Pith. sign in

REVIEW 4 major objections 6 minor 25 references

$\omega$-left approximation dimensions under Stable equivalence

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Stable equivalence preserves ω-left approximation dimensions and thereby carries the Wakamatsu tilting conjecture between Artin algebras without nodes or semisimple direct summands.

desk verdict Theorem 3.5 as stated is false: projective-injective summands are dropped, so the equality fails on an explicit self-injective example; the intended transfer theorem may be repairable but needs major revision. read the letter →

arxiv 2507.09286 v1 pith:EKSWPGHC submitted 2025-07-12 math.RT

Juxiang Sun , Guoqiang Zhao , Junling Zheng This is my paper · ORCID
classification math.RT MSC 16D2016E30
keywords ω-leftapproximationdimensionstableequivalencefaithfulWakamatsutiltingmoduleconjecturerelativen-torsionfreemodulesdominantArtinalgebras
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 proves that $\omega$-left approximation dimensions—a measure of how many steps of a module's best approximation complex by direct sums of a fixed module $\omega$ remain exact—pass unchanged between two Artin algebras that are stably equivalent in the sense of module-category equivalence modulo projectives. The transfer works when the algebras have neither nodes nor semisimple direct summands and the module $\omega$ is self-orthogonal and decomposed into a module without injective summands, an injective-projective part, and a projective-injective part. If the theorem is right, then stable equivalence is strong enough to preserve faithful dimensions, dominant dimensions, basic (Wakamatsu) tilting modules, and the truth of the Wakamatsu tilting conjecture in this class of algebras. The paper's broader message is that a certain tilting-theoretic invariant is a stable-equivalence invariant.

What carries the argument

The load-bearing device is the pair of functors $F: \underline{\mathrm{mod}}\,\Lambda \to \underline{\mathrm{mod}}\,\Gamma$ and $F' = \tau_\Gamma F \tau_\Lambda^{-1}$ that arise from a stable equivalence, together with the module correspondence $\nu = F'(X) \oplus F(I) \oplus Q$ built from the decomposition of $\omega$. These functors biject the subcategories of modules without projective or injective summands and the subcategories of projective-injective modules (Lemma 3.1), and Lemma 3.3 shows $\operatorname{Ext}^n_\Lambda(A,A') \cong \operatorname{Ext}^n_\Gamma(B,B')$ for modules related by this correspondence. Lemma 3.4 is the actual transfer engine: it converts a left $\mathrm{add}\,\omega$-approximation resolution of $M$ into a left $\mathrm{add}\,\nu$-approximation resolution of $N$ term by term, which is exactly what makes the equality of dimensions in Theorem 3.5 go through.

What would settle it

Find a pair of stably equivalent Artin algebras without nodes and without semisimple direct summands, a self-orthogonal module $\omega$ with the stated decomposition, and a module $M$ for which direct computation of the left $\mathrm{add}\,\omega$-approximation resolutions gives different $\omega$-left approximation dimensions on the two sides; the theorem's equality would be false in that case.

Watch

Extended reading notes

Core claim

The central assertion is Theorem 3.5: for stably equivalent Artin algebras $\Lambda$ and $\Gamma$ with neither nodes nor semisimple direct summands, given $\omega = X \oplus I \oplus P$ with $\operatorname{Ext}^1_\Lambda(\omega,\omega)=0$ and a module $M = Y \oplus I' \oplus P'$, the module $\nu = F'(X) \oplus F(I) \oplus Q$ on the $\Gamma$ side satisfies $l.\mathrm{app}_\omega M = l.\mathrm{app}_\nu N$, where $N = F'(Y) \oplus F(I') \oplus Q'$. The proof builds a transfer mechanism in Lemma 3.4 that sends left $\mathrm{add}\,\omega$-approximation sequences to left $\mathrm{add}\,\nu$-approximation sequences, using the fact that the stable equivalence functors $F$ and $F'$ preserve extensions (Lemma 3.3) and biject the relevant subcategories of modules. From this, the paper derives that faithful dimension is preserved (Proposition 3.7), that the maps $\Phi$ and $\Psi$ give one-to-one correspondences between basic Wakamatsu tilting modules and between basic tilting modules (Theorem 3.11), and that the Wakamatsu tilting conjecture holds for $\Lambda$ exactly when it holds for $\Gamma$ (Theorem 3.12).

Load-bearing premise

The whole transfer rests on the pair of algebras being stably equivalent with neither nodes nor semisimple direct summands, and on the module $\omega$ being self-orthogonal.

Editorial extensions

If this is right

  • Faithful dimension is a stable-equivalence invariant: $\operatorname{fadim}_\Lambda \omega = \operatorname{fadim}_\Gamma \nu$ for the corresponding module $\nu$ (Proposition 3.7).
  • Dominant dimension is preserved: $\operatorname{dom.dim} M = \operatorname{dom.dim} N$, and in particular $\operatorname{dom.dim} \Lambda = \operatorname{dom.dim} \Gamma$ (Corollary 3.8).
  • Basic Wakamatsu tilting modules and basic tilting modules correspond one-to-one between $\Lambda$ and $\Gamma$ via $\Phi$ and $\Psi$ (Theorem 3.11).
  • The Wakamatsu tilting conjecture is a stable-equivalence invariant: $\Lambda$ satisfies it if and only if $\Gamma$ does (Theorem 3.12).
  • Relative $n$-torsionfree modules, modules that are $\omega$-$\infty$-torsionfree, and modules with generalized Gorenstein dimension zero transfer to the $\Gamma$ side (Propositions 3.13, Corollary 3.15, Theorem 3.16).

Reading between the lines

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

  • If the theorem is correct, any invariant expressible through $\omega$-left approximation dimensions for modules of the stated decomposable form will transfer automatically, so the result may serve as a template for finding new stable-equivalence invariants beyond those listed.
  • The node-free, semisimple-summand-free hypothesis is exactly where the subcategory bijections used in the proof are available; a natural boundary test is whether a stable equivalence involving an algebra with nodes or a semisimple direct summand can break the equality of approximation dimensions.
  • Because the correspondence matches Wakamatsu tilting modules one-to-one, the class of algebras satisfying the Wakamatsu tilting conjecture is closed under the relevant stable equivalences; this may suggest that a broader class of equivalences transfer the conjecture as well.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper studies the behaviour of ω-left approximation dimensions under stable equivalence of Artin algebras having neither nodes nor semisimple direct summands. Under the standing stable equivalence F and its companion F′, the authors define, for ω=X⊕I⊕P, the module ν=F′(X)⊕F(I)⊕Q, where Q is the direct sum of all indecomposable projective-injective Γ-modules, and claim in Theorem 3.5 that for M=Y⊕I′⊕P′ and N=F′(Y)⊕F(I′)⊕Q′ one has l.app_ω M = l.app_ν N. The paper then derives transfer results for faithful dimension and dominant dimension, gives one-to-one correspondences between basic (Wakamatsu) tilting modules, proves that the Wakamatsu tilting conjecture is preserved under stable equivalence, and formulates analogous statements for relative n-torsionfree modules and generalized Gorenstein dimension.

Significance. If the main theorem were correct, the paper would supply a useful transfer principle for a homological invariant under stable equivalence, with concrete consequences for the Wakamatsu tilting conjecture and for relative homological dimensions. The strategy is well motivated: it uses established stable-equivalence machinery of Auslander–Reiten and Martinez-Villa, and the explicit formulas for ν in terms of ω are natural. The paper is also honest in relying on external theorems rather than inventing ad hoc axioms. However, the central theorem mishandles the projective-injective summand P′, and the infinite-dimensional case of the approximation dimension is not proved; these issues affect the faithful-dimension and tilting applications. The results would be significant after a careful correction of the statement and proof of Theorem 3.5.

major comments (4)
  1. [§3.1, Theorem 3.5] The projective-injective summand P′ plays no role in the proof: the proof constructs an exact sequence for N starting from a resolution of Y only, appending F(I′)⊕Q′ as a split summand, and it never uses P′ or relates Q′ to F′(P′). This is a load-bearing omission. If P′ is not in addω, then no left addω-approximation of P′ can be injective, because an injective submodule of a module in addω splits off and would force P′∈addω; hence no exact truncated complex can start at M with terms in addω, and l.app_ω M is not positive/infinite. On the other hand, Q′ is a projective-injective Γ-module, so Q′∈addν because ν contains the full direct sum Q of all such modules; taking M=P′ and N=Q′ with P′∉addω and Q′≠0 gives l.app_ν N=∞ while l.app_ω M is not positive/infinite, contradicting the asserted equality. A concrete witness is obtained by taking Λ=Γ to be a product of two copies of a suitable self-injective Nakayama algebra, F=F′=id, ω=P1, M=P2 for two nonisomorphic projective-injective modules in different blocks, and Q′=P2. The theorem needs either corrected hypotheses (for example, that P and Q are the full direct sums of all indecomposable projective-injective modules and that Q′ is tied to P′) or a separate argument showing how P′ and Q′ are eliminated; the current proof does not establish either.
  2. [§3.1, proof of Theorem 3.5] The proof treats only the cases l.app_ω M = 1 and l.app_ω M = n for a positive integer n; the case l.app_ω M = ∞ is not addressed. This is not a cosmetic omission, because Proposition 3.7 and Proposition 3.9 need the equality of faithful dimensions in the infinite case to conclude that a Wakamatsu tilting module is transferred to a Wakamatsu tilting module. Since Lemma 3.4(1) is formulated for finite n, an additional argument is required to show that an infinite exact complex with left addω-approximations is transferred to an infinite exact complex with left addν-approximations; the present proof only gives arbitrarily long finite exact complexes, whose compatibility is not shown.
  3. [Lemma 3.4(2)] Lemma 3.4(2) is stated with the sentence 'the proof of (2) is similar' and is used in Proposition 3.10 to transfer the resolving sequence 0→Λ→ω0→…→ωn→0 to Γ. This is a different statement from part (1): it requires a finite exact sequence ending at 0 with all middle terms in addν, not merely a left-approximation sequence of finite length. The truncation and induction used in part (1) do not automatically give the required terminal exactness. A complete proof or a precise reduction to part (1) should be supplied.
  4. [Theorem 3.11] Theorem 3.11 asserts that Φ and Ψ restrict to one-to-one correspondences between WT(Λ) and WT(Γ), and between T(Λ) and T(Γ), but the proof only cites Propositions 3.9 and 3.10, which show that Φ sends each class into the corresponding class. The verification that these maps are inverse to each other—that is, Ψ(Φ(ω))≅ω and Φ(Ψ(ν))≅ν—is not given. This inverse property is used in Theorem 3.12 to conclude that ω is tilting, so the bijectivity claim needs an explicit proof.
minor comments (6)
  1. [Theorem 3.5] The statement of Theorem 3.5 defines N=F′(Y)⊕F(I)⊕Q′, but the proof uses F(I′) and requires I′∈addI(Λ)P; the displayed definition should presumably read F(I′).
  2. [Lemma 3.4(1)] The proof of Lemma 3.4(1) contains several local errors and inconsistencies: 'Definie g1' should be 'Define g1', the notation T′1 appears where T1 or T11 is intended, and the displayed formula for the approximation of F′(Y1)⊕F(J1) uses inconsistent subscripts. These should be corrected throughout.
  3. [Theorem 3.5 proof] In the proof of Theorem 3.5, 'left add V -approximation' should be 'left addν-approximation', and the expressions '1.appνN' and 'l .appνN' should consistently be 'l.appνN'.
  4. [Corollary 3.15] In Corollary 3.15, 'Taking ν = F′(Y)⊕F(I′)⊕Q′' should be 'Taking N = F′(Y)⊕F(I′)⊕Q′', and 'projecitve' is a typo for 'projective'.
  5. [References] References [12], [14], and [20] are all the same arXiv item (Enomoto's 'Maximal self-orthogonal modules and a new generalization of tilting modules') and should be consolidated; [24] is listed as 'preprinted' without a venue or arXiv number.
  6. [Introduction] The phrase 'the Nakamaya conjecture' should be 'the Nakayama conjecture'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's transfer theorems are derived from cited external stable-equivalence results, and no fitted parameter, prediction, or self-citation chain is load-bearing.

full rationale

The derivation chain is self-contained in the relevant sense. The main theorem (Theorem 3.5) is proved by constructing exact sequences and left approximations on the other side of a stable equivalence; the construction uses the cited bijections and Ext-isomorphism results from [3, Section 8], [9, Lemma 4.10], and [22], not the theorem being proved. Theorem 3.12 on the Wakamatsu tilting conjecture is obtained by transferring self-orthogonality, faithful dimension, and projective dimension via Lemma 3.3, Proposition 3.7, Proposition 3.9, and Proposition 3.10, none of which assumes the conjecture. The paper does not fit parameters to data, rename an empirical pattern as a derivation, or import a uniqueness theorem from the authors' prior work. The one self-citation, [24], appears only in the introduction as context and is not used in any proof. Even if Theorem 3.5 has a substantive mathematical gap concerning projective-injective direct summands, as the skeptic's attack suggests, that is a correctness issue, not circularity: the asserted equality is not equivalent to its inputs by construction. Therefore the appropriate circularity finding is a clean zero.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are introduced; the decompositions of ω and M into injective, injective-projective, and projective parts are canonical for basic modules, and Q is the fixed direct sum of all indecomposable projective-injective Γ-modules. The axioms are the standard hypotheses of stable equivalence theory plus the self-orthogonality assumption of the theorem. The paper introduces no new axioms, forces, or entities beyond the constructed modules ν and N, which are determined by the data.

assumptions (4)
  • domain assumption Λ and Γ are Artin algebras with neither nodes nor semisimple direct summands
    Invoked before Lemma 3.1; guarantees the bijections F: modP Λ -> modP Γ and F': modI Λ -> modI Γ and related properties from [3, Section 8] and [9, Lemma 4.10], which the transfer proof relies on.
  • domain assumption F: mod Λ -> mod Γ is an equivalence of stable module categories and F' = τΓ ∘ F ∘ τΛ^{-1}
    Definition of stable equivalence used throughout; F and F' are assumed to commute with direct sums and induce the stated bijections (per [3, Section 8]).
  • domain assumption Ext^1_Λ(ω,ω) = 0 for the module ω in Theorem 3.5
    Core hypothesis of Theorem 3.5 and Proposition 3.7; ensures ν is self-orthogonal and that Ext^1(T,ω)=0 for approximations.
  • standard math Cited stable-equivalence results (e.g., Ext preservation from [22]/[21], Lemma 4.10 from [9])
    The proof of Lemma 3.3 and Lemma 3.1 quotes these external theorems; their correctness is outside the paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of $\omega$-left approximation dimensions under Stable equivalence." pith.science (2026). https://pith.science/paper/EKSWPGHC

@misc{pith2026250709286,
  author       = {Pith},
  title        = {Pith review of: $\omega$-left approximation dimensions under Stable equivalence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EKSWPGHC}},
  note         = {Machine review of arXiv:2507.09286}
}
abstract

In this paper, we investigate some transfer properties of $\omega$-left approximation dimensions of modules of stably equivalent Artin algebras having neither nodes nor semisimple direct summands. As applications, we give a one-to-one correspondence between basic (Wakamatsu) tilting modules, and prove that the Wakamatsu tilting conjecture is preserved under those equivalences.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 24 canonical work pages

  1. [20]

    Enomoto, Maximal self-orthogonal modules and a new generalization of tilting modules, arXiv:2301.13498 [math.AC]

    H. Enomoto, Maximal self-orthogonal modules and a new generalization of tilting modules, arXiv:2301.13498 [math.AC]

    arXiv

  2. [1]

    Bridger, , Stable module theory, Memoirs Amer

    M.Auslander, M., M. Bridger, , Stable module theory, Memoirs Amer. Math. Soc., 94, Provi- dence’Rhode Island: American Mathematical Society, 1969

    work page 1969

  3. [2]

    Auslander, I

    M. Auslander, I. Reiten, Stable equivalence of Artin algebras. In: Proceedings of the Confer- ence on Orders, Group Rings and Related Topics (Ohio State Univ., Columbus, Ohio, 1972), pp. 8-71. Lecture Notes in Math. 353, Springer, Berlin, 1973

    work page 1972

  4. [3]

    Auslander, I

    M. Auslander, I. Reiten, Stable equivalence of dualizing R-varieties, Adv. Math. 12 (1974) 306-366

    work page 1974

  5. [4]

    Auslander, I

    M. Auslander, I. Reiten, and S. Smalo, Representation theory of Artin algebras , CambridgeS- tud. Adv. Math. 36, Cambridge University Press, 1995

    work page 1995

  6. [5]

    Algebra, 1980, 66: 61-22

    Auslander, M., Smal ϕ, S.O., Preprojective modules over Artin algebras, J. Algebra, 1980, 66: 61-22

    work page 1980

  7. [6]

    Auslander and I

    M. Auslander and I. Reiten, Cohen-Macaulay and Gorenstein artin algebras, Progress in Math. (Birkhauser, Basel) 95 (1991), 221-245

    work page 1991

  8. [7]

    Brenner and M

    S. Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gelfand-Ponomarev reflec- tion functors, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., 832, Springer, Berlin (1980), pp. 103-169

    work page 1980

Show all 25 references
  1. [8]

    Beligiannis, I

    A. Beligiannis, I. Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc., 188(883), (2007), viii+207 pp

    2007

  2. [9]

    H.X.Chen, M. Fang, O. Kerner, K.Yamagata. Rigidity dimension of algebras, Math. Proc. Camb. Phil. Soc. (2021), 170, 417–443

    2021

  3. [10]

    Bao, J.F

    Y.H. Bao, J.F. L¨u, Relative torsionfreeness and Frobenius extension, arXiv: 2409.11892v1

    arXiv

  4. [11]

    A. B. Buan and oyvind Solberg, Relative cotilting theory and almost complete cotilting mod- ules, Algebras and modules, II(Geiranger, 1996), 77-92, CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, 1998

    1996

  5. [13]

    Guo, Representation dimension: an invariant under stable equivalence, Trans

    X.Q. Guo, Representation dimension: an invariant under stable equivalence, Trans. Amer. Math. Soc. 357 (8) (2005) 3255–3263. 16

    2005

  6. [15]

    Enochs, O.M.G

    E.E. Enochs, O.M.G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995) 611–633

    1995

  7. [16]

    E. L. Green, I. Reiten and Φ. Solberg, Dualities on generalized Koszul algebras, Mem. Amer. Math. Soc., 159(754), (2002), xvi+67 pp

    2002

  8. [17]

    Huang, ω-k-torsionfree modules and ω-n-left approximation dimension, Science in China (Series A)(2000), 44(2): 184-192

    Z.Y. Huang, ω-k-torsionfree modules and ω-n-left approximation dimension, Science in China (Series A)(2000), 44(2): 184-192

    2000

  9. [18]

    Huang, Extension closure of k-torsionfree modules, Comm

    Z.Y. Huang, Extension closure of k-torsionfree modules, Comm. Algebra 27 (1999), 1457-1464

    1999

  10. [19]

    F. Li, Y.G. Sun, Invariants under stable equivalences of Morita type. Acta Mathematica Scientia 2012, 32B(2):605–618

    2012

  11. [21]

    Miyashita, Tilting modules of finite projective dimension, Math

    Y. Miyashita, Tilting modules of finite projective dimension, Math. Z., 193(1), (1986), 113-146

    1986

  12. [22]

    Martinez-Villa, Properties that are left invariant under stable equivalence

    R. Martinez-Villa, Properties that are left invariant under stable equivalence. Comm. Algebra. (1990) 18 (12): 4141-4169

    1990

  13. [23]

    Mantese and I

    F. Mantese and I. Reiten, Wakamatsu tilting modules, J. Algebra, 278(2), (2004), 532-552

    2004

  14. [24]

    Sun, J.L

    J.X. Sun, J.L. Zheng, G.Q. Zhao, Stable equivalences of tilted algebras, preprinted

  15. [25]

    Wakamatsu, On modules with trivial self-extensions, J

    T. Wakamatsu, On modules with trivial self-extensions, J. Algebra, 114(1), (1988), 106-114

    1988

  16. [26]

    Wei, Auslander generators and homological conjectures, Glasg

    J. Wei, Auslander generators and homological conjectures, Glasg. Math. J., 56(3), (2014), 503-506

    2014

  17. [27]

    Wei, Auslander bounds and homological conjectures, Rev

    J. Wei, Auslander bounds and homological conjectures, Rev. Mat. Iberoam., 27(3), (2011), 871-884. 17

    2011

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.