The Auslander-Gorenstein condition for monomial algebras

Vikt\'oria Kl\'asz

REVIEW 3 major objections 4 minor 2 cited by

The Auslander-Gorenstein condition for monomial algebras

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A monomial algebra is Auslander-Gorenstein exactly when its Auslander-Reiten translate is a well-defined bijection.

desk verdict Plausible and potentially significant abstract, but the full text is unreadable and the reduction step has a real logical gap that needs checking. read the letter →

arxiv 2508.06957 v1 pith:7FWPCYVS submitted 2025-08-09 math.RT

Viktória Klász This is my paper · ORCID

classification math.RT MSC 16E1016E6516G7016P10

keywords Auslander-GorensteinmonomialalgebrasAuslander-ReitentranslatestringgentleNakayamaconjectureGorensteinproperty

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

The paper characterises the Auslander-Gorenstein condition inside the class of monomial algebras, the finite-dimensional path-algebra quotients defined by zero relations. It proves that every Auslander-Gorenstein monomial algebra is a string algebra, classifies the gentle ones combinatorially, and reduces the whole classification to Nakayama algebras by a transformation. Its central result is an equivalence: a monomial algebra is Auslander-Gorenstein iff its Auslander-Reiten translate $\tau=D\operatorname{Tr}$ is a well-defined bijection on the module class where it acts, confirming a conjecture that had been open for this class. The reduction also yields a stronger form of the Auslander-Reiten conjecture for all monomial algebras and a gap theorem: every $2n$-Gorenstein monomial algebra is $(2n+1)$-Gorenstein. If these claims are right, the Auslander-Gorenstein condition becomes a bijection property rather than a homological computation in a large class of examples.

What carries the argument

The carrier of the argument is the Auslander-Reiten translate $\tau=D\operatorname{Tr}$, the duality pairing the Auslander transpose with the vector-space dual; for each indecomposable non-projective module it produces the starting term of the almost split sequence ending at that module. The paper uses the string/band combinatorics of monomial algebras to decide exactly when this translate is total and bijective, and the transform to Nakayama algebras is the device that makes the reduction uniform instead of case-by-case.

What would settle it

Take a monomial algebra whose Auslander-Reiten translate is well-defined and bijective but whose bimodule injective dimension is infinite: such a case would break the main equivalence. Concretely, one could compute the AR translate on all indecomposable modules of a small quiver algebra with different relation lengths and check whether the bijection holds while the algebra fails to be Auslander-Gorenstein. Equivalently, applying the paper's reduction to a 2-Gorenstein monomial algebra that is not Auslander-Gorenstein and obtaining an Auslander-Gorenstein Nakayama algebra would refute the reduc

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Extended reading notes

Core claim

The paper's first structural claim is that an Auslander-Gorenstein monomial algebra cannot be wildly non-string: it must be a string algebra, and among gentle algebras the condition has a simple combinatorial description. The second is a reduction: every 2-Gorenstein monomial algebra can be transformed into a Nakayama algebra, so deciding the Auslander-Gorenstein property for monomial algebras is the same problem as deciding it for Nakayama algebras. The main theorem then states the equivalence mentioned above: Auslander-Gorenstein iff the Auslander-Reiten map is a well-defined bijection. Along the way, the paper proves that all monomial algebras satisfy a stronger form of the Auslander-Reit

Load-bearing premise

The reduction from 2-Gorenstein monomial algebras to Nakayama algebras is load-bearing: the transformation must preserve the Auslander-Gorenstein property in both directions, and the abstract does not state which invariants it preserves.

Editorial extensions

If this is right

Every Auslander-Gorenstein monomial algebra is a string algebra, so its representation theory is governed by strings and bands rather than by arbitrary relations.
The Auslander-Gorenstein classification of monomial algebras is reduced to the same classification for Nakayama algebras, a much smaller and more rigid family.
Every monomial algebra satisfies a stronger form of the Auslander-Reiten conjecture, giving a new positive case of that conjecture.
For gentle algebras, the Auslander-Reiten bijection is explicitly described, so the Gorenstein condition can be read off from the quiver and relations.
For monomial algebras the Gorenstein ladder closes: $2n$-Gorenstein implies $(2n+1)$-Gorenstein for all $n\ge1$.

Reading between the lines

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

One can read the paper as giving a computable criterion: to test whether a monomial algebra is Auslander-Gorenstein, check whether its Auslander-Reiten translate is a well-defined bijection, a question that can be approached with path-algebra combinatorics.
The reduction to Nakayama algebras suggests an algorithmic route—transform, test in the Nakayama case, transfer back—that the paper does not itself spell out as an algorithm.
If the bijection criterion were to hold beyond monomial algebras, it would turn the Auslander-Gorenstein condition into a property of the Auslander-Reiten quiver rather than of the full module category; the string-algebra results here provide a first place to test that.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a math paper whose abstract promises real results: a confirmation of Marczinzik's conjecture for monomial algebras, a classification of gentle Auslander-Gorenstein algebras, and a reduction to Nakayama algebras. If the proofs work, this is a solid contribution to the representation theory of finite-dimensional algebras, and the string-algebra theorem plus the Iwanaga-Fuller generalization are more than routine. The abstract is internally consistent and no red flags appear at the statement level.

The soft spot is real but specific: the reduction procedure is stated only for 2-Gorenstein monomial algebras, yet it is used to classify all Auslander-Gorenstein monomial algebras. The abstract never says every AG monomial algebra is 2-Gorenstein, and the Iwanaga-Fuller lift (2n implies 2n+1) does not bridge that gap. Either the full proof has a separate argument that every AG monomial algebra is 2-Gorenstein, or the reduction is iterated, or the classification claim overreaches. This is exactly the kind of thing a referee should check, not a reason to dismiss the paper out of hand.

The bigger practical problem: the full text we received is garbled mojibake, with even a foreign arXiv header embedded. That makes it impossible to verify any proof, lemma, or definition. This is likely a conversion artifact rather than a flaw in the mathematics, but as it stands the paper cannot be evaluated. I would not cite it until I can read the actual text.

For whom: representation theorists working on Gorenstein properties, Auslander-Reiten theory, or monomial algebras. The gentle-algebra classification and the AR-map bijection would interest that community directly. The Nakayama reduction, if correct, would be a useful tool.

Recommendation: yes, send to peer review, but only after asking the author to resubmit with a readable PDF or TeX source. The abstract alone justifies a referee's time; a referee can then test the reduction step and the proof of the AG-biconnection. My own verdict is unverified, not negative. If the reduction gap is real, the central classification theorem would need to be adjusted; if not, this looks like a strong paper.

Referee Report

3 major / 4 minor

Summary. The paper claims several results about Auslander-Gorenstein monomial algebras. The abstract announces: (1) every Auslander-Gorenstein monomial algebra is a string algebra; (2) a combinatorial classification of Auslander-Gorenstein gentle algebras; (3) a procedure converting any 2-Gorenstein monomial algebra into a Nakayama algebra, reducing the classification of Auslander-Gorenstein monomial algebras to the Nakayama case; (4) a stronger form of the Auslander-Reiten Conjecture for monomial algebras; (5) the main theorem that a monomial algebra is Auslander-Gorenstein if and only if it has a well-defined, bijective Auslander-Reiten map, confirming a conjecture of Marczinzik for monomial algebras; and (6) a generalization of Iwanaga–Fuller, namely that every 2n-Gorenstein monomial algebra is also (2n+1)-Gorenstein. The supplied full text is garbled mojibake, so no definitions, lemmas, proofs, or table contents are legible.

Significance. If the results are correct, they would provide a new homological characterization of the Auslander-Gorenstein condition within monomial algebras via the Auslander-Reiten bijection, together with a useful reduction to Nakayama algebras. The abstract's claims are plausible and internally consistent, and a confirmed Marczinzik conjecture for monomial algebras would be a meaningful contribution. However, the significance cannot presently be assessed: the body of the manuscript is unreadable, and the reduction step as stated has a logical gap. There is no machine-checked proof or code to compensate for the missing textual proofs.

major comments (3)

[Supplied full text] The submitted body is garbled mojibake throughout; no definition, lemma, proof, or table is legible, and the header even shows a different arXiv identifier (2508.06963v1 [cs.AI]) from the announced paper. The central claims—the Auslander-Gorenstein iff bijective Auslander-Reiten map theorem, the gentle-algebra classification, and the reduction procedure—cannot be verified at all. A readable manuscript with complete proofs is a prerequisite for any scientific assessment.
[Abstract] The reduction step is stated only for 'any 2-Gorenstein monomial algebra', but the abstract concludes a reduction of the classification of all Auslander-Gorenstein monomial algebras. No statement asserts that every Auslander-Gorenstein monomial algebra is 2-Gorenstein; the Iwanaga-Fuller-type lift (2n-Gorenstein implies (2n+1)-Gorenstein) even suggests that even Gorenstein dimension may be arbitrarily large. The authors must either prove that all Auslander-Gorenstein monomial algebras are 2-Gorenstein, or describe an iteration/adaptation covering arbitrary Gorenstein dimension. In either case they must also prove that the transformation preserves the Auslander-Gorenstein property in both directions, including the relevant bimodule injective dimensions and Gorenstein projective dimensions, and that it covers all Nakayama targets. Without this bridge, the announced classification and the M
[Abstract (application)] The abstract advertises 'a stronger version of the Auslander-Reiten Conjecture' as an application of the reduction method, but does not state the strengthened assertion, its hypotheses, or how it follows from the Nakayama reduction. Since this is listed as a main outcome, the precise statement and proof must be included; its omission makes the claimed application unverifiable.

minor comments (4)

[Title/header] The full-text header cites arXiv:2508.06963v1 [cs.AI], which does not match the announced article arXiv:2508.06957 (math.RT). The header should be corrected.
[Introduction (where definitions would appear)] The notions '2-Gorenstein', '2n-Gorenstein', and 'Auslander-Gorenstein' for monomial algebras should be defined explicitly, including the role of bimodule injective dimension and the convention for Gorenstein dimension parity.
[Abstract and main theorem] The phrase 'well-defined, bijective Auslander-Reiten map' needs a precise definition: what is the domain and codomain, why well-definedness is nontrivial, and in which category the bijection is taken. Without this, the main theorem is ambiguous.
[Gentle-algebra classification] The abstract promises a 'simple combinatorial classification' of Auslander-Gorenstein gentle algebras, but no combinatorial condition is stated in the abstract or legible text. The classification should be stated explicitly, even as a theorem.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: pure mathematical derivation; the Nakayama reduction is a proof step, not an input.

full rationale

No significant circularity identified. This is a pure mathematics paper: there are no fitted parameters, no data-fitting renamed as prediction, and no empirical calibration. The main theorem (monomial algebra is Auslander-Gorenstein iff it has a well-defined, bijective Auslander-Reiten map) is presented as a confirmation of an external conjecture by Marczinzik, not as a restatement of the paper's own definitions. The reduction of the classification to Auslander-Gorenstein Nakayama algebras is a claimed proof device: the abstract says the procedure transforms 2-Gorenstein monomial algebras into Nakayama algebras, and the paper would need to justify that this covers all Auslander-Gorenstein monomial algebras or that the Gorenstein dimension can be reduced. That is a potential logical gap or correctness concern, but a gap is not circularity: the Nakayama classification is not assumed as the target result, and the reduction is not shown to be equivalent by construction to the conclusion. The supplied full text is badly corrupted, so a line-by-line proof audit is impossible, but nothing in the readable abstract or the recovered fragments exhibits a definitional equivalence, a fitted-input prediction, or a load-bearing self-citation chain. Therefore the derivation chain, as evidenced, is self-contained rather than circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

Pure mathematics: no fitted constants, no free parameters, and no newly postulated objects (no new particles, forces, dimensions, or algebraic entities) are visible in the abstract. The claims rest on standard definitions and on external results named in the abstract (Iwanaga-Fuller, Marczinzik's conjecture, the Auslander-Reiten conjecture). The ledger cannot be fully audited because the full text is unreadable; in particular, hidden auxiliary hypotheses inside the proofs (e.g., finiteness or field assumptions on the base field, admissibility of the quiver) could not be enumerated.

assumptions (5)

standard math Standard definitions of monomial, string, gentle, and Nakayama algebras as quiver-algebra classes
Invoked by every claim in the abstract; these are standard classes in the representation theory of finite-dimensional algebras, but their exact definitions were not restated and the full text was unreadable.
standard math The Auslander-Gorenstein condition (finite injective dimension of the regular bimodule together with the Gorenstein-projective bounds)
The property under study throughout; the abstract assumes the reader's familiarity with the definition and does not restate it.
domain assumption The Auslander-Reiten map is available and well-defined on the relevant module categories
The headline equivalence 'AG iff bijective AR map' presupposes a construction of the AR map for indecomposable modules of these algebras; this is a nontrivial domain fact, not a definition.
domain assumption Prior results assumed: Iwanaga-Fuller theorem, Marczinzik's conjecture, the Auslander-Reiten conjecture
The abstract generalizes the first, confirms the second, and applies the third; these external results are assumed true and are cited by name only.
domain assumption The reduction procedure preserves the Auslander-Gorenstein property
The claimed reduction of the classification to Nakayama algebras depends on this preservation; it is asserted as the paper's own theorem and could not be checked in the unreadable full text.

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Cite this review

Pith. "Pith review of The Auslander-Gorenstein condition for monomial algebras." pith.science (2026). https://pith.science/paper/7FWPCYVS

@misc{pith2026250806957,
  author       = {Pith},
  title        = {Pith review of: The Auslander-Gorenstein condition for monomial algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7FWPCYVS}},
  note         = {Machine review of arXiv:2508.06957}
}

abstract

This paper investigates the Auslander-Gorenstein property for monomial algebras. First, we prove that every Auslander-Gorenstein monomial algebra is a string algebra and present a simple combinatorial classification of Auslander-Gorenstein gentle algebras. Furthermore, we describe a procedure to transform any 2-Gorenstein monomial algebra into a Nakayama algebra, thereby reducing the classification of Auslander-Gorenstein monomial algebras to that of Auslander-Gorenstein Nakayama algebras. As an application of this reduction method, we prove that every monomial algebra satisfies a stronger version of the Auslander-Reiten Conjecture. Our second main result establishes that a monomial algebra is Auslander-Gorenstein if and only if it has a well-defined, bijective Auslander-Reiten map, confirming a conjecture of Marczinzik for monomial algebras. This yields a new homological characterisation of the Auslander-Gorenstein property. Additionally, we provide an explicit description of the Auslander-Reiten bijection in the case of gentle algebras. Along the way, we also generalise a result of Iwanaga and Fuller: We show that every $2n$-Gorenstein monomial algebra is also $(2n+1)$-Gorenstein for every $n\ge 1.$

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Forward citations

Cited by 2 Pith papers

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A Serre-type criterion for $n$-Gorenstein rings and its application to Nakayama algebras
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Over Auslander-Gorenstein Nakayama algebras, Ext^n(S,A) for odd n is always zero or simple, settling the Klász–Kleinau–Marczinzik conjecture via a (G_n) criterion and syzygy filtration.
A survey on Auslander-Gorenstein algebras
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A survey of finite-dimensional Auslander-Gorenstein algebras, including classifications for monomial and incidence algebras and the identification of the Auslander-Reiten permutation with rowmotion, Ringel's homologic...

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