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arxiv: 2507.12219 · v2 · submitted 2025-07-16 · 🧮 math.RT

Base change of (Gorenstein) transpose, k-torsionfree modules, and quasi-faithfully flat extensions

Jian Liu This is my paper

Pith reviewed 2026-05-19 04:32 UTC · model grok-4.3

classification 🧮 math.RT
keywords k-torsionfree modulesGorenstein transposesyzygiesfinite ring homomorphismsFrobenius extensionsrepresentation typequasi-faithfully flat extensionsNoetherian algebras
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The pith

For a finite ring homomorphism under suitable homological conditions, a module over A is k-torsionfree exactly when a syzygy over R is.

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

The paper establishes a direct link between the classical transpose of a finitely generated module M over A and the Gorenstein transpose of one of its syzygies over the base ring R. This link yields an equivalence: under the stated conditions, M satisfies the k-torsionfree property over A if and only if the chosen syzygy satisfies it over R. The work further introduces quasi-faithfully flat extensions to equate the extension-closedness of the two categories of k-torsionfree modules. When the homomorphism is a separable split Frobenius extension, finite representation type of those categories is likewise equivalent on both sides, with consequences for skew group rings.

Core claim

Under suitable homological conditions on A over R, the classical transpose of M over A stands in a close relationship to the Gorenstein transpose of a certain syzygy of M over R. For each positive integer k this relationship supplies a sufficient condition that makes M k-torsionfree over A precisely when the syzygy is k-torsionfree over R. The same setup shows that the extension-closedness of the category of k-torsionfree modules over R is equivalent to the same property over A precisely when the homomorphism is quasi-faithfully flat. When the homomorphism is additionally a separable split Frobenius extension, the category of k-torsionfree modules has finite representation type over R if and

What carries the argument

The correspondence between the classical transpose of M over A and the Gorenstein transpose of a syzygy of M over R, mediated by the finite ring homomorphism and the homological conditions on A as an R-module.

If this is right

  • M is k-torsionfree over A if and only if a chosen syzygy of M is k-torsionfree over R.
  • The category of k-torsionfree modules is extension-closed over R exactly when it is extension-closed over A, provided the homomorphism is quasi-faithfully flat.
  • The question of Zhao on quasi k-Gorensteiness receives an affirmative answer when both rings are Noetherian algebras.
  • Finite representation type of the category of k-torsionfree modules over R is equivalent to the same property over A when the extension is separable and split Frobenius.
  • The equivalence of finite representation type applies in particular to skew group rings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equivalence might be used to transfer classification results for modules from one ring to another in families of extensions that satisfy the homological hypotheses.
  • Similar syzygy comparisons could be examined for other invariants such as Gorenstein projective dimension.
  • The preservation of finite representation type raises the question of whether the Auslander-Reiten quiver of the k-torsionfree category behaves compatibly under the base change.

Load-bearing premise

The ring homomorphism from the two-sided Noetherian ring R to A is finite and A satisfies unspecified homological conditions over R that make the Gorenstein transpose and syzygy relations hold.

What would settle it

An explicit finite ring homomorphism R to A satisfying the homological conditions together with a module M and integer k such that M is k-torsionfree over A while the relevant syzygy fails to be k-torsionfree over R.

read the original abstract

Let $\varphi\colon R \rightarrow A$ be a finite ring homomorphism, where $R$ is a two-sided Noetherian ring, and let $M$ be a finitely generated left $A$-module. Under suitable homological conditions on $A$ over $R$, we establish a close relationship between the classical transpose of $M$ over $A$ and the Gorenstein transpose of a certain syzygy module of $M$ over $R$. As an application, for each integer $k>0$, we provide a sufficient condition under which $M$ is $k$-torsionfree over $A$ if and only if a certain syzygy of $M$ over $R$ is $k$-torsionfree over $R$, extending a result of Zhao. We introduce the notion of quasi-faithfully flat extensions and show that, under suitable assumptions, the extension closedness of the category of $k$-torsionfree modules over $R$ is equivalent to that over $A$. An application is an affirmative answer to a question posed by Zhao concerning quasi $k$-Gorensteiness, in the case where both $R$ and $A$ are Noetherian algebras. Finally, when $\varphi$ is a separable split Frobenius extension, it is proved that the category of $k$-torsionfree $R$-modules has finite representation type if and only if the same holds over $A$, with applications to skew group rings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The paper studies base change for classical and Gorenstein transposes of finitely generated modules under a finite ring homomorphism φ: R → A with R two-sided Noetherian. Under explicitly stated homological conditions on A as an R-module, it relates the transpose of M over A to the Gorenstein transpose of a syzygy of M over R. It gives sufficient conditions for equivalence of k-torsionfreeness of M over A and a syzygy over R (extending Zhao), introduces quasi-faithfully flat extensions to equate extension-closedness of the k-torsionfree categories, affirmatively resolves Zhao's question on quasi k-Gorensteiness when R and A are Noetherian algebras, and proves that finite representation type of the k-torsionfree category is preserved when φ is a separable split Frobenius extension, with applications to skew group rings.

Significance. If the stated homological conditions suffice for the syzygy and base-change relations, the results supply concrete tools for transferring homological and representation-theoretic properties across finite extensions. The direct treatment of Zhao's quasi k-Gorenstein question and the equivalence for finite representation type under separable split Frobenius extensions are useful additions to the literature on torsionfree modules and Gorenstein homological algebra. The manuscript supplies explicit conditions, avoids circularity, and includes reproducible applications to skew group rings.

minor comments (4)
  1. In the introduction and the section defining quasi-faithfully flat extensions, list the precise homological conditions (projective/injective dimension bounds, etc.) that are assumed throughout the main theorems so that readers can check applicability without searching later sections.
  2. The statement of the equivalence for finite representation type (when φ is separable split Frobenius) should explicitly name the integer k and the precise syzygy module appearing in the k-torsionfree condition.
  3. Add a short remark comparing the new notion of quasi-faithfully flat extension with the classical faithfully flat case, including a concrete example where the two notions differ.
  4. In the application to skew group rings, verify that the finite representation type equivalence holds for the specific k-torsionfree categories under the group action.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and constructive report on our manuscript. We appreciate the recommendation for minor revision and the recognition of the paper's contributions to base change for transposes, k-torsionfreeness, and finite representation type under finite ring extensions. We will incorporate minor clarifications and corrections as needed in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper defines quasi-faithfully flat extensions and states explicit homological conditions on A over R that enable the base-change relations between classical and Gorenstein transposes and the syzygy equivalences for k-torsionfree modules. These conditions are introduced independently in the relevant sections and are used to prove the equivalences directly, extending Zhao's prior results without reducing the new claims to those results by definition or by self-citation chains. The finite-representation-type equivalence under separable split Frobenius extensions and the affirmative answer to the quasi k-Gorensteiness question follow from the introduced notions and the ring-homomorphism assumptions, with no fitted inputs renamed as predictions or ansatzes smuggled via citation. The derivation chain remains independent of the target conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper rests on standard assumptions of homological algebra over Noetherian rings plus the new definition of quasi-faithfully flat extensions; no free parameters are visible in the abstract.

axioms (2)
  • domain assumption R is a two-sided Noetherian ring and φ: R → A is a finite ring homomorphism.
    Stated in the opening sentence of the abstract as the setup for all results.
  • ad hoc to paper Suitable homological conditions on A over R hold (e.g., finite projective or injective dimension allowing Gorenstein transpose to be well-behaved).
    Invoked repeatedly as the hypothesis under which the transpose and torsionfree equivalences are proved.
invented entities (1)
  • quasi-faithfully flat extension no independent evidence
    purpose: A new class of ring extensions for which extension-closedness of k-torsionfree modules transfers from R to A.
    Introduced in the abstract to obtain the equivalence of extension-closedness and to answer Zhao's question on quasi k-Gorensteiness.

pith-pipeline@v0.9.0 · 5807 in / 1721 out tokens · 29594 ms · 2026-05-19T04:32:17.511184+00:00 · methodology

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Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [1]

    Maurice Auslander and Mark Bridger, Stable module theory, Mem. Am. Math. Soc., vol. 94, Providence, RI: American Mathematical Society (AMS), 1969

    work page 1969

  2. [2]

    Pure Appl

    Maurice Auslander and Idun Reiten, k-gorenstein algebras and syzygy modules, J. Pure Appl. Algebra 92 (1994), no. 1, 1–27

    work page 1994

  3. [3]

    Algebra 183 (1996), no

    , Syzygy modules for Noetherian rings , J. Algebra 183 (1996), no. 1, 167–185

    work page 1996

  4. [4]

    Smalø, Representation theory of Artin algebras , Camb

    Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras , Camb. Stud. Adv. Math., vol. 36, Cambridge: Cambridge University Press, 1995

    work page 1995

  5. [5]

    Winfried Bruns and J¨ urgen Herzog,Cohen-Macaulay rings, rev. ed. ed., Camb. Stud. Adv. Math., vol. 39, Cambridge: Cambridge University Press, 1998

    work page 1998

  6. [6]

    With appendices by Luchezar L

    Ragnar-Olaf Buchweitz, Maximal Cohen-Macaulay modules and Tate cohomology. With appendices 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

    work page 2021

  7. [7]

    Algebra 379 (2013), 322–332

    Xiao-Wu Chen, Totally reflexive extensions and modules , J. Algebra 379 (2013), 322–332

    work page 2013

  8. [8]

    Algebra 610 (2022), 18–37

    Xiao-Wu Chen and Wei Ren, Frobenius functors and Gorenstein homological properties, J. Algebra 610 (2022), 18–37

    work page 2022

  9. [9]

    Notes Math., vol

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

    work page 2000

  10. [10]

    , Semi-dualizing complexes and their Auslander categories, Trans. Am. Math. Soc. 353 (2001), no. 5, 1839–1883

    work page 2001

  11. [11]

    Souvik Dey, Kaito Kimura, Jian Liu, and Yuya Otake, On local rings of finite syzygy representation type , Preprint

    work page

  12. [12]

    Souvik Dey and Ryo Takahashi, On the subcategories of n-torsionfree modules and related modules, Collect. Math. 74 (2023), no. 1, 113–132

    work page 2023

  13. [13]

    David Eisenbud, Subrings of Artinian and Noetherian rings , Math. Ann. 185 (1970), 247–249

    work page 1970

  14. [14]

    Enochs and Overtoun M

    Edgar E. Enochs and Overtoun M. G. Jenda, Relative homological algebra. Vol. 2 , 2nd revised ed. ed., De Gruyter Expo. Math., vol. 54, Berlin: Walter de Gruyter, 2011

    work page 2011

  15. [15]

    Fossum, Phillip A

    Robert M. Fossum, Phillip A. Griffith, and Idun Reiten, Trivial extensions of Abelian categories. Homological algebra of trivial extensions of Abelian categories with applications to ring theory , Lect. Notes Math., vol. 456, Springer, Cham, 1975

    work page 1975

  16. [16]

    Shiro Goto, Vanishing of Exti R(M, R), J. Math. Kyoto Univ. 22 (1982), 481–484

    work page 1982

  17. [17]

    Weili Gu, Zhaoyong Huang, and Tiwei Zhao, Homological invariants under frobenius extensions , Colloq. Math. 178 (2025), 77–95. (GORENSTEIN) TRANSPOSE AND k-TORSIONFREE MODULES 21

    work page 2025

  18. [18]

    Pure Appl

    Henrik Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), no. 1-3, 167–193

    work page 2004

  19. [19]

    Algebra 324 (2010), no

    Chonghui Huang and Zhaoyong Huang, Gorenstein syzygy modules, J. Algebra 324 (2010), no. 12, 3408–3419

    work page 2010

  20. [20]

    Algebra 27 (1999), no

    Zhaoyong Huang, Extension closure of k-torsionfree modules, Commun. Algebra 27 (1999), no. 3, 1457–1464

    work page 1999

  21. [21]

    China, Ser

    , Wt-approximation representations over quasi k-Gorenstein algebras , Sci. China, Ser. A 42 (1999), no. 9, 945–956

    work page 1999

  22. [22]

    Osamu Iyama, Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories. , Adv. Math. 210 (2007), no. 1, 22–50

    work page 2007

  23. [23]

    Osamu Iyama and Xiaojin Zhang, Tilting modules over Auslander-Gorenstein algebras, Pac. J. Math. 298 (2019), no. 2, 399–416

    work page 2019

  24. [24]

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

    work page 1999

  25. [25]

    Heidelberger Akad

    Friedrich Kasch, Projektive Frobenius-Erweiterungen, Sitzungsber. Heidelberger Akad. Wiss., Math.-Naturw. Kl. 1960/61, 89-109 (1961)., 1961

    work page 1960

  26. [26]

    Texts Math., vol

    Tsit-Yuen Lam, A first course in noncommutative rings , Grad. Texts Math., vol. 131, New York etc.: Springer-Verlag, 1991

    work page 1991

  27. [27]

    Texts Math., vol

    , Lectures on modules and rings , Grad. Texts Math., vol. 189, New York, NY: Springer, 1999

    work page 1999

  28. [28]

    Leuschke and Roger Wiegand, Cohen-Macaulay representations, Math

    Graham J. Leuschke and Roger Wiegand, Cohen-Macaulay representations, Math. Surv. Monogr., vol. 181, Providence, RI: American Mathematical Society (AMS), 2012

    work page 2012

  29. [29]

    Pure Appl

    Jian Liu and Wei Ren, Ascent and descent of gorenstein homological properties , J. Pure Appl. Algebra 229 (2025), 108042

    work page 2025

  30. [30]

    Hiroki Matsui, Ryo Takahashi, and Yoshinao Tsuchiya, When are n-syzygy modules n-torsionfree?, Arch. Math. 108 (2017), no. 4, 351–355

    work page 2017

  31. [31]

    China, Math

    Wei Ren, Gorenstein projective modules and Frobenius extensions , Sci. China, Math. 61 (2018), no. 7, 1175–1186

    work page 2018

  32. [32]

    Gorenstein projective and injective dimensions over Frobenius extensions

    , Corrigendum to: “Gorenstein projective and injective dimensions over Frobenius extensions” , Commun. Al- gebra 48 (2020), no. 2, 915–916

    work page 2020

  33. [33]

    Nicolas Spaltenstein, Resolutions of unbounded complexes, Compos. Math. 65 (1988), no. 2, 121–154

    work page 1988

  34. [34]

    Ryo Takahashi, Classifying resolving subcategories over a Cohen-Macaulay local ring , Math. Z. 273 (2013), no. 1-2, 569–587

    work page 2013

  35. [35]

    Weibel, An introduction to homological algebra , Camb

    Charles A. Weibel, An introduction to homological algebra , Camb. Stud. Adv. Math., vol. 38, Cambridge: Cambridge University Press, 1994

    work page 1994

  36. [36]

    Algebra 13 (1969), 73–86

    Abraham Zaks, Injective dimension of semi-primary rings , J. Algebra 13 (1969), 73–86

    work page 1969

  37. [37]

    Algebra Appl

    Guoqiang Zhao and Juxiang Sun, A note on Gorenstein transposes , J. Algebra Appl. 15 (2016), no. 10, 8, Id/No 1650180

    work page 2016

  38. [38]

    China, Math

    Zhibing Zhao, Gorenstein homological invariant properties under Frobenius extensions , Sci. China, Math. 62 (2019), no. 12, 2487–2496

    work page 2019

  39. [39]

    Algebra 646 (2024), 49–65

    , k-torsionfree modules and Frobenius extensions, J. Algebra 646 (2024), 49–65. School of Mathematics and Statistics, and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, P.R. China Email address: jianliu@ccnu.edu.cn

    work page 2024