{"id":"aef6b12e-07c2-458f-9148-a5288a583ee5","arxiv_id":"2608.03856","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A finite dimensional algebra is Auslander-Gorenstein if and only if it has a well-defined Auslander-Reiten bijection, resolving a conjecture of Marczinzik.","lead":"The authors prove a conjecture in the representation theory of finite dimensional algebras: an algebra is Auslander-Gorenstein exactly when it has a well-defined Auslander-Reiten bijection. This new criterion yields a short proof that a finite lattice with an Auslander-Gorenstein incidence algebra must be distributive.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Final step of Theorem 2.6 invokes an unstated characterization of the Auslander condition; standard but not proved.","rationale":"The stress-test pass confirms the reader's assessment. The main construction is coherent: Lemma 1.4 is a standard duality argument; Lemma 2.2 and Lemma 2.3 support (2) implies (3) solidly; the multiplicity argument in Theorem 2.4 is correct; Theorem 2.6's induction is valid, and the application to distributive lattices is sound under the paper's right-module convention for incidence algebras. The only load-bearing weakness is the final inference from syzygy injective dimensions to the Auslander condition, exactly as the reader identified. This is a standard theorem in the literature, and there is no evidence that it is false, so it does not warrant rejection. However, because the paper does not state or prove the equivalence, a referee should require either a precise citation or a short proof. The verdict of ACCEPT with moderate confidence remains appropriate; the concern is about completeness of presentation, not a detected flaw in the mathematics.","tokens_in":10027,"tokens_out":36392,"duration_ms":333980,"concrete_test":"Write out, from the definitions and the cited sources, the exact lemma used in the final paragraph of Theorem 2.6: for a finite-dimensional algebra A, if P_r(D(A)) is the r-th term in the minimal projective resolution of the right A-module D(A), then idim_A P_r(D(A)) ≤ r for every r implies that A^op satisfies the Auslander condition. In doing so, explicitly identify the duality isomorphism (e.g., flatdim_A^op D(P_i) = idim_A P_i) and verify that no Iwanaga-Gorenstein hypothesis is needed. If the lemma cannot be derived from [AR]/[FGR] verbatim, the final step of Theorem 2.6 is missing support.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the new implication (3) implies (1) in Theorem 2.6 is internally coherent through the induction, but the last two sentences make a nontrivial leap. After proving that every indecomposable direct summand P of P_r(D(A)) satisfies idim P ≤ r, the paper asserts: 'Thus, A^op satisfies the Auslander condition.' This is a characterization of the Auslander condition in terms of the projective resolution of the injective cogenerator D(A), rather than the definition given in the introduction (existence of an injective coresolution of A with flat dimension of the i-th term ≤ i). The equivalence is standard and is plausibly contained in the cited [AR] and [FGR], but it is not stated, proved, or even explicitly cited at that point. If the precise statement requires A or A^op to already be Iwanaga-Gorenstein, or if the duality step transferring injective dimensions of right A-modules to flat dimensions of left A^op-modules has a hidden finiteness or side condition, then the induction alone would not close the argument. No internal inconsistency was found in the earlier steps; this is the single weakest load-bearing assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a new characterization of Auslander-Gorenstein finite dimensional algebras: such algebras are exactly those with a well-defined Auslander-Reiten bijection, and also exactly those that are Iwanaga-Gorenstein with a well-defined grade bijection. This resolves a conjecture of Marczinzik. The proof establishes two new implications, (2) implies (3) and (3) implies (1), with the equivalence (1) iff (2) previously known by Iyama. The paper also gives a short proof that an Auslander-Gorenstein incidence algebra of a finite lattice forces the lattice to be distributive.","tokens_in":10239,"tokens_out":45627,"duration_ms":330320,"significance":"If the result holds, it gives a finite and readily checkable criterion for the Auslander-Gorenstein property: one must verify that every indecomposable injective has a finite minimal projective resolution with an indecomposable projective final term and that the induced map on isomorphism classes is bijective. This is a conceptually clean statement and proves a conjecture. The proofs are largely self-contained, with detailed homological lemmas (Lemmas 1.4, 2.2, 2.3, 2.5). The application to incidence algebras is elegant and gives a new proof of a known theorem.","major_comments":[],"minor_comments":[{"comment":"The final paragraph states 'Thus, A^op satisfies the Auslander condition' after proving idim P_r(D(A)) ≤ r for all r. This uses the standard characterization of the Auslander condition in terms of the injective dimensions of the projective terms in a minimal projective resolution of the injective cogenerator D(A). Since this step is load-bearing for the implication (3) ⇒ (1), the authors should state this equivalence explicitly and give a precise reference (for example [AR] or [FGR]).","section":"§2.2, Theorem 2.6"},{"comment":"In the same final paragraph, the sentence 'which is equivalent to every indecomposable projective A^op-module having finite injective dimension by duality' should be expanded to clarify that this implies idim_{A^op} A^op < ∞, allowing Theorem 1.5(2) to be applied.","section":"§2.2, Theorem 2.6"},{"comment":"The assertion that P(j) is not projective-injective for a join-irreducible element j is used to conclude d = 1, but no proof or citation is given. A brief justification (e.g., that the unique lower cover j_* forces I(j) ≇ P(j)) would make the argument self-contained.","section":"§3, Lemma 3.1"},{"comment":"In the proof of Lemma 2.2, the reference to 'the last assertion of Lemma 1.4' is slightly ambiguous because Lemma 1.4 has several assertions; consider labeling them for clarity.","section":"§1, Lemma 1.4 and §2.1, Lemma 2.2"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid contribution to the representation theory of finite dimensional algebras. The main theorem is significant and the proofs are coherent. The only material request is to make the final step of Theorem 2.6 explicit by stating and citing the standard characterization of the Auslander condition in terms of the projective resolution of D(A). I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hi [Name],\n\nQuick take: this paper proves Marczinzik's conjecture that a finite dimensional algebra is Auslander-Gorenstein iff it has a well-defined Auslander–Reiten bijection. The new implications are (2)⇒(3) and (3)⇒(1), and the proof is largely clean. The multiplicity argument along cycles of the grade permutation in Theorem 2.4 is a genuinely nice idea, and the application to incidence algebras of finite lattices is a short, neat reproof of the known distributivity result.\n\nWhat's actually new: (2)⇒(3) and (3)⇒(1). (1)⇒(2) was Iyama, and the equality ψ(I(S)) = P(h(S)) was in KMT. The grade-bijection characterization (2) is also new. The paper gives a finite, checkable criterion that avoids having to verify injective coresolutions. That has practical value.\n\nThe proofs are detailed and coherent. Lemma 2.2, Lemma 2.3, Theorem 2.4 and the induction in Theorem 2.6 all line up. No circularity: the argument cites Iyama, KMT, AR, FGR for the known directions and derives the rest directly. Self-citations are used only for background and the conjecture statement, which is legitimate.\n\nSoft spots, in proportion. The final step of Theorem 2.6 is compressed. After the induction proves idim P_r(D(A)) ≤ r for every r, the paper asserts \"Thus A^op satisfies the Auslander condition.\" That is a standard duality characterization—the projective resolution of the injective cogenerator dualizes to an injective coresolution of A—but it is not stated or cited at that point. The stress-test concern about hidden side conditions does not land: for finite dimensional algebras the characterization is valid without assuming A or A^op is already Iwanaga-Gorenstein. So this is a presentation gap, not a proof gap. A one-sentence citation would fix it.\n\nMinor stylistic thing: the use of \"grade map\" and \"Auslander–Reiten map\" might trip readers used to the older \"grade bijection\" terminology, but the definitions are clear. The AI-use disclosure is honest, and the proofs are concrete enough that independent checking is straightforward.\n\nWho this is for: anyone working on Auslander-Gorenstein algebras, syzygy modules, or incidence algebras. The main theorem is a clean characterization of an important class, and the lattice application shows the tool works.\n\nBottom line: deserves a serious referee. I'd accept after minor revision, mainly the missing citation in Theorem 2.6 and maybe a bit more detail on the duality step. I'd cite this.","headline":"Settles a conjecture in the expected direction with a clean equivalence between Auslander-Gorenstein and the Auslander-Reiten bijection; the proof is sound and the only real weakness is a compressed standard duality step.","tokens_in":10770,"tokens_out":6149,"would_cite":true,"duration_ms":50156,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G10","16E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Finite dimensional algebras are Auslander-Gorenstein exactly when their Auslander-Reiten map is bijective.","keywords":["Auslander-Gorenstein algebras","Auslander-Reiten bijection","grade bijection","Iwanaga-Gorenstein algebras","incidence algebras","distributive lattices","finite dimensional algebras"],"falsifier":"Find a finite dimensional algebra $A$ whose indecomposable injectives admit a bijective Auslander-Reiten map $\\psi$ but for which some projective term $P_r(D(A))$ in the minimal projective resolution of the injective cogenerator has injective dimension larger than $r$. The theorem predicts no such algebra exists, so any concrete example would refute the implication $(3)\\Rightarrow(1)$.","tokens_in":9838,"feed_emoji":"🔁","tokens_out":10996,"duration_ms":76944,"temperature":0.7,"pith_summary":"The paper proves a conjecture: for a finite dimensional algebra, being Auslander-Gorenstein—a homological regularity condition meaning finite injective dimension together with bounded syzygies—is the same as having a well-defined Auslander-Reiten bijection. That bijection sends each indecomposable injective module to the last nonzero term of its minimal projective resolution, and the claim is that this map exists and is bijective exactly when the algebra is Auslander-Gorenstein. The paper also shows this is equivalent to a second condition phrased through a grade map on simple modules. Since the Auslander-Reiten bijection is finite data, the result turns a homological property into a checkable combinatorial criterion. As an application, the authors give a short new proof that a finite lattice with an Auslander-Gorenstein incidence algebra must be distributive.","feed_headline":"One bijection detects every Auslander-Gorenstein algebra","feed_subtitle":"Finite dimensional algebras are Auslander-Gorenstein exactly when this matching is bijective, settling a conjecture.","key_machinery":"The engine of the proof is the Auslander-Reiten bijection $\\psi$: for an indecomposable injective module $I$ with minimal projective resolution $0 \\to P_d(I) \\to \\cdots \\to P_0(I) \\to I \\to 0$, set $\\psi(I) = P_d(I)$. The paper proves that if $\\psi$ is defined and bijective on all indecomposable injectives, then these last projective terms have injective dimension bounded by their degree, which is exactly the content of the Auslander condition. The companion grade bijection $h$ sends a simple module $S$ to $\\operatorname{top} D\\operatorname{Ext}^{g_S}_A(S,A)$, where $g_S$ is the grade of $S$, and the identity $\\psi(I(S))=P(h(S))$ links the two descriptions.","core_discovery":"The central theorem (Theorem 2.1) states that for a finite dimensional algebra $A$, the following are equivalent: (1) $A$ is Auslander-Gorenstein; (2) $A$ is Iwanaga-Gorenstein and has a well-defined grade bijection $h$; (3) $A$ has a well-defined Auslander-Reiten bijection $\\psi$. Moreover $\\psi(I(S)) = P(h(S))$ for every simple $S$, where $I(S)$ is its injective envelope and $P(S)$ its projective cover. The new implications are $(2)\\Rightarrow(3)$ and $(3)\\Rightarrow(1)$; the latter settles the conjecture. The proof of $(3)\\Rightarrow(1)$ shows by induction that every $r$-th syzygy of an indecomposable injective module has injective dimension at most $r$, which forces the opposite algebra $A^{\\mathrm{op}}$ to satisfy the Auslander condition; duality then transfers this back to $A$ and yields finite self-injective dimension.","pith_inferences":["Because the criterion is finite and combinatorial, it can be implemented by computer search over quiver algebras with relations, offering a practical way to find new Auslander-Gorenstein examples or to test the property in families.","The same bijection viewpoint may extend beyond lattices: the lattice theorem now reduces to an elementary coverage condition drawn from the bijection, and the announced classification of posets with 2-Gorenstein incidence algebras suggests distributive-like posets are only the beginning.","The inductive proof gives a quantitative bonus—each $r$-th syzygy of an indecomposable injective has injective dimension at most $r$—so algebras with the bijection satisfy the Auslander condition with explicit bounds, which could be useful in studying Gorenstein dimensions."],"forward_implications":["The Auslander-Gorenstein property is now decidable from finite data: check that each indecomposable injective's minimal projective resolution ends in an indecomposable projective and that the resulting assignment is a bijection.","The conjecture is settled: every finite dimensional algebra with a well-defined bijective Auslander-Reiten map is Auslander-Gorenstein, going beyond the previously known monomial case.","The two bijections—Auslander-Reiten and grade—coincide on simple modules via $\\psi(I(S))=P(h(S))$, so either can be used to certify the property.","A finite lattice whose incidence algebra is Auslander-Gorenstein must be distributive, now proved without the earlier long case distinction."],"supporting_citations":[{"why":"Supplies the classical result that Auslander-Gorenstein algebras have a well-defined Auslander-Reiten bijection, and Corollary 5.5(b) used to transfer the Auslander condition between $A$ and $A^{\\mathrm{op}}$.","marker":"[AR]"},{"why":"Provides Theorem 3.7, used to show that the Auslander condition passes from $A^{\\mathrm{op}}$ to $A$.","marker":"[FGR]"},{"why":"Shows that Auslander-Gorenstein algebras have a well-defined grade bijection, establishing the direction (1) implies (2).","marker":"[I2]"},{"why":"Proves the equality $\\psi(I(S))=P(h(S))$ linking the Auslander-Reiten and grade bijections, used in Theorem 2.4 and in identifying the maps.","marker":"[KMT]"}],"fun_headline_variants":["One bijection reveals all Auslander-Gorenstein algebras","Bijection criterion settles Marczinzik's conjecture","A single bijection decides Auslander-Gorenstein status","Auslander-Gorenstein algebras bijection is the test"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The induction in the new implication relies on the standard homological equivalence that an algebra satisfies the Auslander condition exactly when the $r$-th projective term in the minimal projective resolution of its injective cogenerator $D(A)$ has injective dimension at most $r$ for every $r$, together with the duality that transfers Iwanaga-Gorenstein between $A$ and $A^{\\mathrm{op}}$; if that equivalence failed, the argument would not force $A$ to be Auslander-Gorenstein.","fun_headline_variants_meta":{"raw":{"variants":["One bijection reveals all Auslander-Gorenstein algebras","Bijection criterion settles Marczinzik's conjecture","A single bijection decides Auslander-Gorenstein status","Auslander-Gorenstein algebras bijection is the test"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000849,"raw_usage":{"total_tokens":3620,"prompt_tokens":800,"completion_tokens":2820,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":416,"completion_tokens_details":{"reasoning_tokens":2754}},"tokens_in":416,"tokens_out":2820,"duration_ms":22912,"temperature":1.0,"reasoning_tokens":2754,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:46:14.810987+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a finite dimensional algebra $A$ whose indecomposable injectives admit a bijective Auslander-Reiten map $\\psi$ but for which some projective term $P_r(D(A))$ in the minimal projective resolution of the injective cogenerator has injective dimension larger than $r$. The theorem predicts no such algebra exists, so any concrete example would refute the implication $(3)\\Rightarrow(1)$.","supporting_citations":[],"review_version":2}