pith. sign in

arxiv: 2405.02921 · v2 · submitted 2024-05-05 · 🧮 math.RT

The Extension dimension of syzygy module categories

Junling Zheng , Lulu Tian , Qianyu Shu This is my paper

Pith reviewed 2026-05-24 01:05 UTC · model grok-4.3

classification 🧮 math.RT
keywords syzygy modulesextension dimensionArtin algebrasderived equivalencestable equivalenceseparable equivalencemodule categories
0
0 comments X

The pith

The extension dimension of syzygy module categories matches for derived equivalent algebras when i is large and stays invariant under stable and separable equivalences for every i.

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

The paper establishes that for sufficiently large i the i-th syzygy module categories of derived equivalent Artin algebras have the same extension dimension. It further shows that this extension dimension is unchanged when the algebra is replaced by any stable or separable equivalent algebra, regardless of how small i is. A reader would care because the result supplies numerical invariants that travel across equivalence classes of algebras and therefore reduce the work of comparing their module-theoretic properties. The claims rest on the fact that the relevant equivalences map syzygy subcategories to one another while preserving extension data.

Core claim

For sufficiently large i, the i-th syzygy module categories of derived equivalent algebras exhibit identical extension dimensions. Furthermore, the extension dimension of the i-th syzygy module category is an invariant under both stable equivalence and separable equivalence for each nonnegative integer i.

What carries the argument

Extension dimension of the i-th syzygy module category, the quantity that records the longest chains of non-split extensions inside the category of i-th syzygies.

If this is right

  • Derived equivalent algebras share the same extension dimension in all sufficiently high syzygy module categories.
  • Stable equivalence forces the extension dimension to agree at every syzygy level.
  • Separable equivalence likewise forces the extension dimension to agree at every syzygy level.
  • The extension dimension supplies a numerical invariant that can be computed in any representative of a stable or separable equivalence class.

Where Pith is reading between the lines

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

  • The result supplies a practical test: if two algebras have different extension dimensions at a high syzygy level, they cannot be derived equivalent.
  • One could try to extend the invariance statements to other notions of equivalence, such as Morita equivalence, though the paper does not address them.
  • The invariants might help partition the set of Artin algebras into equivalence classes by computing a single sequence of numbers rather than full module categories.

Load-bearing premise

The equivalences between algebras induce equivalences or embeddings of the corresponding syzygy module categories that preserve the structure of extensions.

What would settle it

A pair of derived equivalent Artin algebras whose i-th syzygy module categories have different extension dimensions for some sufficiently large i would falsify the first claim.

read the original abstract

In this paper, our primary focus is on investigating the extension dimensions of syzygy module categories associated with Artin algebras, particularly under various equivalences. We demonstrate that, for sufficiently large $i$, the $i$-th syzygy module categories of derived equivalent algebras exhibit identical extension dimensions. Furthermore, we establish that the extension dimension of the $i$-th syzygy module category is an invariant under both stable equivalence and separable equivalence for each nonnegative integer $i$.

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

1 major / 1 minor

Summary. The paper claims that for Artin algebras, the extension dimension of the i-th syzygy module category is invariant under derived equivalence when i is sufficiently large, and is invariant under stable equivalence and under separable equivalence for every nonnegative integer i.

Significance. If the results hold, they supply new numerical invariants preserved by these equivalences, which may help distinguish or classify Artin algebras and their module categories in representation theory. The focus on syzygy subcategories extends existing work on functorial invariants.

major comments (1)
  1. [Abstract] Abstract (and the statements of the main theorems): the invariance claims rest on the assertion that derived, stable, and separable equivalences induce equivalences or embeddings of the relevant syzygy module categories that preserve extension dimension. The paper invokes this as a standard fact without explicit verification that the particular definition of extension dimension (via Ext-groups or minimal extension lengths) is respected by the induced functors; this is load-bearing for all three invariance statements.
minor comments (1)
  1. The abstract would be clearer if it included a one-sentence reminder of the definition of extension dimension used in the paper.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit verification of functorial preservation properties. We address the major comment below and will revise the paper accordingly to strengthen the exposition.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the statements of the main theorems): the invariance claims rest on the assertion that derived, stable, and separable equivalences induce equivalences or embeddings of the relevant syzygy module categories that preserve extension dimension. The paper invokes this as a standard fact without explicit verification that the particular definition of extension dimension (via Ext-groups or minimal extension lengths) is respected by the induced functors; this is load-bearing for all three invariance statements.

    Authors: We agree that an explicit verification would improve clarity, even if the preservation follows from standard properties of the equivalences (e.g., that derived equivalences induce equivalences on the bounded derived category preserving Ext groups, and stable/separable equivalences induce equivalences on the stable category preserving the relevant Ext^1). In the revised manuscript we will add a short lemma (or expanded remark in Section 2) confirming that the induced functors on the i-th syzygy categories preserve the minimal length of extensions in the sense of our definition of extension dimension. This will be placed before the statements of the main theorems. revision: yes

Circularity Check

0 steps flagged

No circularity: invariance proved from definitions of equivalences and extension dimension

full rationale

The paper derives invariance of extension dimensions under derived, stable, and separable equivalences by invoking that such equivalences induce equivalences or embeddings on syzygy module categories preserving extension structure. This follows from standard categorical facts applied to the definitions, without any reduction of the claimed invariants to fitted parameters, self-definitions, or load-bearing self-citations. The derivation chain is self-contained against external benchmarks in representation theory and does not rename known results or smuggle ansatzes via citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure proof paper in homological algebra. No numerical parameters are fitted. The background consists of standard axioms of abelian categories, triangulated categories, and module categories over Artin algebras.

axioms (1)
  • domain assumption Derived, stable, and separable equivalences induce equivalences or embeddings of the syzygy subcategories that preserve non-split extensions.
    This is the structural fact used to transfer extension dimension between equivalent algebras; it is standard but must hold for the invariance statements.

pith-pipeline@v0.9.0 · 5595 in / 1240 out tokens · 19240 ms · 2026-05-24T01:05:02.995652+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages · 1 internal anchor

  1. [1]

    Auslander and I

    M. Auslander and I. Reiten , Representation theory of Artin algebras VI: A functorial approach to almost split sequences, C omm. Algebra 6 (3) (1978) 257-300

    work page 1978

  2. [2]

    Auslander and I

    M. Auslander and I. Reiten,k-Syzygy modules for Noetherian rings, J. Pure Appl. Algebra 92 (1) (1994) 1-27

    work page 1994

  3. [3]

    Auslander and I

    M. Auslander and I. Reiten , Syzygy modules for noetherian rings, J. Algebra 183 (1996) 167-185

    work page 1996

  4. [4]

    Auslander , I

    M. Auslander , I. Reiten and S. O. Smalø , Representation theory of Artin algebras , Cor- rected reprint of the 1995 original, Cambridge Studies in Advanced M athematics 36, Cam- bridge University Press, Cambridge, 1997

    work page 1995

  5. [5]

    Barrios and G

    M. Barrios and G. Mata, On algebras of Ω n-finite and Ω ∞ -infinite representation type, J. Algebra Appl. (2023), https://doi.org/10.1142/S0219498824501 834

    work page doi:10.1142/s0219498824501 2023

  6. [6]

    the representation dimension of Artin al- gebras

    A. Beligiannis , Some ghost lemmas, survey for “the representation dimension of Artin al- gebras”, Bielefeld, http://www.mathematik.uni-bielefeld.de/ ∼ sek/2008/ghosts.pdf, 2008

    work page 2008

  7. [7]

    P. E. Bergh and K. Erdmann , The representation dimension of Hecke algebras and sym- metric groups, Adv. Math. 228 (2011) 2503–2521. 27

    work page 2011

  8. [8]

    B ¨ohmler and R

    B. B ¨ohmler and R. Marczinzik , On the extension-closed property for the subcategory Tr(Ω2(mod−A)), Algebr. Represent. Theory 26 (5) (2023) 1433-1440

    work page 2023

  9. [9]

    Bondal, A

    A. Bondal, A. and M. van den Bergh, Generators and represent ability of functors in commu- tative and noncommutative geometry, Mosc. Math. J. 3 (1) (2003) 1–36

    work page 2003

  10. [10]

    H. X. Chen , M. F ang, O. Kerner , S. Koenig and K. Yamagata, Rigidity dimension of algebras, Math. Proc. Cambridge Philos. Soc. 170 (2) (2021) 417-443

    work page 2021

  11. [11]

    H. L. Dao and R. Takahashi, The radius of a subcategory of modules, Algebra & Number Theory 8 (1) (2014) 141-172

    work page 2014

  12. [12]

    Dugger and B

    D. Dugger and B. Shipley , K-theory and derived equivalences, Duke Math. J. 124 (3) (2004) 587-617

    work page 2004

  13. [13]

    Gabriel and A

    P. Gabriel and A. V. Roˇiter, Representations of finite dimensional algebras , Encyclopedia Math. Sci. Vol 73, Springer-Verlag, 1992. Chapter 12, Derivation and Tilting, by B. K eller

    work page 1992

  14. [14]

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

    work page 2005

  15. [15]

    Goto and R

    S. Goto and R. Takahashi , Extension closedness of syzygies and local Gorensteinness of commutative rings, Algebr. Represent. Theory 19 (3) (2016) 511–521

    work page 2016

  16. [16]

    Derived dimensions of representation-finite algebras

    Y. Han , Derived dimensions of representation-finite algebras, Preprint, 1-4, arXiv:0909.0330

    work page internal anchor Pith review Pith/arXiv arXiv

  17. [17]

    D. Happel , Triangulated categories in the representation theory of fin ite-dimensional alge- bras, London Mathematical Society Lecture Note Series 119, Cambridge University Press, Cambridge, 1988

    work page 1988

  18. [18]

    Happel, Reduction techniques for homological conjectures, Tsukuba J

    D. Happel, Reduction techniques for homological conjectures, Tsukuba J. Math. 17 (1) (1993) 115-130

    work page 1993

  19. [19]

    Heller , The loop-space functor in homological algebra, Trans

    A. Heller , The loop-space functor in homological algebra, Trans. Amer. Math. Soc. 96 (1960) 382-394

    work page 1960

  20. [20]

    Hu and S

    W. Hu and S. Y. Pan , Stable functors of derived equivalences and Gorenstein project ive modules, Math. Nachr. 290 (10) (2017) 1512-1530

    work page 2017

  21. [21]

    Hu and C

    W. Hu and C. C. Xi , Derived equivalences and stable equivalences of Morita type, I, Nagoya Math. J. 200 (2010) 107-152

    work page 2010

  22. [22]

    Z. Y. Huang , Syzygy modules for quasi k-Gorenstein rings, J. Algebra 299 (1) (2006) 21-32

    work page 2006

  23. [23]

    Huybrechts , Fourier-Mukai transforms in algebraic geometry , Oxford Mathematical Monographs

    D. Huybrechts , Fourier-Mukai transforms in algebraic geometry , Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxf ord, 2006. viii+307 pp

    work page 2006

  24. [24]

    Kadison , On global dimension, tower of algebras and Jones index, Preprint, U.C

    L. Kadison , On global dimension, tower of algebras and Jones index, Preprint, U.C. Berkeley and Roskilde University Centre, 1991, http://milne.ruc.dk/imfufate kster/pdf/210.pdf

    work page 1991

  25. [25]

    Kadison , On split, separable subalgebras with counitality condition, Hokkaido Math

    L. Kadison , On split, separable subalgebras with counitality condition, Hokkaido Math. J. 24 (1995) 527–549

    work page 1995

  26. [26]

    Keller , Deriving DG categories, Ann

    B. Keller , Deriving DG categories, Ann. Sci. ´Ecole Norm. Sup. 27 (4) (1994) 63-102. 28

    work page 1994

  27. [27]

    Keller , Invariance and localization for cyclic homology of DG-algebras, J

    B. Keller , Invariance and localization for cyclic homology of DG-algebras, J. Pure Appl. Algebra 123 (1-3) (1998) 223-273

    work page 1998

  28. [28]

    Krause , and G

    H. Krause , and G. Zwara, Grzegorz, Stable equivalence and generic modules, Bull. London Math. Soc. 32 (5) (2000) 615–618

    work page 2000

  29. [29]

    Krause and D

    H. Krause and D. Kussin , Rouquier’s theorem on representation dimension, Contemp. Math. 406 (2006) 95-103

    work page 2006

  30. [30]

    Linckelmann , Finite generation of Hochschild cohomology of Hecke algebras of fin ite classical typein characteristic zero

    M. Linckelmann , Finite generation of Hochschild cohomology of Hecke algebras of fin ite classical typein characteristic zero. B ull. Lond. Math. Soc. 43 (2011), 871–885

    work page 2011

  31. [31]

    Y. M. Liu and C. C. Xi , Constructions of stable equivalences of Morita type for finite dimension, III, J. London Math. Soc. 76 (2) (2007) 567-585

    work page 2007

  32. [32]

    Y. M. Liu and C. C. Xi , Construction of stable equivalences of Morita type for finite- dimensional algebras, I, Trans. Amer. Math. Soc. 358 (6) (2006) 2537–2560

    work page 2006

  33. [33]

    X. Ma , Y. Y. Peng , and Z. Y. Huang , The Extension Dimension of Subcategories and Recollements of Abelian Categories, Acta. Math. Sin. (English Ser) (4) 40 (2024) 1042-1058

    work page 2024

  34. [34]

    Mart´ınez-Villa, Algebras stably equivalent to l-hereditary

    R. Mart´ınez-Villa, Algebras stably equivalent to l-hereditary. In: Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 197 9), 396-431, Lecture Notes in Math. 832, Springer, Berlin, 1980

    work page 1980

  35. [35]

    Mart ´ınez-Villa, Properties that are left invariant under stable equivalence, Comm

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

    work page 1990

  36. [36]

    Oppermann , Lower bounds for Auslander’s representation dimension, Duke Math

    S. Oppermann , Lower bounds for Auslander’s representation dimension, Duke Math. J. 148 (2) (2009) 211-249

    work page 2009

  37. [37]

    S. Y. Pan and C. C. Xi , Finiteness of finitistic dimension is invariant under derived equiva- lences, J. Algebra 322 (1) (2009) 21-24

    work page 2009

  38. [38]

    S. E. Peacock , Separable equivalence, complexity and representation type., J. Algebra 490 (2017) 219-240

    work page 2017

  39. [39]

    Rickard, Morita theory for derived categories, J

    J. Rickard, Morita theory for derived categories, J. London Math. Soc. 39 (3) (1989) 436-456

    work page 1989

  40. [40]

    Rickard , Derived categories and stable equivalence, J

    J. Rickard , Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (3) (1989) 303-317

    work page 1989

  41. [41]

    Rickard , Derived equivalences as derived functors, J

    J. Rickard , Derived equivalences as derived functors, J. Lond. Math. Soc. (2) 43 (1) (1991) 37-48

    work page 1991

  42. [42]

    Rouquier , Representation dimension of exterior algebras, Invent

    R. Rouquier , Representation dimension of exterior algebras, Invent. Math. 165 (2) (2006) 357-367

    work page 2006

  43. [43]

    Rouquier , Dimensions of triangulated categories, J

    R. Rouquier , Dimensions of triangulated categories, J. K-theory 1 (2) (2008) 193-256

    work page 2008

  44. [44]

    J. L. Verdier , Cat´ egories d´ eriv´ ees, etat O,Lecture Notes in Mathematics 569, Springer- Verlag, Berlin, 1977, 262-311. 29

    work page 1977

  45. [45]

    J. Q. Wei , Derived invariance by syzygy complexes. Mathematical Proceedings of the Cam- bridge PhilosophicalSociety, 164(2018) 325-343

    work page 2018

  46. [46]

    C. C. Xi , Representation dimension and quasi-hereditary algebras, Adv. Math. 168 (2) (2002) 193-212

    work page 2002

  47. [47]

    C. C. Xi , Derived equivalences of algebras, Bull. London Math. Soc. 50 (6) (2018) 945-985

    work page 2018

  48. [48]

    C. C. Xi and J. B. Zhang , New invariants of stable equivalences of algebras, Preprint, 1-20 , arXiv:2207.10848

    work page arXiv

  49. [49]

    J. B. Zhang and J. L. Zheng , Extension dimensions of derivd and stable equivalent algebras, J. Algebra 646 (2024) 17-48

    work page 2024

  50. [50]

    J. L. Zheng and Z. Y. Huang , The derived and extension dimensions of abelian categories, J. Algebra 606 (2022) 243-265

    work page 2022

  51. [51]

    J. L. Zheng , X. Ma and Z. Y. Huang , The extension dimension of abelian categories, Algebr. Represent. Theory 23 (3) (2020) 693-713

    work page 2020

  52. [52]

    G. D. Zhou and A. Zimmermann , On singular equivalences of Morita type, J. Algebra 385 (2013) 64-79

    work page 2013

  53. [53]

    Zimmermann-Huisgen , Homological domino effects and the first finitistic dimension con- jecture, Invent

    B. Zimmermann-Huisgen , Homological domino effects and the first finitistic dimension con- jecture, Invent. Math. 108 (2) (1992) 369–383. 30

    work page 1992