{"id":"179cd933-d3dd-414d-a204-173381d2cdd0","arxiv_id":"2508.06957","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A monomial algebra is Auslander-Gorenstein if and only if its Auslander-Reiten map is well-defined and bijective, confirming Marczinzik's conjecture for this class of algebras.","lead":"A math paper proves that for monomial algebras, a homological regularity property (being Auslander-Gorenstein) is equivalent to having a well-defined bijective Auslander-Reiten map, confirming a conjecture of Marczinzik in this class. It also gives a combinatorial classification for gentle algebras and reduces the general monomial case to Nakayama algebras.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reduction to Nakayama algebras is only stated for 2-Gorenstein algebras, yet it is used to classify all Auslander-Gorenstein monomial algebras.","rationale":"The reader's weakest assumption also targets the reduction, but emphasizes preservation of the Auslander-Gorenstein property under the transformation, whereas my concern is about the domain of the transformation: it is only defined for 2-Gorenstein algebras, so it may fail to cover all AG monomial algebras unless a separate theorem ensures every AG monomial algebra is 2-Gorenstein. Both concerns are real and both are unverifiable from the garbled full text. Because the paper cannot currently be read, the appropriate verdict remains UNVERDICTED. If the correct full text reveals that the reduction indeed applies only to 2-Gorenstein algebras and that higher-dimensional AG monomial algebras exist, the classification claim would be seriously weakened; if it includes the missing bridge, the objection is resolved. The Iwanaga-Fuller statement in the abstract is also suspiciously trivial under the standard 'injective dimension at most n' reading, but I do not make that the primary attack because the central claim's viability hinges more directly on the reduction's coverage.","tokens_in":22220,"tokens_out":8311,"duration_ms":79406,"concrete_test":"Obtain the correct full text (e.g., from arXiv source files) and check for a theorem stating that every Auslander-Gorenstein monomial algebra is 2-Gorenstein, or an inductive/iterative extension of the reduction to higher Gorenstein dimension. Alternatively, construct or locate an explicit Auslander-Gorenstein monomial algebra with injective dimension >2 (for example among the gentle algebras the paper classifies) and test whether the reduction procedure accepts it; if it does not, the reduction cannot underpin the full classification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's reduction step is load-bearing: it transforms only 2-Gorenstein monomial algebras into Nakayama algebras, but the announced classification and the Marczinzik-conjecture confirmation concern all Auslander-Gorenstein monomial algebras. For the reduction to cover the full class, one of the following must hold: (i) every Auslander-Gorenstein monomial algebra is 2-Gorenstein, or (ii) the procedure can be iterated or adapted to arbitrary Gorenstein dimension. The abstract states neither. It only says every AG monomial algebra is a string algebra and gives an Iwanaga-Fuller-type lift (2n-Gorenstein implies (2n+1)-Gorenstein), which suggests the Gorenstein dimension may be arbitrarily large even. Without a bridge from 2-Gorenstein to the full AG class, the classification and the 'AG iff bijective Auslander-Reiten map' theorem do not follow from the stated reduction. This is a concrete logical gap in the abstract's implication structure, not merely a missing detail.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22303,"tokens_out":4360,"duration_ms":46560,"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":[{"comment":"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.","section":"Supplied full text"},{"comment":"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","section":"Abstract"},{"comment":"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.","section":"Abstract (application)"}],"minor_comments":[{"comment":"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.","section":"Title/header"},{"comment":"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.","section":"Introduction (where definitions would appear)"},{"comment":"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.","section":"Abstract and main theorem"},{"comment":"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.","section":"Gentle-algebra classification"}],"recommendation":"major_revision","confidential_remarks":"I cannot certify the mathematics because the body of the manuscript as supplied is completely garbled. If this is a pipeline artifact, the editor should obtain a clean PDF before further review; as it stands the paper is not reviewable. Independently of the garbling, the abstract's reduction from 2-Gorenstein monomial algebras to Nakayama algebras does not obviously cover all Auslander-Gorenstein monomial algebras, and the advertised stronger Auslander-Reiten Conjecture is not stated. These issues should be addressed before the paper is considered further."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nFor 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.\n\nRecommendation: 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.","headline":"Plausible and potentially significant abstract, but the full text is unreadable and the reduction step has a real logical gap that needs checking.","tokens_in":22915,"tokens_out":1395,"would_cite":false,"duration_ms":17020,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E10","16E65","16G70","16P10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A monomial algebra is Auslander-Gorenstein exactly when its Auslander-Reiten translate is a well-defined bijection.","keywords":["Auslander-Gorenstein","monomial algebras","Auslander-Reiten translate","string algebras","gentle algebras","Nakayama algebras","Auslander-Reiten conjecture","Gorenstein property"],"falsifier":"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","tokens_in":21958,"feed_emoji":"🔁","tokens_out":15757,"duration_ms":167461,"temperature":0.7,"pith_summary":"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.","feed_headline":"Monomial algebras: Gorenstein iff AR translate is bijective","feed_subtitle":"This reduces the classification to Nakayama algebras and proves a stronger Auslander-Reiten conjecture.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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 $n\\ge1$."],"supporting_citations":[],"fun_headline_variants":["Gorenstein monomial iff AR map is a bijection","Monomial Gorenstein algebras reduce to Nakayama case","AR bijection characterizes Gorenstein monomials","Marczinzik conjecture proven for monomial algebras","Gentle Gorenstein monomials get combinatorial test"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Gorenstein monomial iff AR map is a bijection","Monomial Gorenstein algebras reduce to Nakayama case","AR bijection characterizes Gorenstein monomials","Marczinzik conjecture proven for monomial algebras","Gentle Gorenstein monomials get combinatorial test"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0006,"raw_usage":{"total_tokens":2650,"prompt_tokens":761,"completion_tokens":1889,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":505,"completion_tokens_details":{"reasoning_tokens":1809}},"tokens_in":505,"tokens_out":1889,"duration_ms":15264,"temperature":1.0,"reasoning_tokens":1809,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T22:26:04.542365+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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","supporting_citations":[],"review_version":1}