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arxiv: 2506.23277 · v5 · pith:HZON332Pnew · submitted 2025-06-29 · 🧮 math.RA · math.KT

Totally acyclic complexes and homological invariants over arbitrary rings

Jian Wang , Yunxia Li , Jiangsheng Hu , Haiyan zhu This is my paper

Pith reviewed 2026-05-22 00:57 UTC · model grok-4.3

classification 🧮 math.RA math.KT
keywords totally acyclic complexesIwanaga-Gorenstein ringsNakayama conjecturesilpsplisflinon-commutative ringshomological invariants
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The pith

Over any ring, every acyclic complex of projectives, injectives or flats being totally acyclic is equivalent to equalities among the invariants silp, spli and sfli and characterizes Iwanaga-Gorenstein rings in the non-commutative case.

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

The paper establishes equivalent characterizations for the property that every acyclic complex of projective, injective or flat modules over an arbitrary ring R is totally acyclic. It connects these properties directly to the finiteness or vanishing of the homological invariants silp(R), spli(R) and sfli(R), and supplies sufficient conditions under which spli(R) equals silp(R). The central extension shows that prior characterizations of Iwanaga-Gorenstein rings carry over to non-commutative rings and simultaneously furnish new equivalent conditions for the Nakayama conjecture.

Core claim

For an arbitrary ring R the conditions that every acyclic complex of projective modules is totally acyclic, that every acyclic complex of injective modules is totally acyclic, and that every acyclic complex of flat modules is totally acyclic are equivalent to one another and to the coincidence or finiteness of the invariants silp(R), spli(R) and sfli(R); these equivalences extend the commutative characterizations of Iwanaga-Gorenstein rings and supply additional equivalent statements for the Nakayama conjecture.

What carries the argument

The distinction between acyclic complexes and totally acyclic complexes of projective, injective and flat modules, together with the three homological invariants silp(R), spli(R) and sfli(R).

If this is right

  • spli(R) equals silp(R) whenever the listed sufficient conditions on the ring hold.
  • The non-commutative Iwanaga-Gorenstein property admits several new equivalent formulations in terms of totally acyclic complexes.
  • The Nakayama conjecture receives additional equivalent characterizations via the same complex conditions.
  • The three invariants silp, spli and sfli are tied together by the single requirement that acyclic complexes are totally acyclic.

Where Pith is reading between the lines

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

  • The same pattern of equivalences may be testable for other classes of complexes or for higher homological dimensions.
  • Results previously known only for commutative rings can now be re-examined directly in the non-commutative setting without extra hypotheses.
  • The link to the Nakayama conjecture suggests that computational checks of the invariants could decide the conjecture for specific families of algebras.

Load-bearing premise

The usual definitions and functorial properties of acyclic and totally acyclic complexes continue to make sense and behave as expected when the ring is allowed to be non-commutative.

What would settle it

A concrete non-commutative ring R together with an explicit acyclic complex of projectives that is not totally acyclic, yet silp(R) equals spli(R) and the ring satisfies the other listed conditions.

read the original abstract

In this paper, we investigate equivalent characterizations of the condition that every acyclic complex of projective, injective, or flat modules is totally acyclic over a general ring R. We provide examples to illustrate relationships among these conditions and show that several are closely tied to the homological invariants silp(R), spli(R) and sfli(R). We also give sufficient conditions for the equality spli(R) = silp(R), thereby refining results due to Ballas-Chatzistavridis and Wang-Yang. Further, we extend a result of Christensen-Foxby-Holm on characterizations of Iwanaga-Gorenstein rings to the non-commutative setting. This generalizes a theorem of Estrada-Fu-Iacob, offering additional equivalent characterizations under a general assumption while also yielding characterizations of the Nakayama conjecture.

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

2 major / 2 minor

Summary. The paper investigates equivalent characterizations of the condition that every acyclic complex of projective, injective, or flat modules over an arbitrary ring R is totally acyclic. It relates these conditions to the invariants silp(R), spli(R), and sfli(R), provides examples, gives sufficient conditions for spli(R) = silp(R) that refine prior results, and extends characterizations of Iwanaga-Gorenstein rings from Christensen-Foxby-Holm to the non-commutative setting while generalizing Estrada-Fu-Iacob and yielding characterizations of the Nakayama conjecture.

Significance. If the central equivalences hold for arbitrary rings, the work provides a useful non-commutative extension of standard results in homological algebra, with concrete ties to well-studied invariants and the Nakayama conjecture. The examples and sufficient conditions for equality of invariants add practical value, and the generalization beyond commutative cases broadens applicability to general ring theory.

major comments (2)
  1. [§4, Theorem 4.3] §4, Theorem 4.3 (extension of Christensen-Foxby-Holm): the proof that acyclicity of Hom(P, I) implies total acyclicity for arbitrary R invokes functorial properties of Hom and tensor without explicitly verifying that left/right module asymmetries do not obstruct the transfer of exactness from the commutative case; this is load-bearing for the non-commutative claim.
  2. [§5, Proposition 5.1] §5, Proposition 5.1 (characterizations of Nakayama conjecture): the equivalence between the totally acyclic condition and the conjecture is stated under a 'general assumption' whose precise statement and verification for non-Noetherian rings is not detailed, weakening the claimed additional characterizations.
minor comments (2)
  1. [§2] Notation for left/right modules is occasionally ambiguous in §2; explicit use of _R or R_ subscripts would improve clarity.
  2. [Theorem 3.2] The statement of Theorem 3.2 cites Ballas-Chatzistavridis but does not indicate which of their results is being refined by the new sufficient condition for spli = silp.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§4, Theorem 4.3] §4, Theorem 4.3 (extension of Christensen-Foxby-Holm): the proof that acyclicity of Hom(P, I) implies total acyclicity for arbitrary R invokes functorial properties of Hom and tensor without explicitly verifying that left/right module asymmetries do not obstruct the transfer of exactness from the commutative case; this is load-bearing for the non-commutative claim.

    Authors: The proof of Theorem 4.3 is written for an arbitrary associative unital ring R and explicitly tracks left and right module structures: projective complexes are taken in the category of right R-modules while injective complexes are taken in the category of left R-modules (or vice versa, depending on the Hom or tensor functor under consideration). The exactness transfer uses only the general adjointness isomorphism Hom_R(M ⊗_R N, P) ≅ Hom_R(M, Hom_R(N, P)) and the fact that Hom_R(-, I) is exact when I is injective, both of which hold without commutativity. Nevertheless, to remove any ambiguity about the passage from the commutative case, we will insert a short clarifying paragraph immediately after the statement of the theorem that records the precise module categories and confirms that no additional commutativity is used. revision: partial

  2. Referee: [§5, Proposition 5.1] §5, Proposition 5.1 (characterizations of Nakayama conjecture): the equivalence between the totally acyclic condition and the conjecture is stated under a 'general assumption' whose precise statement and verification for non-Noetherian rings is not detailed, weakening the claimed additional characterizations.

    Authors: The general assumption in Proposition 5.1 is the standing hypothesis, introduced at the beginning of Section 5, that every acyclic complex of projective (respectively injective or flat) modules is totally acyclic; this is precisely the condition whose equivalent characterizations are developed in Sections 3 and 4. Because the definitions of silp(R), spli(R) and sfli(R) make sense for any ring and the proofs in Section 5 invoke only these definitions together with the adjointness and exactness properties already established for arbitrary rings, the equivalences hold without any Noetherian hypothesis. To address the referee’s concern we will (i) restate the assumption verbatim inside the proposition and (ii) add a one-sentence verification that the argument remains valid when R is non-Noetherian. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations rely on external prior theorems and standard definitions

full rationale

The paper extends characterizations from Christensen-Foxby-Holm and Estrada-Fu-Iacob to non-commutative rings using standard definitions of acyclic/totally acyclic complexes, silp/spli/sfli invariants, and functorial properties of Hom and tensor. These are invoked as established in the literature rather than redefined or fitted within the paper. No step reduces a claimed equivalence or prediction to a quantity defined from the paper's own inputs or self-citations; the central claims build on cited external results without self-referential closure. The derivation chain remains independent of the present work's equations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Operates entirely within the standard framework of homological algebra over rings; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract. All content rests on established definitions of modules, complexes, and the three invariants.

axioms (2)
  • standard math Projective, injective, and flat modules over an arbitrary ring R form complexes whose acyclicity and total acyclicity are well-defined via standard Hom and tensor functors.
    Invoked as the foundation for all equivalent characterizations.
  • domain assumption The invariants silp(R), spli(R), and sfli(R) are defined as suprema of appropriate homological dimensions and are comparable across arbitrary rings.
    Central to linking the totally acyclic condition to these numerical invariants.

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Works this paper leans on

46 extracted references · 46 canonical work pages · 2 internal anchors

  1. [1]

    Auslander, I

    M. Auslander, I. Reiten, k-Gorenstein algebras and syzygy modules, J. Pure Appl. Algebra 92 (1994) 1-27

    work page 1994

  2. [2]

    Auslander, I

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

    work page 1995

  3. [3]

    Ballas, On certain homological finiteness conditions , Comm

    D. Ballas, On certain homological finiteness conditions , Comm. Algebra 45 (2017) 481-492

    work page 2017

  4. [4]

    Ballas, C

    D. Ballas, C. Chatzistavridis, On certain homological invariants for coherent rings , 2022, http://users.uoa.gr/ emmanoui/files/silp spli coherent.pdf

    work page 2022

  5. [5]

    Bazzoni, M

    S. Bazzoni, M. Cort´es-Izurdiaga, S. Estrada, Periodic modules and acyclic complexes , Algebr. Represent. Theory 23 (2020) 1861-1883. 11

    work page 2020

  6. [6]

    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. Algebra 288 (2005) 137-211

    work page 2005

  7. [7]

    Beligiannis, I

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

    work page 2007

  8. [8]

    Bennis, Rings over which the class of Gorenstein flat modules is closed under extensions , Comm

    D. Bennis, Rings over which the class of Gorenstein flat modules is closed under extensions , Comm. Algebra 37 (2009) 855-868

    work page 2009

  9. [9]

    Bennis and N

    D. Bennis and N. Mahdou, Global Gorenstein dimensions, Proc. Amer. Math. Soc. 138 (2010) 461-465

    work page 2010

  10. [10]

    The stable module category of a general ring

    D. Bravo, J. Gillespie, M. Hovey, The stable module category of a general ring , arXiv: 1405.5768v1

    work page internal anchor Pith review Pith/arXiv arXiv

  11. [11]

    Celikbas, L.W

    O. Celikbas, L.W. Christensen, L. Liang, G. Piepmeyer, Stable homology over associative rings , Trans. Amer. Math. Soc. 369 (2017) 8061-8086

    work page 2017

  12. [12]

    Cheatham, D.R

    T.J. Cheatham, D.R. Stone, Flat and projective character modules , Proc. Amer. Math. Soc. 81 (1981) 175-177

    work page 1981

  13. [13]

    Christensen, S

    L.W. Christensen, S. Estrada, P. Thompson, Gorenstein weak global dimension is symmetric , Math. Nachr. 294 (2021) 2121-2128

    work page 2021

  14. [14]

    Christensen, H.B

    L.W. Christensen, H.B. Foxby, H. Holm, Derived Category Methods in Commutative Algebra , Springer Nature, Switzerland AG 2024

    work page 2024

  15. [15]

    Christensen, K

    L.W. Christensen, K. Kato, Totally acyclic complexes and locally Gorenstein rings , J. Algebra Appl. 17 (2018) 1850039 (6 pages)

    work page 2018

  16. [16]

    Colby, Rings which have flat injective modules , J

    R.R. Colby, Rings which have flat injective modules , J. Algebra 35 (1975) 239-252

    work page 1975

  17. [17]

    Dalezios, I

    G. Dalezios, I. Emmanouil, Homological dimension based on a class of Gorenstein flat modules , Comptes Rendus - S´erie Math´ematique 361 (2023) 1429-1448

    work page 2023

  18. [18]

    Dembegioti, O

    F. Dembegioti, O. Talelli, On a relation between certain cohomological invariants , J. Pure Appl. Algebra 212 (2008) 1432-1437

    work page 2008

  19. [19]

    Z.X. Di, L. Liang and J.P. Wang, Virtually Gorenstein rings and relative homology of complexes , J. Pure Appl. Algebra 227 (2023) 107127

    work page 2023

  20. [20]

    Ding, J.L

    N.Q. Ding, J.L. Chen, The flat dimensions of injective modules , Manuscripta Math. 78 (1993) 165-177

    work page 1993

  21. [21]

    Emmanouil, On certain cohomological invariants of groups , Adv

    I. Emmanouil, On certain cohomological invariants of groups , Adv. Math. 225 (2010) 3446-3462

    work page 2010

  22. [22]

    Emmanouil, On the finiteness of Gorenstein homological dimensions , J

    I. Emmanouil, On the finiteness of Gorenstein homological dimensions , J. Algebra 372 (2012) 376-396

    work page 2012

  23. [23]

    Emmanouil, O

    I. Emmanouil, O. Talelli, On the flat length of injective modules , J. London Math. Soc. 84 (2011) 408-432

    work page 2011

  24. [24]

    Enochs, M

    E.E. Enochs, M. Cort´ es-Izurdiaga, B. Torrecillas, Gorenstein conditions over triangular matrix rings , J. Pure Appl. Algebra 218 (2014) 1544-1554

    work page 2014

  25. [25]

    Enochs, A

    E.E. Enochs, A. Iacob, Gorenstein injective covers and envelopes over Noetherian rings , Proc. Amer. Math. Soc. 143 (2015) 5-12

    work page 2015

  26. [26]

    Enochs, O.M.G

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

    work page 1995

  27. [27]

    Enochs, O.M.G

    E.E. Enochs, O.M.G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin-New York, 2000

    work page 2000

  28. [28]

    Enochs, O.M.G

    E.E. Enochs, O.M.G. Jenda, B. Torrecillas, Gorenstein flat modules , Nanjing Daxue Xuebao Shuxue Bannian Kan 10 (1993) 1-9

    work page 1993

  29. [29]

    Estrada, X.H

    S. Estrada, X.H. Fu, A. Iacob, Totally acyclic complexes , J. Algebra 470 (2017) 300-319

    work page 2017

  30. [30]

    Gedrich, K.W

    T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups , Topology Appl. 25 (1987) 203-223

    work page 1987

  31. [31]

    Holm, Gorenstein homological dimensions, J

    H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004) 167-193

    work page 2004

  32. [32]

    Huang, On Auslander-type conditions of modules , Publ

    Z.Y. Huang, On Auslander-type conditions of modules , Publ. Res. Inst. Math. Sci. 59 (2023) 57-88

    work page 2023

  33. [33]

    Iwanaga, On rings with finite self-injective dimension , Comm

    Y. Iwanaga, On rings with finite self-injective dimension , Comm. Algebra 7 (1979) 393-414

    work page 1979

  34. [34]

    Iwanaga, On rings with finite self injective dimension II , Tsukuba J

    Y. Iwanaga, On rings with finite self injective dimension II , Tsukuba J. Math. 4 (1980) 107-113

    work page 1980

  35. [35]

    Iyengar, H

    S. Iyengar, H. Krause, Acyclicity versus total acyclicity for complexes over noetherian rings , Doc. Math. 11 (2006) 207-240

    work page 2006

  36. [36]

    C. U. Jensen, Les foncteurs d´ eriv´ es delim← −et leurs applications an th´ eorie des modules, Lecture Notes in Mathematics 254, Springer, Berlin, 1972

    work page 1972

  37. [37]

    Jensen, On the vanishing of lim← − (i), J

    C.U. Jensen, On the vanishing of lim← − (i), J. Algebra 15 (1970) 151-166. 12

    work page 1970

  38. [38]

    Maddox, Absolutely pure modules, Proc

    B.H. Maddox, Absolutely pure modules, Proc. Amer. Math. Soc. 18 (1967) 155-158

    work page 1967

  39. [39]

    M¨ uller,The classification of algebras by dominant dimension , Canad

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

    work page 1968

  40. [40]

    Murfet, S

    D. Murfet, S. Salarian, Totally acyclic complexes over noetherian schemes , Adv. Math. 226 (2011) 1096- 1133

    work page 2011

  41. [41]

    Nakayama, On algebras with complete homology , Abh

    T. Nakayama, On algebras with complete homology , Abh. Math. Semin. Univ. Hambg 22 (1958) 300-307

    work page 1958

  42. [42]

    ˇSaroch, J

    J. ˇSaroch, J. ˇSt’ov´ ıˇ cek,Singular compactness and definability for Σ-cotorsion and Gorenstein modules , Sel. Math. New Ser. 26:23 (2020)

    work page 2020

  43. [43]

    Stenstr¨ om,Coherent rings and F P-injective modules, J

    B. Stenstr¨ om,Coherent rings and F P-injective modules, J. London Math. Soc. 2 (1970) 323-329

    work page 1970

  44. [44]

    On purity and applications to coderived and singularity categories

    J. ˇSt’ov´ ıˇ cek,On purity and applications to coderived and singularity categories , arXiv:1412.1615v1

    work page internal anchor Pith review Pith/arXiv arXiv

  45. [45]

    Talelli, A characterization of cohomological dimension for a big class of groups , J

    O. Talelli, A characterization of cohomological dimension for a big class of groups , J. Algebra 326 (2011) 238-244

    work page 2011

  46. [46]

    J.P. Wang, G. Yang, A balance result over generalized coherent rings , Publ. Math. Debrecen 107/1-2 (2025) 139-154. 13

    work page 2025