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Virtually Gorenstein Artin algebras are weakly Gorenstein

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Pith's one-line read Over virtually Gorenstein Artin algebras, every semi-Gorenstein-projective module is Gorenstein projective.

desk verdict The main theorem is probably true and worth publishing, but the paper as submitted rests on an omitted proof of the crucial Ext-criterion, so a referee should demand a full write-up before acceptance. read the letter →

arxiv 2608.10049 v2 pith:G5FTW2YJ submitted 2026-08-10 math.RA math.ACmath.RT

Weiqing Li This is my paper · ORCID
classification math.RAmath.ACmath.RT MSC 16G1018G25
keywords virtuallyGorensteinArtinalgebrasweaklyprojectivemodulessemi-Gorenstein-projectiveAuslander-GorensteinConjectureStrongNakayamaAuslander-Reitencotorsiontheory
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 over any virtually Gorenstein Artin algebra, the classes of semi-Gorenstein-projective and Gorenstein projective modules coincide: a module $M$ with $\operatorname{Ext}^i_\Lambda(M,\Lambda)=0$ for all $i\ge1$ is Gorenstein projective ($\perp^\infty\Lambda=\mathcal{GP}$). The proof derives this from a new dimension formula, computing the Gorenstein projective dimension of $M$ as $\inf\{n\mid \operatorname{Ext}^{n+l}_\Lambda(M,\Lambda)=0\text{ for all }l\ge1\}$. If correct, the result settles a 2020 open question for this class and makes every virtually Gorenstein Artin algebra weakly Gorenstein. It also yields the Strong Nakayama Conjecture, the Auslander-Gorenstein Conjecture, and Tachikawa's First Conjecture for these algebras, along with the Auslander-Reiten Conjecture for algebras all of whose Gorenstein projective modules are projective.

What carries the argument

The load-bearing device is the cotorsion theory $(\perp\mathcal{GI},\mathcal{GI})$ -- a pair of module classes orthogonal under $\operatorname{Ext}^1$ -- which, for virtually Gorenstein algebras, is part of a hereditary cotorsion triple $(\mathcal{GP},\perp\mathcal{GI},\mathcal{GI})$. The paper builds a set $X$ of kernels and images taken from injective resolutions of the indecomposable injective modules and from the minimal injective resolution of $\Lambda$, and proves that $\perp\mathcal{GI}=\perp(X^\perp)$ and that $\perp X=\mathcal{GP}$. The dimension formulas of Theorems 3.6 and 3.7 then convert vanishing of Ext against $\Lambda$ into a bound on Gorenstein projective dimension, and the equality $\perp^\infty\Lambda=\mathcal{GP}$ follows.

What would settle it

Construct a virtually Gorenstein Artin algebra $\Lambda$ and a module $M$ with $\operatorname{Ext}^i_\Lambda(M,\Lambda)=0$ for all $i\ge1$ such that $M$ is not Gorenstein projective; this directly contradicts $\perp^\infty\Lambda=\mathcal{GP}$. Equivalently, exhibit a module for which the formula in Theorem 3.7(2) gives a value strictly below the true Gorenstein projective dimension.

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Extended reading notes

Core claim

The central claim is Theorem 3.10: if $\Lambda$ is a virtually Gorenstein Artin algebra, then $\perp^\infty\Lambda=\mathcal{GP}$. In words, a right module $M$ over $\Lambda$ has $\operatorname{Ext}^i_\Lambda(M,\Lambda)=0$ for every $i\ge1$ if and only if $M$ is Gorenstein projective. Since the opposite algebra of a virtually Gorenstein algebra is again virtually Gorenstein, the statement holds on both sides, so $\Lambda$ is weakly Gorenstein. The proof passes through a characterization of modules of virtually finite injective dimension and through new formulas (Theorems 3.6 and 3.7) that compute injective, projective, and Gorenstein-projective dimensions by vanishing of Ext against a single test module: the algebra $\Lambda$ itself or the character module $D(\Lambda^{\mathrm{op}})$.

Load-bearing premise

The argument rests on the unproved half of Theorem 3.7, which says that over these algebras, the Gorenstein projective dimension of every module $M$ is the least $n$ such that $\operatorname{Ext}^{n+l}_\Lambda(M,\Lambda)=0$ for all $l\ge1$; the text points out only that the proof is similar to the dual statement, and if this formula fails then the equality $\perp^\infty\Lambda=\mathcal{GP}$ need not follow.

Editorial extensions

If this is right

  • Every virtually Gorenstein Artin algebra is weakly Gorenstein: the containment $\mathcal{GP}\subseteq\perp^\infty\Lambda$ becomes an equality, so semi-Gorenstein-projective modules over these algebras are exactly the Gorenstein projective ones.
  • The Strong Nakayama Conjecture holds for all modules over virtually Gorenstein Artin algebras: if $\operatorname{Ext}^n_\Lambda(M,\Lambda)=0$ for all $n\ge0$, then $M=0$.
  • The Generalized Nakayama Conjecture, the Nakayama Conjecture, the Auslander-Gorenstein Conjecture, and Tachikawa's First Conjecture follow for virtually Gorenstein Artin algebras.
  • For an Artin algebra in which the Gorenstein projective modules are precisely the projective modules, the Auslander-Reiten Conjecture holds for all modules, and projective dimension is given by the Ext-vanishing formula.
  • The Gorenstein Symmetry Conjecture is reproved for virtually Gorenstein Artin algebras by identifying five numerical invariants of the algebra.

Reading between the lines

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

  • The proof identifies Ext-vanishing against the ring itself as the only test needed for Gorenstein projectivity over virtually Gorenstein algebras; one could test whether the same is true with other test modules $C$ in place of $\Lambda$, which would produce a family of weak Gorenstein conditions relative to $C$.
  • The paper does not consider whether the equality $\perp^\infty\Lambda=\mathcal{GP}$ is preserved under common algebra constructions such as one-point extensions, tensor products, or iterated tilts; checking this would show how far the theorem extends beyond the examples discussed.
  • The new projective-dimension criterion may feed into the finitistic dimension problem: over virtually Gorenstein algebras, bounding the relevant Ext-vanishing index for finitely generated modules of finite projective dimension would bound finitistic dimension directly.
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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

3 major / 4 minor

Summary. The paper studies virtually Gorenstein Artin algebras and claims that for such algebras every semi-Gorenstein-projective module (i.e., every module M with Ext^i(M,Λ)=0 for all i≥1) is Gorenstein projective. This is stated as Theorem 3.10, from which the paper derives weak Gorensteinness, the Strong Nakayama Conjecture, the Auslander-Gorenstein Conjecture, Tachikawa's First Conjecture, and related statements. The central technical tool is Theorem 3.7, which gives a criterion for projective and Gorenstein-projective dimensions in terms of vanishing of Ext against GP or against Λ, respectively. The paper also contains a characterization of modules of virtually finite injective dimension (Theorem 2.8) and applications to the Gorenstein Symmetry Conjecture.

Significance. If the main result is correct, it resolves a natural case of Ringel and Zhang's Question 1.4 and shows that several classical homological conjectures hold for virtually Gorenstein Artin algebras. The paper also proposes new dimension criteria that could be useful beyond this class. However, the decisive arguments are delegated: Theorem 3.6 is said to follow by adapting a lemma of Beligiannis, and Theorem 3.7(2) is dismissed with 'the proof is similar'. Since the main theorem and its corollaries rest on these unproved assertions, the current manuscript does not yet provide a verifiable proof of its central claim.

major comments (3)
  1. [§3, Theorem 3.7(2)] The proof of the Gorenstein projective dimension criterion is omitted; the text states only 'We only prove (1); the proof of (2) is similar.' This equality is load-bearing: Theorem 3.10 and Corollary 3.8 use precisely the case n=0 of (2), i.e., the assertion that ⊥∞Λ ⊆ GP for all modules in Mod-Λ. This is not a routine dual of (1): proving it requires constructing a complete projective resolution for an arbitrary (not necessarily finitely generated) module M with Ext^i(M,Λ)=0 for all i≥1, or invoking a previously established theorem of the same strength. The author should supply the full argument, or state and prove the precise known result being adapted, with all finiteness conditions on the modules involved made explicit.
  2. [§3, Theorem 3.6] Both equalities in Theorem 3.6 are delegated: (1) is said to follow by adapting [8, Lemma 5.1], and (2) is dismissed as 'similar'. These criteria for injective and Gorenstein injective dimension are used in the proof of Theorem 3.7(1) (via the duality argument) and in Theorem 3.14. Since the proof of Theorem 3.6 is not written out, the reader cannot check whether the adaptation works for infinitely generated modules, which is essential for the claimed scope of the paper. Please provide the complete proof or a precise statement of the quoted lemma with the necessary modifications.
  3. [§3, Corollary 3.8] The proof that Ext^i(M,Λ)=0 for all i≥0 forces M=0 is not supplied; the text says only that it is 'a direct consequence of Theorem 3.7'. From Theorem 3.7(2) one obtains Gpd(M)=0, i.e., M is Gorenstein projective. The additional step is to show that a Gorenstein projective module with Hom(M,Λ)=0 must be zero. This can be proved by embedding M into a projective module P via a complete resolution and noting that the cokernel is Gorenstein projective, hence Ext^1(coker,Λ)=0, so Hom(P,Λ)→Hom(M,Λ) is surjective; if Hom(M,Λ)=0 then every map P→Λ vanishes on M, and since projective modules are separated by maps to Λ, M=0. This argument (or an equivalent one) should be written out explicitly.
minor comments (4)
  1. [§1, Definition 1.2] The notation ⊥∞Λ is used for ⊥∞{Λ}, but the convention at the beginning defines ⊥∞C for a class C; writing ⊥∞{Λ} or defining the singleton case would avoid ambiguity.
  2. [§2, proof of Theorem 2.8] There is a duplicated word 'from from' in the sentence 'Hence we see from from the exact sequence (♣) that im(hi)∈ ⊥∞(X⊥).'
  3. [§3, proof of Lemma 3.3] The phrase 'dimension shfting' appears twice and should be 'dimension shifting'. In the same proof, 'Ext ≥i Λ (V,Λ)=0' should presumably be 'Ext^≥i_Λ(V,M)=0', since the vanishing is against M, not against Λ.
  4. [§3, proof of Lemma 3.3] The sentence 'Since the set S finite' should read 'Since S is finite'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main equality is a substantive characterization, not a restatement of inputs.

full rationale

The derivation chain is self-contained in the relevant sense. Theorem 3.10 concludes ⊥∞Λ=GP from Theorem 3.7(2), which asserts Gpd_Λ(M)=inf{n | Ext^{n+l}_Λ(M,Λ)=0 for all l≥1}. Setting n=0 makes the semi-Gorenstein-projective condition Ext^i(M,Λ)=0 (i≥1) equivalent to Gpd(M)=0, and hence to M∈GP. This is not circular: the virtually Gorenstein hypothesis GP⊥=⊥GI and the definition of Gorenstein projective dimension do not contain the conclusion ⊥∞Λ⊆GP; the paper derives the equivalence from prior cotorsion-triple, duality, and filtration results. The one self-citation, [24, Lemma 3.6(2)], is used only in the proof of Theorem 3.7(1) to bound pd(M) by pd(DD(M)) for a pure submodule; it is a published, parameter-free lemma whose assumptions do not include the target result, and it does not drive Theorem 3.10. No fitted parameter is renamed as a prediction, no uniqueness theorem from the authors' prior work is imported to force a choice, and no ansatz is smuggled in via citation. The obvious concern is that the proof of Theorem 3.7(2) is omitted ('the proof of (2) is similar'), and Corollary 3.8 additionally needs a short argument that Hom(M,Λ)=0 with M∈GP forces M=0; but an omitted proof is a completeness/correctness issue, not a circular reduction, since no displayed equation makes the target follow from its own assumption. Therefore no circular step is established.

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

No free parameters or invented entities appear. The proof relies on standard cotorsion theory, duality, and filtration results from the cited literature. The only self-cited supporting lemma is [24, Lemma 3.6(2)], used for the projective dimension of a pure submodule, which is a standard fact and not circular.

assumptions (4)
  • standard math The pair (⊥GI, GI) is a hereditary complete cotorsion theory for any ring.
    Used in Theorem 2.8 and Lemma 3.4 to obtain special precover and preenvelope sequences; cited from Saroch and Stovicek, Theorem 5.6.
  • standard math Ext-Tor duality isomorphisms for character modules over Artin algebras.
    Used in proofs of Theorems 3.6, 3.7 and 3.14 to transfer Ext computations across the duality D; cited from Goebel and Trlifaj, Lemma 2.16(b).
  • standard math Every finitely generated module over an Artin algebra is finitely filtered by the simple modules.
    Used in Lemma 3.3 to lift vanishing of Ext^t(S,M) from simples to all finitely generated modules; cited from Anderson and Fuller, Proposition 11.1.
  • standard math Existence of injective covers over right Noetherian rings.
    Used in Theorem 2.8 to construct the injective resolution of M; cited from Enochs and Jenda.

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Pith. "Pith review of Virtually Gorenstein Artin algebras are weakly Gorenstein." pith.science (2026). https://pith.science/paper/G5FTW2YJ

@misc{pith2026260810049,
  author       = {Pith},
  title        = {Pith review of: Virtually Gorenstein Artin algebras are weakly Gorenstein},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G5FTW2YJ}},
  note         = {Machine review of arXiv:2608.10049}
}
read the original abstract

We prove that all semi-Gorenstein-projective modules over a virtually Gorenstein Artin algebra are Gorenstein projective. It turns out that the Auslander-Gorenstein Conjecture, the (Strong) Nakayama Conjecture and Tachikawa's First Conjecture hold for virtually Gorenstein Artin algebras. We also establish a new criterion for determining the projective (injective) dimensions of modules over virtually Gorenstein Artin algebras. This allows us to show the validity of the Auslander-Reiten Conjecture for (infinitely generated) modules over Artin algebras such that all Gorenstein projective modules are projective.

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Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    Asensio Mayor, J

    J. Asensio Mayor, J. Martinez Hernandez, On flat and projective envelopes,J. Algebra160 (1993), 434-440

    work page 1993

  2. [2]

    F. W. Anderson, K. R. Fuller,Rings and Categories of Modules, 2nd. Springer-Verlag, New York, 1992

    work page 1992

  3. [3]

    Auslander, I

    M. Auslander, I. Reiten, On a generalized version of Nakayama conjecture,Proc. Amer. Math. Soc.52 (1975), 69-74

    work page 1975

  4. [4]

    Auslander, I

    M. Auslander, I. Reiten, Applications of contravariantly finite subcategories,Adv. Math.86 (1991), 111-152. 10

    work page 1991

  5. [5]

    Auslander, I

    M. Auslander, I. Reiten,k-Gorenstein algebras and syzygy modules,J. Pure Appl. Algebra92 (1994), 1-27

    work page 1994

  6. [6]

    Auslander, I

    M. Auslander, I. Reiten, S. O. Smalø,Representation Theory of Artin Algebras.Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge Univ. Press, Cambridge, 1995

    work page 1995

  7. [7]

    Beligiannis, Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras,J

    A. Beligiannis, Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras,J. Al- gebra288 (2005), 137-211

    work page 2005

  8. [8]

    Beligiannis, On algebras of finite Cohen-Macaulay type,Adv

    A. Beligiannis, On algebras of finite Cohen-Macaulay type,Adv. Math.226 (2011), 1973-2019

    work page 2011

Show all 32 references
  1. [9]

    Beligiannis, H

    A. Beligiannis, H. Krause, Thick subcategories and virtually gorenstein algebras,Ill. J. Math.52(2) (2008), 551-562

    2008

  2. [10]

    Beligiannis, I

    A. Beligiannis, I. Reiten, Homological and homotopical aspects of torsion theories,Mem. Amer. Math. Soc. 188(883) (2007), 1-207

    2007

  3. [11]

    Cummings, Left-right symmetry of finite finitistic dimension,Bull

    C. Cummings, Left-right symmetry of finite finitistic dimension,Bull. Lond. Math. Soc.56(2) (2024), 624-633

    2024

  4. [12]

    H. X. Chen, M. Fang, C. C. Xi, Tachikawa¡¯s second conjecture, derived recollements, and gendo- symmetric algebras,Compos. Math.160(11), (2024), 2704-2734

    2024

  5. [13]

    R. R. Colby, K. R. Fuller, A note on the Nakayama Conjecture,Tsukuba J. Math.14 (1990), 343-352

    1990

  6. [14]

    L. W. Christensen, H. Holm, Algebras that satisfy Auslander’s condition on vanishing of cohomology, Math. Z.265 (2010), 21-40

    2010

  7. [15]

    H. X. Chen, C. C. Xi, Homological theory of self-orthogonal modules,Trans. Amer. Math. Soc. 378(10) (2025), 7287-7335

    2025

  8. [16]

    H. X. Chen, C. C. Xi, Virtually Gorenstein algebras of infinite dominant dimension,J. Pure Appl. Algebra230(4) (2026), 108224

    2026

  9. [17]

    E. E. Enochs, O. M. G. Jend, Gorenstein injective and projective modules,Math. Z. 220(4) (1995), 611-633

    1995

  10. [18]

    E. E. Enochs, O. M. G. Jend,Relative Homological Algebra, Berlin, Walter de Gruyter, 2000

    2000

  11. [19]

    Gillespie, A

    J. Gillespie, A. Iacob, Duality pairs, generalized Gorenstein modules, and Ding injective envelopes, Comptes Rendus Math´ematique360 (2022), 381-398

    2022

  12. [20]

    G ¨obel, J

    R. G ¨obel, J. Trlifaj,Approximations and Endomorphism Algebras of Modules, de Gruyter Exp. Math. vol. 41, 2 revised and extended edn. de Gruyter, Berlin, (2012)

    2012

  13. [21]

    Holm, Rings with finite gorenstein injective dimension,Proc

    H. Holm, Rings with finite gorenstein injective dimension,Proc. Amer. Math. Soc.132(5) (2003), 1279-1283

    2003

  14. [22]

    Z. Y . Huang, Selforthogonal modules with finite injective dimension II,J. Algebra264 (2003), 262- 268

    2003

  15. [23]

    Kimura, Y

    K. Kimura, Y . Mifune, Y . Otake, R. Takahashi On stronglyG-regular rings, arXiv:2608.08228. 11

    arXiv

  16. [24]

    W. Q. Li, OnG-(n,d)-rings andn-coherent rings,Int. Electron. J. Algebra37 (2025), 147-178

    2025

  17. [25]

    L. X. Mao, N. Q. Ding, Relative cotorsion modules and relative flat modules,Comm. Algebra34 (2006), 2303-2317

    2006

  18. [26]

    B. J. M ¨ueller, The classification of algebras by dominant dimension,Canad. J. Math.20 (1968), 398- 409

    1968

  19. [27]

    Moradifar, J

    P. Moradifar, J. ˇSaroch, J. ˇSˇtov´ıˇcek, Finitistic dimension conjectures via Gorenstein projective dimen- sion,J. Algebra591 (2022), 15-35

    2022

  20. [28]

    Nakayama, On algebras with complete homology,Abh

    T. Nakayama, On algebras with complete homology,Abh. Math. Sem. Univ. Hamburg22 (1958), 300-307

    1958

  21. [29]

    C. M. Ringel, P. Zhang, Gorenstein-projective and semi-Gorenstein-projective modules,Algebra& Number Theory14(1) (2020), 1-36

    2020

  22. [30]

    C. M. Ringel, P. Zhang, Gorenstein-projective modules over short local algebras.J. Lond. Math. Soc. 106(2) (2022), 528-589

    2022

  23. [31]

    ˇSaroch, J

    J. ˇSaroch, J. ˇSˇtov´ıˇcek, Singular compactness and definability forP-cotorsion and Gorenstein modules, Sel. Math. New Ser.26:23 (2020)

    2020

  24. [32]

    Tachikawa,Quasi-Frobenius rings and generalizations: QF-3and QF-1rings, Notes by Claus Michael Ringel

    H. Tachikawa,Quasi-Frobenius rings and generalizations: QF-3and QF-1rings, Notes by Claus Michael Ringel. Lecture Notes in Mathematics, 351, Berlin-Heidelberg New York, 1973. 12

    1973

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