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Cotorsion pairs, thick subcategories, and finitely generated Gorenstein projective modules

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Pith's one-line read The authors prove that, for a noetherian algebra over a Cohen-Macaulay ring with a canonical module, the finitely generated Gorenstein projective modules are precisely those with vanishing Ext^1 against every module in the thick subcategory

desk verdict A solid, carefully written paper that proves a genuinely new characterization of Gproj and a hereditary cotorsion pair in mod(R), though the main theorem leans on an external result that should be stated explicitly. read the letter →

arxiv 2603.01424 v2 pith:WCFZWUR3 submitted 2026-03-02 math.RA math.ACmath.RT

Souvik Dey , Jian Liu , Xue-Song Lu This is my paper · ORCID
classification math.RAmath.ACmath.RT MSC 13C1413D0716D9016E0516E30
keywords Cohen-MacaulayringsGorensteinprojectivemodulesthicksubcategoriescotorsionpairscanonicalweaklymaximalnoetherianalgebras
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 an exact description of the finitely generated Gorenstein projective modules over any noetherian algebra R over a Cohen-Macaulay ring S that admits a canonical module ω, provided R itself is maximal Cohen-Macaulay as an S-module. It proves that a finitely generated R-module is Gorenstein projective if and only if it has vanishing Ext^1 against every module in the thick subcategory generated by the finitely generated projectives together with Hom_S(R,ω). From this equality, the authors derive a hereditary cotorsion pair in the category of finitely generated modules, a characterization of when R is weakly Gorenstein in terms of a single short exact sequence, and a criterion for Cohen-Macaulay local rings to be Gorenstein. The payoff is that a subtle homological property becomes a single equality of module classes that can be checked explicitly.

What carries the argument

The load-bearing object is the thick subcategory Thick(proj(R) ∪ {R†}) in mod(R), where R† = Hom_S(R,ω); a thick subcategory is the smallest class of modules closed under direct summands and under the two-out-of-three rule in short exact sequences that contains all finitely generated projectives and the dual module R†. The central mechanism is the identification of the left Ext^1-orthogonal class of this subcategory with Gproj(R). Two auxiliary tools carry the proof: the change-of-rings isomorphism Ext^i_S(M,ω) ≅ Ext^i_R(M,R†), and a criterion stating that a finitely generated module is Gorenstein projective if it has vanishing positive Ext against R and admits a projective coresolution whos

What would settle it

A decisive counterexample would be a noetherian S-algebra R satisfying the hypotheses (S Cohen-Macaulay with canonical module ω, R maximal Cohen-Macaulay over S) together with a finitely generated module M for which Ext^1_R(M,T)=0 for every T in Thick(proj(R) ∪ {Hom_S(R,ω)}) but M is not Gorenstein projective. Alternatively, a Cohen-Macaulay local ring with Gproj(R)⊥1 = P<∞(R) that is not Gorenstein would falsify the final criterion.

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Extended reading notes

Core claim

The central discovery is Theorem 3.6: if S is a Cohen-Macaulay ring admitting a canonical module ω and R is a noetherian S-algebra with R ∈ MCM(S) as an S-module, then Gproj(R) = ⊥1 Thick(proj(R) ∪ {Hom_S(R,ω)}). That is, the finitely generated Gorenstein projective R-modules are exactly those modules X for which Ext^1_R(X,T)=0 for every T in the smallest thick subcategory containing the finitely generated projectives and the module Hom_S(R,ω). The forward direction shows every Gorenstein projective module is maximal Cohen-Macaulay over S and uses the change-of-rings isomorphism Ext^i_S(M,ω) ≅ Ext^i_R(M,Hom_S(R,ω)) to get orthogonality. The reverse direction uses a Cohen-Macaulay approximati

Load-bearing premise

The main equality depends on a cited Cohen-Macaulay approximation theorem that supplies, for every finitely generated module, a short exact sequence whose kernel lies in the thick subcategory and whose middle term is maximal Cohen-Macaulay over S; if that theorem does not hold under exactly the stated hypotheses, the characterization collapses.

Editorial extensions

If this is right

  • For every ring R satisfying the hypotheses, (Gproj(R), Gproj(R)⊥1) is a hereditary cotorsion pair in mod(R), so the finitely generated Gorenstein projective modules form the left half of an approximation-theoretic pair.
  • If S has finite Krull dimension, then Gproj(R) = ⊥1 Thick(proj(R) ∪ I<∞(R)), so the same characterization can be stated with modules of finite injective dimension in place of Hom_S(R,ω).
  • The ring R is left weakly Gorenstein if and only if Thick(proj(R) ∪ {R†}) is contained in the double Ext-orthogonal of R; equivalently, a single module X defined by a short exact sequence 0→X→P→R†→0 suffices to test the property.
  • For a Cohen-Macaulay local ring, R is Gorenstein if and only if Gproj(R)⊥1 = P<∞(R), and this is also equivalent to (Gproj(R), P<∞(R)) being a cotorsion pair.
  • For Artin algebras, a semi-Gorenstein projective module with finite Auslander bound is automatically Gorenstein projective.

Reading between the lines

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

  • The equality Gproj(R) = ⊥1 Thick(proj(R) ∪ {R†}) reduces a global acyclicity condition to a single family of Ext^1-vanishing conditions, which could make Gorenstein projectivity checkable by computations once R† and its syzygies are known.
  • Because the characterization is phrased purely in terms of a thick subcategory, it may transfer to derived or singularity categories over the same class of algebras, potentially giving new descriptions of the stable category Gproj(R).
  • The weakly Gorenstein test suggests that, in examples where a presentation of R† is explicit, the single module X gives a finite certificate for weak Gorensteinness, which could be pushed to algorithmic verification for concrete algebras.
  • If the cited Cohen-Macaulay approximation theorem could be replaced by a weaker splitting condition, the main theorem might extend beyond the current hypotheses to algebras that are not finite over S or lack a canonical module.
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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 / 5 minor

Summary. The paper studies finitely generated Gorenstein projective modules over noetherian algebras R that are module-finite over a Cohen-Macaulay ring S with canonical module ω and are maximal Cohen-Macaulay over S. Its central result, Theorem 3.6, identifies Gproj(R) with the left Ext^1-orthogonal class of Thick(proj(R) ∪ {Hom_S(R,ω)}). From this the authors derive a hereditary cotorsion pair (Gproj(R), Gproj(R)^⊥1) in mod(R) (Corollary 3.10), characterizations of left weakly Gorenstein rings (Theorem 4.5), and, for Cohen-Macaulay local rings, an equivalence between Gorensteinness, Gproj(R)^⊥1 = P^{<∞}(R), and the cotorsion-pair condition (Theorem 5.6). The paper also contains results on Artin algebras, including Proposition 3.13 and Corollary 3.15, and remarks on virtually Gorenstein algebras. The overall structure is clear, and the main chain of implications is coherent, but the proof of the central theorem relies on an unstated external theorem, and one assertion in the proof of Theorem 5.6 is insufficiently justified.

Significance. If the results are correct, they are valuable. Theorem 3.6 gives a finitely generated analogue of the recent cotorsion-pair theorem of Cortés-Izurdiaga and Šaroch and yields the desired hereditary cotorsion pair in mod(R). The weakly Gorenstein criteria and the Gorensteinness criterion for Cohen-Macaulay local rings are clean, testable statements; Theorem 5.6 addresses a question highlighted in the abstract as due to Takahashi. The paper is well organized, gives illustrative examples, and the internal line of reasoning is transparent. There is no apparent circularity and no parameter-fitting or definitional reduction. The main risks are external black boxes: the unstated Miyachi theorem used in Theorem 3.6 and the unproved existence of a finite-injective-dimension module used in Corollary 5.5. These are load-bearing, so the paper needs revision before the claims can be fully accepted.

major comments (3)
  1. [§3, Theorem 3.6] The converse containment in Theorem 3.6 depends completely on [33, Theorem 1.2(c)], which is cited but never stated. The proof needs, for every X ∈ ⊥1 Thick(proj(R)∪{R†}), a short exact sequence 0→K→M→X→0 with K ∈ Thick(proj(R)∪{R†}) and M maximal Cohen-Macaulay over S. This is what forces X to be maximal Cohen-Macaulay and is then used for the embedding via Lemma 3.4 and for the vanishing of Ext^i_R(X,R†). Please state Miyachi's theorem precisely, with all hypotheses, and verify that it applies under the paper's assumptions: S a Cohen-Macaulay ring with canonical module, R a noetherian S-algebra, R ∈ MCM(S), with no local/complete hypothesis and no order assumption. In particular, confirm that K lies in the stated thick subcategory and that M is maximal Cohen-Macaulay over S. The dependency propagates to Corollary 3.10, Remark 3.7, and Theorem 4.5, so this is not an isolated presentatio
  2. [§5, Corollary 5.5 and Theorem 5.6] Corollary 5.5 asserts that every Cohen-Macaulay local ring admits a nonzero finitely generated module of finite injective dimension, citing only the vague 'discussion following [9, Corollary 9.6.2]'. This is load-bearing for the implication (2)⇒(1) in Theorem 5.6. The assertion is not a formal consequence of the definition; canonical modules, which are the standard source of such modules, are known not to exist for every Cohen-Macaulay local ring unless the ring is a homomorphic image of a Gorenstein local ring (see, for example, Bruns–Herzog, Theorem 3.3.6). Please supply a precise reference with a statement, or prove the existence directly under the hypotheses of Corollary 5.5. Without this, the proof of Theorem 5.6 has a gap.
  3. [§2.11, §3 (proof of Theorem 3.6), §4 (Lemma 4.4)] The proof of Theorem 3.6 begins by taking a left proj(R)-approximation f0:X→Q−1, referring to 2.11. But 2.11 only records covariantly finite add(M) for Artin algebras and commutative noetherian rings, while the theorem allows an arbitrary noetherian S-algebra. Existence of left proj(R)-approximations is not automatic from the existence of projective covers. It can be proved for any two-sided noetherian R by choosing a finite generating set of the right R-module Hom_R(X,R) and forming the corresponding map X→R^n, but this argument is absent. The same existence is needed in Lemma 4.4. Please add this argument or a reference covering the noncommutative case.
minor comments (5)
  1. [Title] The title on the first page of the manuscript differs from the arXiv metadata. Please align the two.
  2. [§3, proof of Theorem 3.6] In the final paragraph of the proof, the phrase 'as X∈Gproj(R)' is misleading: X is not yet known to be Gorenstein projective at that point. The intended justification is that X was shown to be maximal Cohen-Macaulay as an S-module by the splitting argument; please reword.
  3. [§3, Proposition 3.13] In cases (2) and (3) of the proof, the notation 'Hom_Λ(d0, Ki)' and 'Hom_Λ(d0, Ki), i≥0' should refer to the relevant test module in S_Λ ∪ T_Λ; the symbols Ki and Li are mixed. Please correct the displayed morphisms and the concluding sentences.
  4. [References] References [26] and [27] appear to be duplicates: both are listed as Iyengar–Krause, 'The Nakayama functor and its completion for Gorenstein algebras', Bull. Soc. Math. Fr. 150 (2022), no. 2, 347–391. Please remove the duplicate.
  5. [§3, Lemma 3.5] The proof says the isomorphism follows from the derived tensor–hom adjunction and that R† is quasi-isomorphic to RHom_S(R,ω). This is terse; since the point is used in several places, please include a direct argument via an injective resolution of ω over S, or at least explain why Hom_S(R,I) gives an injective resolution of R† over R.

Circularity Check

1 steps flagged · score 1.0 of 10

Local circular phrasing in Theorem 3.6's converse, but not load-bearing; the central derivation is otherwise independent of its own conclusions.

  1. other [Theorem 3.6, converse direction, paragraph beginning 'In order to show that X∈Gproj(R)']
    "As being shown in the first paragraph of the proof,Xis maximal Cohen–Macaulay as anS-module asX∈Gproj(R), and it follows that Ext i R(X, R†) = Ext i S(X, ω) = 0 fori >0 by Lemma 3.5 and Lemma 3.3 (1)."

    At this point in the converse direction, X is only assumed to lie in ⊥1 Thick(proj(R)∪{R†}); the proof is still trying to prove X∈Gproj(R). Read literally, the sentence uses the target conclusion X∈Gproj(R) to justify the MCM property and Ext-vanishing needed for that same conclusion. However, the step is not load-bearing: immediately before, the split of Miyachi's sequence had already shown that X is a direct summand of an S-maximal Cohen–Macaulay module, so X∈MCM(S), and hence Ext^i_R(X,R†)=Ext^i_S(X,ω)=0 follows from Lemmas 3.3 and 3.5 without invoking X∈Gproj(R). Thus this is a local phrasing slip rather than a derivation that reduces to its own output.

full rationale

Apart from the local circular locution in Theorem 3.6, the paper's derivation chain is not circular. The two sides of the main equality, Gproj(R) and ⊥1 Thick(proj(R)∪{R†}), have independent definitions: the former via totally acyclic complexes of finitely generated projectives, and the latter via Ext1-orthogonality to a thick subcategory. The proof bridges them with external results, chiefly Miyachi [33, Theorem 1.2(c)] for a Cohen–Macaulay approximation sequence, Lemma 3.4 for canonical-module biduality, and Lemma 3.5 for the change-of-rings Ext isomorphism. The fact that [33, Theorem 1.2(c)] is cited but not stated is a correctness/verification risk, not circularity, since it is prior external work rather than the paper's own theorem. The paper contains no self-citations by the authors, no fitted parameters passed off as predictions, and no uniqueness or ansatz imported from the authors' earlier work. Later results (Corollary 3.10, Theorems 4.5 and 5.6) are derived from Theorem 3.6 together with standard literature (Hovey, Foxby, etc.), not from their own conclusions. The isolated sentence invoking 'as X∈Gproj(R)' is a non-load-bearing typo, so the overall circularity score is minimal.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

This is a data-free pure-mathematics paper. The central claims rest on standard homological algebra background, the stated hypotheses on S and R, and one external approximation theorem (Miyachi) that is cited but not stated. There are no fitted numerical parameters and no newly postulated objects.

assumptions (6)
  • domain assumption S is a Cohen-Macaulay ring admitting a canonical module ω
    Stated in Theorems 1.1, 1.2, 3.6, and 4.5; needed to define R† = Hom_S(R,ω) and for the canonical-module duality of maximal Cohen-Macaulay modules (Lemma 3.3).
  • domain assumption R is a noetherian S-algebra that is maximal Cohen-Macaulay as an S-module
    Stated throughout; needed for Lemma 3.4 (projective coresolution by copies of R†), Lemma 3.5 (Ext adjunction), and for proving that every Gorenstein projective R-module is MCM over S.
  • domain assumption Miyachi's Cohen-Macaulay approximation theorem [33, Theorem 1.2(b),(c)]
    Used in Theorem 3.6 to obtain the short exact sequence 0→K→M→X→0 with K ∈ Thick(proj(R)∪{R†}) and M maximal Cohen-Macaulay as an S-module; this is load-bearing for showing every X in the left orthogonal is MCM.
  • standard math Canonical-module duality for CM rings: MCM(S) = ⊥∞ω and M ≅ Hom_S(Hom_S(M,ω),ω) for M ∈ MCM(S) [9, Theorem 3.3.10]
    Used in Lemmas 3.3 and 3.4; standard in Cohen-Macaulay ring theory.
  • standard math Foxby's criterion: a nonzero finitely generated module over a local ring with finite projective and finite injective dimension forces the ring to be Gorenstein [19, Corollary 4.4]
    Used in Lemma 5.3 to prove Theorem 5.6 (2)⇒(1).
  • standard math Hovey's theorem: for Iwanaga-Gorenstein rings, (Gproj(R), P<∞(R)) is a cotorsion pair in mod(R) [22, Theorem 8.3]
    Used in Theorem 5.6 (1)⇒(3).

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Pith. "Pith review of Cotorsion pairs, thick subcategories, and finitely generated Gorenstein projective modules." pith.science (2026). https://pith.science/paper/WCFZWUR3

@misc{pith2026260301424,
  author       = {Pith},
  title        = {Pith review of: Cotorsion pairs, thick subcategories, and finitely generated Gorenstein projective modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WCFZWUR3}},
  note         = {Machine review of arXiv:2603.01424}
}
abstract

Let $R$ be a noetherian algebra over a Cohen--Macaulay ring $S$ admitting a canonical module $\omega$, and assume that $R$ is maximal Cohen--Macaulay over $S$. We prove that the category of finitely generated Gorenstein projective $R$-modules coincides with the left $\mathrm Ext$-orthogonal class of the thick subcategory generated by $R$ and ${\mathrm Hom}_S(R,\omega)$. As an application, finitely generated Gorenstein projective $R$-modules form the left half of a hereditary cotorsion pair. In the case of Cohen--Macaulay local rings, this yields an affirmative answer to a question of R. Takahashi. We further characterize when $R$ is left weakly Gorenstein. Finally, we prove that a Cohen--Macaulay local ring is Gorenstein if and only if the right $\mathrm Ext$-orthogonal class of finitely generated Gorenstein projective modules coincides with the category of finitely generated modules of finite projective dimension.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On strongly G-regular rings

    math.AC 2026-08 conditional novelty 8.0 of 10

    The authors construct commutative local artin algebras that are G-regular and weakly Gorenstein but admit infinitely generated non-projective Gorenstein projective modules, refuting Chen's Problems A, B, and C.

Reference graph

Works this paper leans on

39 extracted references · 2 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Tokuji Araya and Olgur Celikbas,Reducing invariants and total reflexivity, Ill. J. Math.64 (2020), no. 2, 169–184

    2020

  2. [2]

    Tokuji Araya and Ryo Takahashi,On reducing homological dimensions over Noetherian rings, Proc. Am. Math. Soc.150(2022), no. 2, 469–480

    2022

  3. [3]

    Auslander and Sverre O

    M. Auslander and Sverre O. Smalø,Preprojective modules over Artin algebras, J. Algebra66 (1980), no. 1, 61–122. MR 591246

    1980

  4. [4]

    Samuel (1966-1967), tome 1, notes by: M

    Maurice Auslander,Anneaux de Gorenstein et torsion en alg´ ebre commutative, S` eminaire d’alg´ ebre commutative dirig´ e par P. Samuel (1966-1967), tome 1, notes by: M. Mangeney, C. Peskine, L. Szpiro, Secr´ etariat Math´ ematique, Paris, 2–69, 1967

    1966

  5. [5]

    Maurice Auslander and Ragnar-Olaf Buchweitz,The homological theory of maximal Cohen- Macaulay approximations, M´ em. Soc. Math. Fr., Nouv. S´ er.38(1989), 5–37

    1989

  6. [6]

    Algebra288(2005), no

    Apostolos Beligiannis,Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras., J. Algebra288(2005), no. 1, 137–211

    2005

  7. [7]

    Apostolos Beligiannis and Henning Krause,Thick subcategories and virtually Gorenstein al- gebras., Ill. J. Math.52(2008), no. 2, 551–562

    2008

  8. [8]

    Apostolos Beligiannis and Idun Reiten,Homological and homotopical aspects of torsion theo- ries, Mem. Am. Math. Soc., vol. 883, Providence, RI: American Mathematical Society (AMS), 2007. 20 SOUVIK DEY, JIAN LIU, AND XUE-SONG LU

    2007

Show all 39 references
  1. [9]

    J¨ urgen Herzog,Cohen-Macaulay rings, Cambridge Studies in Ad- vanced Mathematics, vol

    Winfried Bruns and H. J¨ urgen Herzog,Cohen-Macaulay rings, Cambridge Studies in Ad- vanced Mathematics, vol. 39, Cambridge University Press, 1998

    1998

  2. [10]

    With ap- pendices by Luchezar L

    Ragnar-Olaf Buchweitz,Maximal Cohen-Macaulay modules and Tate cohomology. With ap- pendices by Luchezar L. Avramov, Benjamin Briggs, Srikanth B. Iyengar and Janina C. Letz, Math. Surv. Monogr., vol. 262, Providence, RI: American Mathematical Society (AMS), 2021

    2021

  3. [11]

    Xiao-Wu Chen,Gorenstein Homological Algebra of Artin Algebras, Preprint, arXiv:1712.04587, 2017

    2017 arXiv

  4. [12]

    Notes Math., vol

    Lars Winther Christensen,Gorenstein dimensions, Lect. Notes Math., vol. 1747, Berlin: Springer, 2000

    2000

  5. [13]

    Z.265(2010), no

    Lars Winther Christensen and Henrik Holm,Algebras that satisfy Auslander’s condition on vanishing of cohomology, Math. Z.265(2010), no. 1, 21–40. MR 2606948

    2010

  6. [14]

    Math.218(2008), no

    Lars Winther Christensen, Greg Piepmeyer, Janet Striuli, and Ryo Takahashi,Finite Goren- stein representation type implies simple singularity, Adv. Math.218(2008), no. 4, 1012–1026

    2008

  7. [15]

    Manuel Cort´ es-Izurdiaga and Jan ˇSaroch,The cotorsion pair generated by the Gorenstein projective modules andλ-pure-injective modules, Isr. J. Math. (2025), 1–30

    2025

  8. [16]

    Pure Appl

    Zhenxing Di, Li Liang, and Junpeng Wang,Virtually Gorenstein rings and relative homology of complexes, J. Pure Appl. Algebra227(2023), no. 1, 15, Id/No 107127

    2023

  9. [17]

    Enochs and Overtoun M

    Edgar E. Enochs and Overtoun M. G. Jenda,Gorenstein injective and projective modules, Math. Z.220(1995), no. 4, 611–633

    1995

  10. [18]

    ,Relative homological algebra. Vol. 2, 2nd revised ed. ed., De Gruyter Expo. Math., vol. 54, Berlin: Walter de Gruyter, 2011

    2011

  11. [19]

    Scand.40(1977), 5–19

    Hans-Bjorn Foxby,Isomorphisms between complexes with applications to the homological the- ory of modules, Math. Scand.40(1977), 5–19

    1977

  12. [20]

    Nan Gao, Pu Zhang, and Shijie Zhu,Weakly gorensteinness of tensor algebras and morita algebras, Preprint,arXiv:2512.16072, 2025

    2025

  13. [21]

    Dieter Happel,Triangulated categories in the representation theory of finite dimensional alge- bras, Lond. Math. Soc. Lect. Note Ser., vol. 119, Cambridge (UK) etc.: Cambridge University Press, 1988

    1988

  14. [22]

    Mark Hovey,Cotorsion pairs, model category structures, and representation theory, Math. Z. 241(2002), no. 3, 553–592. MR 1938704

    2002

  15. [23]

    Chonghui Huang and Zhaoyong Huang,Torsionfree dimension of modules and self-injective dimension of rings, Osaka J. Math. (2012), 21–35

    2012

  16. [24]

    Zhaoyong Huang,Auslander-type conditions and weakly Gorenstein algebras, Bull. Lond. Math. Soc.56(2024), no. 11, 3382–3399

    2024

  17. [25]

    Math.11(2006), 207–240

    Srikanth Iyengar and Henning Krause,Acyclicity versus total acyclicity for complexes over noetherian rings, Doc. Math.11(2006), 207–240

    2006

  18. [26]

    Iyengar and Henning Krause,The Nakayama functor and its completion for Gorenstein algebras, Bull

    Srikanth B. Iyengar and Henning Krause,The Nakayama functor and its completion for Gorenstein algebras, Bull. Soc. Math. Fr.150(2022), no. 2, 347–391

    2022

  19. [27]

    ,The Nakayama functor and its completion for Gorenstein algebras, Bull. Soc. Math. Fr.150(2022), no. 2, 347–391

    2022

  20. [28]

    Lars Kadison,New examples of Frobenius extensions, Univ. Lect. Ser., vol. 14, Providence, RI: American Mathematical Society, 1999

    1999

  21. [29]

    J.212(2013), 87–96

    Henning Krause and Greg Stevenson,A note on thick subcategories of stable derived cate- gories, Nagoya Math. J.212(2013), 87–96

    2013

  22. [30]

    Algebra526(2019), 104–111

    Ren´ e Marczinzik,On a new formula for the Gorenstein dimension, J. Algebra526(2019), 104–111

    2019

  23. [31]

    ,Gendo-symmetric algebras, dominant dimensions and gorenstein homological alge- bra, Preprint, arXiv:1608.04212, 2021

    2021 arXiv

  24. [32]

    Hideyuki Matsumura,Commutative ring theory. Transl. from the Japanese by M. Reid, Camb. Stud. Adv. Math., vol. 8, Cambridge University Press, Cambridge, 1986

    1986

  25. [33]

    Algebra 26(1998), no

    Jun-ichi Miyachi,Cohen-Macaulay approximations and Noetherian algebras, Comm. Algebra 26(1998), no. 7, 2181–2190. MR 1626578

    1998

  26. [34]

    Claus Michael Ringel and Pu Zhang,Gorenstein-projective and semi-Gorenstein-projective modules, Algebra Number Theory14(2020), no. 1, 1–36

    2020

  27. [35]

    Algebra301 (2006), no

    Ryo Takahashi,Remarks on modules approximated byG-projective modules, J. Algebra301 (2006), no. 2, 748–780

    2006

  28. [36]

    Jiaqun Wei,Auslander bounds and homological conjectures., Rev. Mat. Iberoam.27(2011), no. 3, 871–884. SOME CRITERIA FOR GORENSTEINNESS 21

    2011

  29. [37]

    Proceedings of the NATO advanced research workshop, Sinaia, Romania, September 17–22, 2002, Dordrecht: Kluwer Academic Publishers, 2003, pp

    Yuji Yoshino,Modules of G-dimension zero over local rings with the cube of maximal ideal be- ing zero, Commutative algebra, singularities and computer algebra. Proceedings of the NATO advanced research workshop, Sinaia, Romania, September 17–22, 2002, Dordrecht: Kluwer Academi...

    2002

  30. [38]

    Algebra13(1969), 73–86

    Abraham Zaks,Injective dimension of semi-primary rings, J. Algebra13(1969), 73–86

    1969

  31. [39]

    Pure Appl

    Fatemeh Zareh-Khoshchehreh, Mohsen Asgharzadeh, and Kamran Divaani-Aazar,Gorenstein homology, relative pure homology and virtually Gorenstein rings, J. Pure Appl. Algebra218 (2014), no. 12, 2356–2366. Department of Mathematical Sciences, 850 West Dickson Street, University of ...

    2014

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