On an example concerning the second rigidity theorem

Arash Sadeghi; Hiroki Matsui; Olgur Celikbas

REVIEW 2 major objections 5 minor 22 references

On an example concerning the second rigidity theorem

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that, for a hypersurface ring, reflexivity of a tensor product forces both factors to be reflexive once the second factor is locally free in codimension one.

desk verdict A solid, honest paper that sharpens the boundary of Huneke–Wiegand's Second Rigidity Theorem, with only a fixable typo in the graph-connectedness proof. read the letter →

arxiv 1908.02167 v1 pith:DO5YVBP5 submitted 2019-08-06 math.AC

Olgur Celikbas , Hiroki Matsui , Arash Sadeghi This is my paper · ORCID

classification math.AC MSC 13D0713H1013D0513C12

keywords reflexivityoftensorproductsmodulesTor-rigiditycondition(S2)syzygyvanishingTorminimal-primegraphcompleteintersectiondimensionhypersurfacering

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

This paper establishes a complement to the Second Rigidity Theorem: a reflexive tensor product can force both factors to be reflexive, not merely the factor with finite projective dimension. Over a hypersurface ring, if $M$ has finite projective dimension and $N$ is locally of finite projective dimension at every prime of height at most one, then reflexivity of $M \otimes_R N$ implies reflexivity of both $M$ and $N$. The proof is carried by a connectedness theorem for the graph of minimal prime ideals, which gives $N$ a well-defined rank under the condition $(S_2)$, together with the depth formula for modules of finite complete intersection dimension. It also shows that a previously constructed counterexample, where the tensor product and $N$ are reflexive but $M$ is not, escapes exactly because $N$ has infinite projective dimension at a height-one prime. A broader theorem replaces reflexivity by the Serre-type condition $(\tilde S_n)$ and replaces finite projective dimension of $M$ by Tor-rigidity, yielding vanishing of all positive Tor modules.

What carries the argument

The load-bearing object is the graph $G(R)$ whose vertices are the minimal prime ideals of $R$, with an edge between $\mathfrak p$ and $\mathfrak q$ exactly when $\operatorname{height}(\mathfrak p+\mathfrak q) \leq 1$. Under condition $(S_2)$, this graph is connected, and connectedness is used to show that a module which is free at every prime of height at most one has constant rank across all minimal primes. That rank, combined with the depth formula for modules of finite complete intersection dimension, lets the authors upgrade local freeness to the Serre-type condition on both modules. The other mechanical ingredient is Tor-rigidity: if $M$ is Tor-rigid, meaning that vanishing of $\operatorname{Tor}_1^R(M,N)$ forces vanishing of $\operatorname{Tor}_2^R(M,N)$, then vanishing of $\operatorname{Ext}^1_R(\operatorname{Tr}N,M)$ propagates to vanishing of all positive Tor groups through the exact sequences of section 2.2; this converts reflexivity of the tensor product into a rank statement.

What would settle it

Localize the known counterexample at the height-one prime $\mathfrak q=(x,y)$ and resolve $N_{\mathfrak q}$: the minimal free resolution alternates multiplication by $x$ and $y$ forever, so $\operatorname{pd}_{R_{\mathfrak q}}(N_{\mathfrak q})=\infty$, verifying that this is precisely where the added hypothesis excludes the example. A single example satisfying all hypotheses of Theorem 1.2 with $M$ not reflexive would refute the claim; a computational search over small hypersurface presentations, checking the biduality map $M \to M^{**}$ explicitly, could be used to look for one.

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Extended reading notes

Core claim

Theorem 1.2 asserts: let $R$ be a hypersurface ring, that is, a quotient of an unramified regular local ring by a regular element, and let $M$ and $N$ be nonzero finitely generated modules with $\operatorname{pd}_R(M)<\infty$ and $\operatorname{pd}_{R_{\mathfrak p}}(N_{\mathfrak p})<\infty$ for every prime $\mathfrak p$ of height at most one. If $M \otimes_R N$ is reflexive, then both $M$ and $N$ are reflexive. The main Theorem 3.1 is more general: for a Noetherian local ring, if $M$ is Tor-rigid, $N$ has finite complete intersection dimension, $M \otimes_R N$ satisfies $(\tilde S_n)$, and $\operatorname{Tor}_i^R(M,N)$ is torsion for all sufficiently large $i$, then $\operatorname{Tor}_i^R(M,N)=0$ for all $i \geq 1$ and $N$ satisfies $(\tilde S_n)$. The known counterexample with reflexive tensor product but non-reflexive $M$ fails the height-one local finiteness hypothesis, so Theorem 1.2 removes precisely that escape.

Load-bearing premise

The argument depends on the graph of minimal primes being connected under condition $(S_2)$; if that graph were disconnected, a module locally free in height one would not have to have constant rank, and the step forcing $M$ to be reflexive would break.

Editorial extensions

If this is right

The known counterexample cannot be adjusted to satisfy the height-one local finiteness condition; the height-one prime with infinite projective dimension is essential to the counterexample.
Under the hypotheses of Theorem 3.1, the tensor product being an $n$-th syzygy forces $N$ to be an $n$-th syzygy and kills all positive Tor groups, so the conclusion applies not only to reflexivity but to higher syzygy behavior.
In Corollary 4.4, the same hypotheses plus local freeness on $Y_1(R)$ upgrade the conclusion to both $M$ and $N$ satisfying $(\tilde S_n)$; in the case $n=2$, both modules are reflexive.
The sharpness remark shows that the torsion hypothesis on high Tor groups alone cannot replace Tor-rigidity: a 2-Tor-rigid example with torsion high Tor has a non-reflexive conclusion.
The paper leaves open whether, with only torsion of high Tor assumed and no Tor-rigidity, a reflexive tensor product forces at least one factor to be reflexive; it notes that for domains of dimension at least two the answer is affirmative.

Reading between the lines

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

The connectedness-of-graph argument is not tied to reflexivity: the same graph should govern other statements where local freeness in codimension one is promoted to global rank, such as questions about higher syzygies of tensor products.
A natural extension is to replace the height-one condition by a higher-codimensional analogue: if $N$ is locally free outside a closed set of codimension at least $c$, one might expect reflexivity of $M \otimes_R N$ to force both factors to satisfy higher $(\tilde S_n)$ conditions.
The explicit presentation of the tensor product in the counterexample could be used to search for similar examples in higher dimensions where the local finiteness condition holds; the paper's method predicts that none exist over $(S_2)$ hypersurfaces.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a solid paper with a clean new criterion. It doesn't settle the reflexivity question for M in Huneke–Wiegand's Second Rigidity Theorem, but it shows that the Celikbas–Takahashi counterexample cannot occur once N has finite projective dimension locally on the height-one primes. The main theorem holds up.

What is new: Theorem 3.1 is a genuinely general statement—Tor-rigid M, finite CI-dimension N, tensor product satisfying (~S_n), and torsion high Tor force all positive Tor to vanish and N to satisfy (~S_n). Corollary 4.4 upgrades the conclusion to both M and N when N is locally free on Y1(R), using connectedness of the Hochster–Huneke graph. Theorem 1.2 is the hypersurface packaging, and it is exactly the right boundary condition to rule out the [10] example. The new explicit presentation of M ⊗ N in Example 4.5 is a useful addition. The proof strategy is coherent, and the graph-connectedness lemma is proven in the text rather than taken for granted.

Soft spots are minor. The stress-test note is right: in Proposition 4.3(i), the sentence 'Then u is R-regular' is a typo—when a is the unit in the argument, the regular element is v, not u. The contradiction still goes through (v regular and in J forces I = 0), so this is cosmetic, not load-bearing. Also, in Theorem 3.1 the step where depth(R_q) ≥ depth(N_q) is used to conclude CI-dim(N_q)=0 relies on the standard inequality depth ≤ depth R for modules of finite CI-dimension; it is valid but not explicitly cited. Finally, Example 4.5's verification depends on a Macaulay2 computation via an unpublished note [21]; that is acceptable for an example, and the main theorem does not depend on it. The citation pattern is clean: several references are by overlapping authors, but they are distinct published results and the central argument does not lean on the example.

Who it's for: commutative algebraists working on tensor products, reflexivity, and rigidity of Tor. It deserves a serious referee; I would send it out. After a fix of the typo, I'd be happy with acceptance.

Referee Report

2 major / 5 minor

Summary. The paper revisits an example of Celikbas and Takahashi concerning the failure of both modules in a tensor product to be reflexive in the Second Rigidity Theorem of Huneke and Wiegand. The authors prove a general result, Theorem 3.1, stating that if M is Tor-rigid, N has finite complete intersection dimension, M⊗N satisfies the condition (~S_n), and Tor_i(M,N) is torsion for all i≫0, then all positive Tor modules vanish and N satisfies (~S_n). Under the additional hypotheses that R satisfies (S2) and N is locally free on Y_1(R), Corollary 4.4 strengthens the conclusion to both M and N satisfying (~S_n). This yields Theorem 1.2 for hypersurface rings: if M has finite projective dimension and N has finite projective dimension after localization at every prime of height at most one, then reflexivity of M⊗N forces both M and N to be reflexive. The paper also analyzes the Hochster–Huneke graph and proves connectedness without a completeness assumption, and it revisits the Celikbas–Takahashi example to demonstrate sharpness.

Significance. If correct, the paper gives a clean positive result that rules out the known counterexample to the Second Rigidity Theorem under a mild local finiteness condition on N. The main theorem is more general than the hypersurface setting, working with complete intersection dimension and the (~S_n) conditions, and the proof is largely self-contained, with useful preliminary lemmas on transposes, pushforwards, and Tor-rigidity. The paper also supplies a proof of connectedness of the Hochster–Huneke graph in the non-complete case, and it provides an explicit presentation of the tensor product in the Celikbas–Takahashi example. These are genuine contributions to the homological algebra of tensor products and rigidity.

major comments (2)

[§2.1 and proof of Theorem 3.1] The definition of Tor-rigidity in §2.1 is the weak one: Tor^R_1(M,N)=0 implies Tor^R_2(M,N)=0. However, the proof of Theorem 3.1 uses the standard strong form of Tor-rigidity: after showing Tor^R_1(M,N_1)=0, the text concludes 'As M is Tor-rigid, we have Tor^R_i(M,N)=0 for each i≥1.' This inference requires the strong property Tor^R_i=0 ⇒ Tor^R_{i+1}=0 for all i≥1, not merely the i=1 case stated in §2.1. Since the examples cited in §2.1 (Lichtenbaum, Huneke–Wiegand) do satisfy the strong form, the gap is easily fixed by amending the definition, but as written the proof is not valid under the stated definition.
[§4, Proposition 4.3(i)] In the final paragraph of the proof, after assuming that a is a unit, the element that is shown to be regular is v=a(u+v), not u. The sentence 'Then u is R-regular, and the equality uJ=0 shows that J=0' is therefore not literally correct. The contradiction still follows because v is R-regular and v∈J, so for every u∈I the relation uv=0 forces u=0, giving I=0. This is a typographical slip rather than a substantive gap, but it should be fixed.

minor comments (5)

[§4, Proposition 4.3(ii)] In the proof of part (ii), the module is N, but several occurrences read 'M_{p_i}', 'M_{q_{i+1}}', and 'rank_{R_p}(M_p)'. These should be N throughout to avoid confusion.
[Example 4.5] The displayed exact sequence is hard to parse because the matrices and the module M⊗N are not clearly connected by arrows, and the phrase 'rightmost matrix' is ambiguous. Please rewrite the sequence with labeled maps and indicate explicitly which cokernel is M⊗R N. Also, since the verification relies on unpublished Macaulay2 code [21], it would be helpful to include the code or a reproducible computation.
[Lemma 3.3] The reduction to CI-dim_{R_p}(N_p)=0 for p∈Y_0(R) uses the fact that finite CI-dimension satisfies the Auslander–Buchsbaum type formula CI-dim(N_p)=depth(R_p)-depth(N_p). This is standard, but it would be good to cite it explicitly, since the argument in that lemma depends on it.
[§2.1] The examples of Tor-rigid modules listed in §2.1 should state explicitly that they satisfy the strong Tor-rigidity property used in Theorem 3.1, not merely the weak form given in the definition.
[General] There are a few typographical issues: 'syzgy' should be 'syzygy' in the text after 2.2, 'The inequality in (3.1.5) are due to' should be 'is due to', and 'enviroment' in the acknowledgments should be 'environment'.

Circularity Check

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No significant circularity; the main theorem is proved from independent homological results and a self-contained graph-connectedness argument.

full rationale

The derivation of Theorem 1.2 runs through Corollary 4.4, which relies on Theorem 3.1 and Proposition 4.3. Theorem 3.1 is proved from Lemmas 3.2–3.4 using standard Ext–Tor exact sequences, Tor-rigidity, finite CI-dimension, the depth formula, and pushforwards; none of these steps is equivalent to the conclusion by definition, and no parameter is fitted to the target result. Proposition 4.3, whose connectedness of the Hochster–Huneke graph is the load-bearing geometric input, is proved in the paper rather than merely assumed. The cited example of Celikbas and Takahashi is used only in Example 4.5 and Remark 4.6 to test sharpness, not to prove Theorem 1.2. Several references are by overlapping authors (e.g., [8], [9], [10], [12]), but they are distinct published theorems with independent hypotheses and do not contain the paper's conclusion; they therefore do not make the central derivation circular. The typographical slip in Proposition 4.3(i), where the regular element is printed as u rather than v, affects the wording of a contradiction argument but does not change the fact that the proof is independent of the result being established. Overall, the central claim is self-contained against standard background results, so there is no circularity.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It relies on standard homological background results, several of which are cited from the authors' own prior work, but these are independent published theorems. The only implicit assumption is the standard fact that every module over a complete intersection has finite CI-dimension, used in the proof of Theorem 1.2.

assumptions (8)

domain assumption R is a commutative Noetherian local ring with maximal ideal m; all modules are finitely generated.
Stated in Section 1; the entire theory is over such rings.
standard math Auslander-Bridger exact sequence (2.2): 0 to Ext^1(Tr Omega^n M, -) to Tor_n^R(M, -) to Hom(Ext^n(M,R), -) to Ext^2(Tr Omega^n M, -).
Used throughout the proof of Theorem 3.1 to relate Tor and Ext via transpose; quoted from [2, 2.8].
standard math Depth formula (2.7): if CI-dim(M) is finite or CI-dim(N) is finite and Tor_i(M,N)=0 for all i at least 1, then depth(M)+depth(N)=depth(R)+depth(M tensor_R N).
Used in (3.1.5) and in Corollary 4.4; quoted from [1, 2.5].
standard math Ext-Tor duality (2.8): if CI-dim(N)=0 then Ext^i(TrN,M)=0 for all i at least 1 iff Tor_i(M,N)=0 for all i at least 1.
Used in Lemma 3.3 and in the proof of Theorem 3.1; quoted from [9, 3.2].
standard math Equivalence of Serre-type condition (~S_n), n-torsion-freeness, and n-th syzygy under finite G-dimension (2.6).
Used to translate the goal 'N satisfies (~S_n)' into vanishing of Ext^i(TrN,M); quoted from [12, 2.4] and [13, 3.8].
standard math Every finitely generated module over a complete intersection (in particular a hypersurface) has finite CI-dimension.
Implicitly used to pass from Theorem 1.2 to Corollary 4.4, which requires CI-dim(N) finite; the paper does not state this explicitly.
standard math Lichtenbaum's theorem: over a hypersurface that is a quotient of an unramified regular local ring, every module of finite projective dimension is Tor-rigid.
Used in the proof of Theorem 1.2 to satisfy condition (i) of Corollary 4.4; quoted in 2.1(i) from [16, 2.4] and [19, Theorem 3].
standard math Hochster-Huneke graph connectedness (Theorem 4.3, proven in the paper) for rings satisfying (S2).
Proven in Section 4; used in Corollary 4.4 to show N has rank.

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Pith. "Pith review of On an example concerning the second rigidity theorem." pith.science (2026). https://pith.science/paper/DO5YVBP5

@misc{pith2026190802167,
  author       = {Pith},
  title        = {Pith review of: On an example concerning the second rigidity theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DO5YVBP5}},
  note         = {Machine review of arXiv:1908.02167}
}

read the original abstract

In this paper we revisit an example of Celikbas and Takahashi concerning the reflexivity of tensor products of modules. We study Tor-rigidity and the Hochster--Huneke graph with vertices consisting of minimal prime ideals, and determine a condition with which the aforementioned example cannot occur. Our result, in particular, corroborates the Second Rigidity Theorem of Huneke and Wiegand.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [10]

Olgur Celikbas and Ryo Takahashi, On the second rigidity theorem of Huneke and Wiegand , Proc. Amer. Math. Soc. (2019)

work page 2019

[2] [21]

Frank Moore, Macaulay2 code to compute the biduality map and the pushforw ard module, unpublished (2015)

W. Frank Moore, Macaulay2 code to compute the biduality map and the pushforw ard module, unpublished (2015)

work page 2015

[3] [1]

Algebra 26 (1998), no

Tokuji Araya and Y uji Y oshino, Remarks on a depth formula, a grade inequality and a conjectu re of Auslan- der, Comm. Algebra 26 (1998), no. 11, 3793–3806. MR MR1647079 (99h:13010)

work page 1998

[4] [2]

94, American Mathematical Society, Providence, R.I., 1 969

Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1 969

work page

[5] [3]

Maurice Auslander and Idun Reiten, Applications of contravariantly ﬁnite subcategories , Adv. Math. 86 (1991), 111–152

work page 1991

[6] [4]

Avramov, Inﬁnite free resolutions , six lectures on commutative algebra (Bellaterra, 1996), P rogr

Luchezar L. Avramov, Inﬁnite free resolutions , six lectures on commutative algebra (Bellaterra, 1996), P rogr. Math., vol. 166, Birkh¨ auser, Basel, 1998, pp. 1–118

work page 1996

[7] [5]

Avramov and Ragnar-Olaf Buchweitz, Support varieties and cohomology over complete inter- sections, Invent

Luchezar L. Avramov and Ragnar-Olaf Buchweitz, Support varieties and cohomology over complete inter- sections, Invent. Math. 142 (2000), no. 2, 285–318. MR MR1794064 (2001j:13017)

work page 2000

[8] [6]

39, Cambridge University Press, Cambridge, 1993

Winfried Bruns and J¨ urgen Herzog, Cohen-Macaulay rings , Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993

work page 1993