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Quasilifting of hulls and depth of tensor product of modules

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2412.01392 v2 pith:GVZDVMGF submitted 2024-12-02 math.AC

Sutapa Dey , Amit Tripathi This is my paper · ORCID
classification math.AC MSC 13D0713C1413C15
keywords tensorproductsofmodulesdepthformulacompleteintersectionringsTETor-vanishingcomplexityGorensteindimensionhullsandapproximations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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).

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Lemma 4.2(b)] 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.
  2. [Lemma 4.3] 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).
minor comments (5)
  1. [Lemma 4.2, proof] 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.
  2. [Lemma 4.3] 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.
  3. [Lemma 4.3, proof] 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.
  4. [Theorem 1.2, proof, part (b)] 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.
  5. [Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are derived from external results and existing tools, and the recovery of [14,1.2] is explicitly presented as a special case.

full rationale

The derivation chain in this paper is self-contained against external benchmarks rather than circular. Theorem 1.2 is proved by reducing to the implication (4.12) -> Tor-vanishing, using the complexity-reduction lemmas 4.3 and 4.5, which rely on the quasilifting construction of Huneke-Jorgensen-Wiegand [18], the asymptotic Ext result of Celikbas-Dao [12,2.4], the Avramov-Buchweitz support-variety theory [6,III], and the characterization of TE rings in [23,4.3]. None of these cited results assumes the target theorem, and none of them is by the present authors. Part (b) is explicitly a recovery of the known theorem [14,1.2], not a renaming of it, while part (a) is a genuinely different TE-ring statement. The paper also provides an independent counterexample (Example 5.1) to a question from [14]. The only notable defect is a writing-level typo in the proof of Lemma 4.2(b), where M1 = Omega^r_R M is asserted to be x-regular even though R = Q/(x) annihilates every R-module; the intended construction is evidently Omega^r_Q. This is a correctness gap in the self-contained proof, not circularity, and the paper itself notes that the TE case was independently proved in [24,2.5(3)], an external preprint. No fitted inputs are renamed as predictions, no self-citations are load-bearing, and no definition is engineered to make the conclusion hold by construction.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

This is pure mathematics, so no data fitting occurs and the free-parameters list is empty. The central claims rest on standard homological algebra (Auslander-Bridger formula, depth formula, Rees isomorphism, acyclicity lemma, change-of-rings spectral sequence), on the Avramov-Buchweitz theorem that complete intersections are TE, on Jorgensen-Sega's characterization of TE rings, and on two black-box results ([12,2.4] for complexity reduction and [14,3.4] for Tor vanishing). One additional unstated assumption is a reduction to the m-adic completion in Lemma 4.3; the text 'M is isomorphic to hat M' is not literally true for non-complete R. No new entities are postulated.

assumptions (9)
  • domain assumption Rings are commutative Noetherian and modules are finitely generated throughout.
    Global setting stated in Section 2.
  • standard math Auslander-Bridger formula: G-dim_R(M) + depth_R(M) = depth_R(R).
    Used in Proposition 3.5 and Lemma 5.2; cited to [3].
  • standard math Depth formula for Tor-independent pairs with one module of finite complete intersection dimension.
    Invoked in Section 2.6, Example 5.1, Lemma 5.2, and Theorem 4.10; cited to [1,22].
  • standard math Avramov-Buchweitz: local complete intersection rings are TE, so eventual vanishing of Ext and Tor coincide.
    Used in Theorem 1.2(b) and in the proofs of Lemmas 4.3 and 4.4; cited to [6,III].
  • standard math Jorgensen-Sega: a ring is TE if and only if it is Gorenstein and ET.
    Used in Lemma 4.2 and in the proof of Theorem 1.2; cited to [23,4.3].
  • domain assumption [12,2.4]: a deformation factorization exists with complexity drop cx_Q(M,N) = cx_R(M,N)-1 while preserving vpd.
    Load-bearing for Lemmas 4.3 and 4.4; the proof is not reproduced, only invoked 'as in [12,2.4]'.
  • domain assumption [14,3.4]: Tor vanishing up to complexity plus one yields full Tor vanishing under length finiteness assumptions.
    Used in the proof of Theorem 1.3; the statement is not included in the paper.
  • standard math The change-of-rings spectral sequence collapses to the long exact sequence (2.5).
    Used in Lemma 4.5 and Theorem 1.2; cited to [29,5.6.6].
  • domain assumption The m-adic completion preserves depth, Tor-length finiteness, complexity, TE, and complete intersection properties, and the paper silently reduces to complete rings.
    Needed to make Lemma 4.3 coherent because vpd is defined via deformations of the completion; not stated in the theorem.

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Pith. "Pith review of Quasilifting of hulls and depth of tensor product of modules." pith.science (2026). https://pith.science/paper/GVZDVMGF

@misc{pith2026241201392,
  author       = {Pith},
  title        = {Pith review of: Quasilifting of hulls and depth of tensor product of modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GVZDVMGF}},
  note         = {Machine review of arXiv:2412.01392}
}
abstract

We investigate the depth of the tensor product of finitely generated modules over local rings. One of the main ingredients of our approach is a lifting construction introduced by Huneke, Jorgensen, and Wiegand. We recover a result of Celikbas, Sadeghi, and Takahashi for local complete intersection rings. Additionally, we provide a negative answer to a question they asked and establish a corresponding lower bound. We derive a result on the depth of the tensor product of certain modules over local complete $\mathcal{TE}$ rings. Some general conditions on the existence of hulls and approximations are also studied.

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