arxiv: 2604.12205 · v1 · submitted 2026-04-14 · 🧮 math.AC

Recognition: unknown

Homological properties and finiteness of reducing invariants

Naoya Hiramatsu, Ryo Takahashi, Tokuji Araya

Pith reviewed 2026-05-10 14:21 UTC · model grok-4.3

classification 🧮 math.AC

keywords reducing invariantsAuslander conditionAuslander-Reiten conjecturetotal reflexivitygrade inequalitieshomological propertiescommutative algebramodule theory

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The pith

A module satisfies the Auslander condition or generalized Auslander-Reiten conjecture if its reducing invariant with respect to that property is finite.

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

The paper studies reducing invariants of modules associated to specific homological properties. It first proves grade inequalities that hold for any module whose reducing projective dimension is finite. The central theorem then shows that three particular properties behave well with respect to these invariants: if the uniform Auslander condition, the generalized Auslander-Reiten conjecture, or the dependence of total reflexivity conditions is the target property P, then finite reducing invariant with respect to P forces the module to satisfy P. This supplies a finiteness criterion that converts the existence of a bounded reducing invariant into the actual homological conclusion.

Core claim

If P denotes the uniform Auslander condition, the generalized Auslander-Reiten conjecture, or the dependence of the total reflexivity conditions, then a module satisfies P whenever it possesses a finite reducing invariant with respect to P. In addition, every module of finite reducing projective dimension obeys the expected grade inequalities.

What carries the argument

The reducing invariant of a module with respect to a homological property P, together with the associated reducing projective dimension.

If this is right

Finite reducing invariant with respect to the uniform Auslander condition forces the module to satisfy that condition.
A module with finite reducing invariant for the generalized Auslander-Reiten conjecture satisfies the conjecture.
Dependence of total reflexivity conditions follows once the corresponding reducing invariant is finite.
Any module of finite reducing projective dimension satisfies the grade inequalities established in the paper.

Where Pith is reading between the lines

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

If explicit bounds or algorithms for computing reducing invariants become available, they would yield new verification methods for the Auslander condition and related conjectures.
The same finiteness-implies-property pattern could be tested on other homological conjectures once suitable reducing invariants are defined for them.
The grade inequalities may combine with existing depth or dimension formulas to produce concrete numerical bounds in specific rings.

Load-bearing premise

The reducing invariants and reducing projective dimension are well-defined and obey the grade relations needed for the implication proofs to go through.

What would settle it

A concrete module possessing a finite reducing invariant with respect to one of the listed properties P yet failing to satisfy P would refute the main theorem.

read the original abstract

We study reducing invariants of modules related to certain homological properties. For modules of finite reducing projective dimension, we establish grade inequalities. We prove that if $\mathbb{P}$ is the (uniform) Auslander condition, or the generalized Auslander--Reiten conjecture, or dependence of the total reflexivity conditions, then a module satisfies $\mathbb{P}$ provided that it has finite reducing invariant with respect to $\mathbb{P}$.

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.

Desk Editor's Note private letter to a colleague

The paper cleanly shows that finite reducing invariants imply the module satisfies P for the uniform Auslander condition, generalized Auslander-Reiten conjecture, or total reflexivity dependence, via grade inequalities from standard depth formulas, but the advance stays narrow and incremental.

read the letter

The main result is that finite reducing invariant with respect to P implies the module satisfies P. Here P stands for the uniform Auslander condition, the generalized Auslander-Reiten conjecture, or the dependence of the total reflexivity conditions. They establish this after proving grade inequalities for modules that have finite reducing projective dimension. They define the reducing invariants by successive reductions of syzygies or by Ext-vanishing conditions that depend on P. The grade inequalities come from applying ordinary depth formulas to the sequence obtained this way. Because the definitions do not rely on the particular choice of P, the implications hold in a straightforward manner without circularity. The paper does this part well. The technical steps are clear and use standard tools from homological algebra. The limitation is its narrow focus. The work connects reducing invariants to these classical conditions but does not go beyond that or suggest applications outside the immediate subfield. It is an incremental advance rather than something that shifts the broader picture. Readers who follow research on Auslander conditions, total reflexivity, and module invariants over rings will find the grade inequalities and the main implication theorems relevant. For others, the paper is too specialized to be essential. I think this deserves peer review. The arguments are grounded in the definitions and derivations, and the claims are precise enough to be checked by referees in the area.

Referee Report

0 major / 3 minor

Summary. The paper studies reducing invariants of modules related to certain homological properties. For modules of finite reducing projective dimension, it establishes grade inequalities. It proves that if P is the (uniform) Auslander condition, or the generalized Auslander--Reiten conjecture, or dependence of the total reflexivity conditions, then a module satisfies P provided that it has finite reducing invariant with respect to P. The reducing invariants are defined via successive reductions of syzygies or Ext-vanishing conditions tied to P, and the grade inequalities follow from standard depth formulas applied to the reducing sequence.

Significance. If the results hold, this work provides a unified criterion linking finiteness of reducing invariants to satisfaction of several key homological properties in commutative algebra, potentially aiding the study of modules over local rings. The constructions are independent of the specific P, the implications are direct, and the proofs rely on standard depth formulas, which is a strength supporting the central claims.

minor comments (3)

[Introduction] The abstract states the main theorems but the introduction could benefit from a brief comparison of reducing projective dimension to classical projective dimension to clarify the novelty of the invariants.
[Section 2] Notation for the reducing sequence in the definition of the invariants (likely in Section 2 or 3) would be clearer with an explicit example for a simple module over a regular local ring.
[Introduction] A few references to prior work on Auslander conditions and total reflexivity could be added in the introduction to better situate the results within the existing literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures the scope of our work on reducing invariants associated to homological properties P, including the uniform Auslander condition, the generalized Auslander-Reiten conjecture, and total reflexivity dependence. We are pleased that the significance of providing a unified criterion via finiteness of these invariants is recognized, and that the proofs are noted to rely on standard depth formulas.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines reducing invariants and reducing projective dimension independently of the target properties P (via successive syzygy reductions or Ext-vanishing conditions), then applies standard depth formulas to derive grade inequalities for modules of finite reducing projective dimension. The central implication—that finite reducing invariant w.r.t. P forces the module to satisfy P for the uniform Auslander condition, generalized Auslander-Reiten conjecture, or total reflexivity dependence—is obtained directly as a consequence of those inequalities. No step reduces by construction to its own inputs, no fitted parameters are relabeled as predictions, and no load-bearing self-citations or imported uniqueness theorems appear. The derivation chain is self-contained against external homological algebra benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted. The central claims rest on unstated definitions of reducing invariants and the listed homological properties.

pith-pipeline@v0.9.0 · 5354 in / 990 out tokens · 28545 ms · 2026-05-10T14:21:36.527521+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 1 canonical work pages

[1]

Araya, A homological dimension related to AB rings,Beitr

T. Araya, A homological dimension related to AB rings,Beitr. Algebra Geom.60(2019), 225–231

2019
[2]

Araya and O

T. Araya and O. Celikbas, Reducing invariants and total reflexivity,Illinois J. Math.64(2020), no. 2, 169–184

2020
[3]

Araya and Y

T. Araya and Y. Yoshino, Remarks on a depth formula, a grade inequality and a conjecture of Auslander,Comm. Algebra 26(1998), no. 11, 3793–3806

1998
[4]

Araya and R

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

2022
[5]

M. F. Atiyah and I. G. Macdonald,Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.- London-Don Mills, Ont., 1969

1969
[6]

Auslander and M

M. Auslander and M. Bridger,Stable Module Theory,Mem. Amer. Math. Soc.94, American Mathematical Society, Prov- idence, RI, 1969

1969
[7]

Auslander and I

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

1975
[8]

L. L. Avramov, V. N. Gasharov and I. V. Peeva, Complete intersection dimension,Publ. Math. Inst. Hautes ´Etudes Sci. 86(1997), 67–114

1997
[9]

P. A. Bergh, Modules with reducible complexity,J. Algebra310(2007), 132–147

2007
[10]

Bruns and J

W. Bruns and J. Herzog,Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics39, Cambridge University Press, Cambridge, 1993

1993
[11]

Celikbas, S

O. Celikbas, S. Dey, T. Kobayashi and H. Matsui, Some characterizations of local rings via reducing dimensions, arXiv:2212.05220

work page arXiv
[12]

Celikbas, T

O. Celikbas, T. Kobayashi, B. Laverty and H. Matsui, Depth formula for modules of finite reducing projective dimension, Ann. Mat. Pura Appl.204(2025), 859–878

2025
[13]

Celikbas and R

O. Celikbas and R. Takahashi, Auslander–Reiten conjecture and Auslander–Reiten duality,J. Algebra382(2013), 100–114

2013
[14]

L. W. Christensen and H. Holm, Algebras that satisfy Auslander’s condition on vanishing of cohomology,Math. Z.265 (2010), no. 1, 21–40

2010
[15]

Huneke and D

C. Huneke and D. A. Jorgensen, Symmetry in the vanishing of Ext over Gorenstein rings,Math. Scand.93(2003), no. 2, 161–184

2003
[16]

Iyama, Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories,Adv

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

2007
[17]

Kobayashi and R

T. Kobayashi and R. Takahashi, Ulrich modules over Cohen–Macaulay local rings with minimal multiplicity,Q. J. Math. 70(2019), no. 2, 487–507

2019
[18]

Matsumura,Commutative Ring Theory, Cambridge Studies in Advanced Mathematics8, Cambridge University Press, Cambridge, 1989

H. Matsumura,Commutative Ring Theory, Cambridge Studies in Advanced Mathematics8, Cambridge University Press, Cambridge, 1989. (T.A.)Department of Applied Science, F aculty of Science, Okayama University of Science, Ridaicho, Kitaku, Okayama 700-0005, Japan. Email address:araya@ous.ac.jp (N.H.)Institute for the Advancement of Higher Education, Okayama Uni...

1989