duality statements used in the proofs of Theorems 1.1 and 1.2.","marker":"[IMSW21]"},{"why":"Is the formula for $q_R(M,N)$ that Theorem 1.3 extends from finite complete intersection dimension to the local derived (dep) hypothesis.","marker":"[Jor99]"},{"why":"Supplies the trivial-vanishing rings used in Example 5.15, which satisfy (dep), (UBC), and (tr) globally but fail them at a localization.","marker":"[LMn20]"},{"why":"Provides the canonical-module duality and MCM approximation facts invoked in the (rdep) arguments.","marker":"[LW12]"}],"fun_headline_variants":["Tensor-depth bound matches ring dimension","Auslander bound equals dimension for CM rings","Uniform Auslander condition yields depth formula","Depth formula arises from uniform bound","New conditions unify tensor-product depth"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the reduction to a complete ring with a canonical module is free—specifically, the cited completion lemmas [KLOT23, Lemma 2.10] and [CH10, Remark 5.7] apply as stated—and that Lemma 3.9, proved for $f_M \\otimes N$, also yields its conclusion for $f_M \\otimes f_N$ in the step from (3) to (4).","fun_headline_variants_meta":{"raw":{"variants":["Tensor-depth bound matches ring dimension","Auslander bound equals dimension for CM rings","Uniform Auslander condition yields depth formula","Depth formula arises from uniform bound","New conditions unify tensor-product depth"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000884,"raw_usage":{"total_tokens":3921,"prompt_tokens":1153,"completion_tokens":2768,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":769,"completion_tokens_details":{"reasoning_tokens":2708}},"tokens_in":769,"tokens_out":2768,"duration_ms":21754,"temperature":1.0,"reasoning_tokens":2708,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:44:14.458950+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a Cohen-Macaulay local ring $R$ of dimension $d>0$ with two maximal Cohen-Macaulay modules $M,N$ satisfying $q_R(M,N)=1$: nonzero Tor in degree 1 and vanishing beyond. Theorem 1.1 predicts that such a ring cannot satisfy derived (ldep), because condition (4) would force $q_R(M,N)=0$. So exhibiting such a ring that does satisfy derived (ldep)—or merely checking whether a ring known to satisfy condition (3) admits such a pair—would settle the equivalence; equivalently, for any candidate ring, computing the Auslander bound $b_R$ and checking whether $b_R=d$ gives a direct numerical test.","supporting_citations":[],"review_version":1}