{"id":"2537969f-6fd5-433a-9a9c-74f08806fda1","arxiv_id":"1908.02912","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Irreducible morphisms and Auslander-Reiten triangles in the stable category of modules over a repetitive algebra take three canonical shapes, and every such triangle is induced by an Auslander-Reiten sequence.","lead":"This paper classifies the possible shapes of irreducible morphisms and of Auslander-Reiten triangles in the stable category of modules over a repetitive algebra. It proves that every such triangle is induced by a short exact sequence, which restricts the component morphisms to three canonical forms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.7's proof that a projective summand P forces M≅radP and M''≅P/socP uses a false submodule-intersection equality; Corollary 2.5's proof also has an unjustified irreducibility step.","rationale":"The reader identified the no-projective-summand hypothesis as the weakest assumption; my primary concern is not that hypothesis itself but the proof of the bridge theorem that this hypothesis feeds. The invalid intersection equality is a clear mathematical error in a central proof, independent of any disagreement with the literature. Since the conclusion may be standard, I do not ask for rejection, but the paper needs a rigorous repair of Theorem 2.7 and Corollary 2.5 before an unqualified ACCEPT is warranted. The Corollary 2.5 gap is secondary but reinforces the need for a careful rewrite of Section 2.3. This is a correctness risk in presentation, not a demonstrated falsehood of the main theorems.","tokens_in":17534,"tokens_out":39414,"duration_ms":444388,"concrete_test":"Independently re-prove the 'Moreover' part of Theorem 2.7 without the false intersection equality: (a) use essentiality of α to identify P with the injective hull of M, so indecomposability of M gives indecomposability of P; (b) use the right almost split inclusion radP→P to obtain λ:M→radP with α=ι∘λ, and deduce M≅radP from indecomposability of M and λ split mono. Separately, in the Section 4 example, compute the AR sequence ending at P/socP for a projective-injective with non-simple HomΛ(Q,I) and verify it has the exact form (2.9) with M′=radP/socP, h an epimorphism and h′ a monomorphism. If these checks pass, the theorem is true but needs a rewritten proof; if they fail, Theorems 3.3(ii)-(iii) are unsupported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The main results depend on Theorem 2.7, and its proof contains an invalid step. After showing that α:M→P is an essential monomorphism, the authors assume P=P′⊕P″ and assert M≅Imα=(Imα∩P′)⊕(Imα∩P″). A submodule of a direct sum need not be the direct sum of its intersections; a diagonal submodule is a counterexample. This equality is what proves P indecomposable, then M≅radP and M″≅P/socP, and hence the reduction of Theorem 3.3 to AR sequences of the special form (2.9). The conclusion may be recoverable by the standard fact that an essential monomorphism from an indecomposable module into an injective makes that injective an indecomposable injective hull, but the argument as written is invalid. A second gap appears in Corollary 2.5: from h and h′ irreducible in Λ̂-mod and h−h′ factoring through a projective, the proof asserts the difference is irreducible because Irr is a k-vector space; in fact the difference could lie in rad^2, so the claimed uniqueness of module representatives—used to define stably smonic/sepic/sirreducible—is not established by the given argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the stable category of finitely generated modules over the repetitive algebra Λ̂ of a finite-dimensional algebra Λ. It claims a trichotomy for irreducible morphisms in Λ̂-mod (smonic, sepic, sirreducible) under a no-projective-direct-summands hypothesis, and it describes the shape of Auslander-Reiten triangles by proving that every AR triangle whose three terms have no projective summands is induced by an AR sequence in Λ̂-mod. The engine is Theorem 2.7, which asserts that the inducing AR sequence has middle term M′⊕P with P projective, and that if P≠0 then P is indecomposable and the sequence has the form 0→radP→M′⊕P→P/socP→0. Section 3 uses this to classify the stable irreducible morphisms appearing in AR triangles, and Section 4 gives a gentle-algebra example.","tokens_in":17726,"tokens_out":17102,"duration_ms":190269,"significance":"If correct, the results provide a concrete structural description of irreducible morphisms and AR triangles in the stable category of a repetitive algebra, connecting the module-theoretic AR sequences of Hughes–Waschbüsch with Happel’s triangulated stable category. The paper follows a plausible strategy based on standard Frobenius-category and AR-theory methods, and it includes a substantial example verifying the claimed classification. However, the central results currently rest on two proof steps that are invalid as written: an asserted submodule equality in the proof of Theorem 2.7, and a vector-space argument in Corollary 2.5. Both steps are load-bearing for Theorems 3.2 and 3.3, so the manuscript requires substantive revision before the claims can be regarded as established.","major_comments":[{"comment":"In the proof of Theorem 2.7, after establishing that α:M→P is an essential monomorphism, the authors assume P=P′⊕P″ and assert Imα=(Imα∩P′)⊕(Imα∩P″). This equality is false in general: a submodule of a direct sum need not be the direct sum of its intersections with the summands, with a diagonal submodule as a counterexample. This step is the only argument given to prove that P is indecomposable, and it is also used to reach the conclusions M≅radP and M″≅P/socP, which in turn are needed to reduce Theorem 3.3 to AR sequences of the special form (2.9). The theorem may still be true, since the isomorphism δ:I(M)→P obtained earlier, together with the standard fact that the injective hull of an indecomposable module is indecomposable, directly yields that P is indecomposable. The authors should replace the faulty equality with a correct proof.","section":"§2.3, Theorem 2.7"},{"comment":"The proof of Corollary 2.5 contains two invalid steps. First, from h_k0,h′_k0 ∈ Irr(M,M′_k0) and the fact that Irr is a k-vector space, the authors conclude that v_k0∘u_k0∈Irr(M,M′_k0). This does not follow: elements of Irr are classes in rad/rad², and the difference of two nonzero classes can be zero, so the representative v_k0∘u_k0 may lie in rad² and need not be irreducible. Second, the assertion that each component h_k−h′_k factors through an indecomposable projective summand P_k is not justified; a morphism factoring through a projective module P need not factor through a single indecomposable summand of P. Corollary 2.5 is used to make Definition 3.1 well-defined and to justify replacing h′ by h″ in the proof of Theorem 3.3, so this gap directly affects Theorems 3.2 and 3.3. The authors need either a valid proof of Corollary 2.5 or a reformulation that avoids relying on it.","section":"§2.2, Corollary 2.5"}],"minor_comments":[{"comment":"The introduction refers to 'Theorem 3.2' twice where the second mention should clearly be Theorem 3.3.","section":"Introduction"},{"comment":"There is a typo in the phrase 'Let ι_M : M → I(M) the the injective Λ̂-hull of M'; 'the the' should be 'be the'.","section":"§2.3, Theorem 2.7"},{"comment":"The sentence 'such that h′ is equal to h″ module an isomorphism' should read 'modulo an isomorphism'.","section":"§3, proof of Theorem 3.3"},{"comment":"In part (ii), the statement 'M′_i = 0 for all n≠i0,i0+1' uses the variable n where i is intended.","section":"§2.3, Corollary 2.8"},{"comment":"The reference 'Remark rem1.1' in part (i) is broken; it should be 'Remark 2.1'.","section":"§3, Lemma 3.5"},{"comment":"The quiver and the Auslander-Reiten quiver in Section 4 are extremely difficult to read; the labels around Figure 1 appear garbled. A redrawn, clean version would greatly improve the exposition.","section":"§4, Figure 1"}],"recommendation":"major_revision","confidential_remarks":"The two major issues identified are genuine and load-bearing, but they do not appear to be hopeless. The Theorem 2.7 gap is readily repairable via the standard indecomposability of injective hulls. The Corollary 2.5 gap is more delicate: if Corollary 2.5 is actually false, then Definition 3.1 and Theorem 3.2 would be ill-founded. I would ask the authors to either give a correct proof of Corollary 2.5 or restructure the paper so that the main theorems do not depend on it. The paper is within the journal's scope and the overall strategy is plausible, so I recommend a major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper does what it says—transfers the shape classification of irreducible morphisms and AR triangles from Λ-hat-mod to its stable category—and the example is a genuine check. But the stress-test note is right: two steps in the load-bearing proofs are invalid, so this is not just a typo-fixing round.\n\nWhat's new: Theorem 3.2 is an explicit transfer of Giraldo [7, Thm. 26] to the stable category using Theorem 2.4, and Theorem 3.3 generalizes [15, §4.1] from the bounded derived category to Λ-hat-mod with an indecomposability condition. The authors are upfront about the provenance. The Section 4 example is useful and appears to check all four cases of Theorem 3.3.\n\nNow the soft spots. The first stress-test point lands. In the proof of Theorem 2.7, after showing α: M → P is an essential monomorphism, the authors assume that if P = P' ⊕ P'', then Im α = (Im α ∩ P') ⊕ (Im α ∩ P''). That equality is not valid for arbitrary essential submodules; the essentiality condition alone does not force it. The diagonal submodule in a direct sum is not essential, so it is not a counterexample in the essential case, but the equality still does not follow. The conclusion that P is indecomposable might be recoverable if one proves M is uniform or uses a different injective-hull argument, but the paper doesn't supply that. Since Theorem 2.7 underpins the reduction of Theorem 3.3 to the special AR sequences (2.9), this gap matters.\n\nThe second point also lands. In Corollary 2.5, from h and h' irreducible and h − h' factoring through a projective, the proof claims the difference is irreducible because Irr is a k-vector space. That is not valid: the class of the difference in rad/rad^2 can be zero even when the morphism is nonzero, so the difference need not be irreducible. The corollary's conclusion—that the stable class determines the module representative—is used to define stably smonic/sepic/sirreducible and in the proof of Theorem 3.3. It may be true, but the given argument doesn't establish it.\n\nOverall: the mathematics looks plausible and the results are modest, useful extensions. The proof gaps are not typos; they are invalid inference steps. The paper deserves a serious referee—not a desk reject—but the authors need to fix these proofs or supply alternative arguments.\n\nAudience: representation theorists working on repetitive algebras, AR theory, and stable categories. I would engage with it after revision; right now, the proofs as written do not fully support the main theorems.","headline":"Transfers the irreducible-shape classification to the stable category, but two load-bearing proof steps are invalid and need real revision.","tokens_in":18343,"tokens_out":14944,"would_cite":false,"duration_ms":158873,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G10","16G20","20C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Irreducible morphisms in the stable category of modules over a repetitive algebra have exactly three canonical shapes, and every Auslander-Reiten triangle whose terms avoid projective summands is induced by an Auslander-Reiten sequence.","keywords":["repetitive algebras","stable category","Auslander-Reiten triangles","irreducible morphisms","Auslander-Reiten sequences","triangulated categories","Frobenius categories","gentle algebras"],"falsifier":"Take the gentle algebra in Section 4 and inspect the displayed stable Auslander-Reiten quiver for an arrow whose source and target have no projective summands. The theorem predicts that this arrow is stably smonic, stably sepic, or stably sirreducible and that the surrounding triangle is induced by an Auslander-Reiten sequence with at most one indecomposable projective summand; an arrow of any other shape, or a triangle whose end terms are not $\\operatorname{rad}\\hat P$ and $\\hat P/\\operatorname{soc}\\hat P$ when a projective summand appears, would disprove the claim.","tokens_in":17244,"feed_emoji":"🔺","tokens_out":14356,"duration_ms":133562,"temperature":0.7,"pith_summary":"This paper pins down the possible shapes of irreducible morphisms and Auslander-Reiten triangles in the stable category of finitely generated modules over a repetitive algebra. It proves that, for objects with no projective direct summands, an irreducible stable morphism is always one of three types: all component maps are split monomorphisms, all are split epimorphisms, or exactly one component is irreducible. It then proves that every Auslander-Reiten triangle whose three terms avoid projective summands is induced by an ordinary Auslander-Reiten sequence of modules, possibly padded by a single indecomposable projective summand; when that summand is present, the two end terms are forced to be its radical and its top. These results turn questions about the triangulated stable category back into module-category problems, where Auslander-Reiten sequences are easier to construct and recognize.","feed_headline":"Irreducible morphisms have exactly three shapes","feed_subtitle":"The result turns triangulated questions back into module-category problems.","key_machinery":"The machinery is the pair consisting of the stable category $\\widehat{\\Lambda}$-$\\underline{\\mathrm{mod}}$, a triangulated category whose translation is the first cosyzygy functor $\\Omega^{-1}$, and the transfer theorem (Theorem 2.4) that compares irreducibility in the stable category with irreducibility in the module category $\\widehat{\\Lambda}$-$\\mathrm{mod}$. The repetitive algebra $\\widehat{\\Lambda}$ is the doubly infinite matrix algebra built from $\\Lambda$ and its injective cogenerator, and its module category is a Frobenius category in the sense that projective and injective objects coincide, so passing to the stable category produces the triangulated structure. The load-bearing transfer says that when $\\hat M$ and $\\hat M'$ have no projective direct summands, a morphism's stable class is split mono, split epi, or irreducible exactly when a representative is. The proof of Theorem 2.7 then starts from the Auslander-Reiten sequence beginning at $\\hat M$ with middle term $\\hat M'\\oplus \\hat Y$, uses the source-morphism property to cancel any non-projective summand of $\\hat Y$, and concludes $\\hat Y$ must be a projective $\\hat P$; irreducibility of $\\hat M\\to \\hat P$ forces $\\hat M\\simeq \\operatorname{rad}\\hat P$ and dually $\\hat M''\\simeq \\hat P/\\operatorname{soc}\\hat P$.","core_discovery":"On the paper's own terms, the discovery is a structural trichotomy plus a lifting theorem. If $\\hat h\\colon \\hat M\\to \\hat M'$ is an irreducible morphism in $\\widehat{\\Lambda}$-$\\underline{\\mathrm{mod}}$ with $\\hat M$ and $\\hat M'$ free of projective direct summands and at least one of them indecomposable, then $\\hat h$ is stably smonic, stably sepic, or stably sirreducible, meaning its class in $\\widehat{\\Lambda}$-$\\underline{\\mathrm{mod}}$ has all split-monomorphism components, all split-epimorphism components, or exactly one irreducible component. In an Auslander-Reiten triangle $\\hat M \\xrightarrow{\\hat h} \\hat M' \\xrightarrow{\\hat h'} \\hat M'' \\to \\Omega^{-1}\\hat M$ with all three terms projective-summand-free, $\\hat h$ and $\\hat h'$ must pair as follows: smonic with sepic, sepic with sirreducible, and sirreducible with smonic or sirreducible. The enabling fact is Theorem 2.7: such a triangle is induced by an Auslander-Reiten sequence $0\\to \\hat M \\to \\hat M'\\oplus \\hat P \\to \\hat M''\\to 0$ in $\\widehat{\\Lambda}$-$\\mathrm{mod}$, where $\\hat P$ is projective, and if $\\hat P\\neq 0$ then $\\hat P$ is indecomposable with $\\hat M\\simeq \\operatorname{rad}\\hat P$ and $\\hat M''\\simeq \\hat P/\\operatorname{soc}\\hat P$.","pith_inferences":["Because the transfer theorem is invoked only when no projective summands are present, the classification may need a separate bookkeeping for Auslander-Reiten triangles in which some term does contain a projective summand; a natural extension is to strip off the projective part and then apply the trichotomy to the remaining stable morphism.","The proof identifies the projective summand in the inducing sequence as the unique correction needed to turn a source morphism in the module category into one in the stable category, which suggests that any Frobenius category whose stable category is triangulated should admit a parallel 'one projective correction' description whenever a transfer theorem of the same kind holds.","For finite-dimensional algebras of infinite global dimension, Theorem 3.3 offers a way to read off the shape of derived-category Auslander-Reiten triangles directly from module-theoretic data, which could simplify explicit computations of stable Auslander-Reiten quivers in examples beyond gentle algebras.","A testable extension is to check whether the trichotomy survives when only one of the two objects is assumed free of projective summands; Lemma 2.3 gives partial statements in that direction, but the full classification may require an additional case."],"forward_implications":["Every Auslander-Reiten triangle in $\\widehat{\\Lambda}$-$\\underline{\\mathrm{mod}}$ whose terms avoid projective summands can be constructed from an ordinary Auslander-Reiten sequence in $\\widehat{\\Lambda}$-$\\mathrm{mod}$, so such triangles can be studied through short exact sequences and projectives.","When the inducing sequence needs a nonzero projective summand $\\hat P$, the triangle is completely pinned down by $\\hat P$: it starts at $\\operatorname{rad}\\hat P$, ends at $\\hat P/\\operatorname{soc}\\hat P$, and its middle term is the middle term of the corresponding Auslander-Reiten sequence plus $\\hat P$.","The two middle maps of an Auslander-Reiten triangle have locked shapes: a smonic map forces its successor to be sepic, a sepic map forces its successor to be sirreducible, and a sirreducible map forces its successor to be smonic or sirreducible.","For algebras where the bounded derived category and $\\widehat{\\Lambda}$-$\\underline{\\mathrm{mod}}$ are equivalent as triangulated categories, the same trichotomy describes the shape of Auslander-Reiten triangles in the derived category.","The gentle algebra worked out in Section 4 shows that all four pairings allowed by Theorem 3.3 actually occur in a single stable Auslander-Reiten component of type $\\mathbb{Z}A_\\infty$."],"supporting_citations":[{"why":"Supplies the transfer theorem and the smonic/sepic/sirreducible classification of irreducible morphisms in the module category of the repetitive algebra.","marker":"[7]"},{"why":"Provides the triangulated structure of the stable category, the definition and formal properties of Auslander-Reiten triangles, and the fact that the two non-translation maps in such a triangle are irreducible.","marker":"[9]"},{"why":"Establishes the existence of Auslander-Reiten sequences in the category of modules over the repetitive algebra and introduces the repetitive-algebra setting.","marker":"[12]"},{"why":"Provides the general additive-category facts about irreducible morphisms under split projections used in Lemma 2.3.","marker":"[4]"},{"why":"Gives the embedding of the bounded derived category into the stable category of the repetitive algebra, the motivation for studying Auslander-Reiten triangles in this setting.","marker":"[10]"},{"why":"Gives the companion shape classification of Auslander-Reiten triangles in the bounded derived category that this paper recovers and extends.","marker":"[15]"},{"why":"Supplies the standard definitions and radical formalism for Auslander-Reiten sequences and irreducible morphisms.","marker":"[16]"}],"fun_headline_variants":["Irreducible morphisms: three shapes, one theorem","Stable category irreducible maps: smonic, sepic, or sirreducible","Trichotomy for irreducible morphisms in stable module category","AR triangles in stable category lift to module sequences","Three canonical forms for irreducible morphisms in stable category"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the assumption that none of the three terms in the triangle contains a projective summand; if one does, the bridge between stable and module irreducibility breaks, so the classification could fail.","fun_headline_variants_meta":{"raw":{"variants":["Irreducible morphisms: three shapes, one theorem","Stable category irreducible maps: smonic, sepic, or sirreducible","Trichotomy for irreducible morphisms in stable module category","AR triangles in stable category lift to module sequences","Three canonical forms for irreducible morphisms in stable category"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000337,"raw_usage":{"total_tokens":1917,"prompt_tokens":1052,"completion_tokens":865,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":668,"completion_tokens_details":{"reasoning_tokens":782}},"tokens_in":668,"tokens_out":865,"duration_ms":8702,"temperature":1.0,"reasoning_tokens":782,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:29:44.250276+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the gentle algebra in Section 4 and inspect the displayed stable Auslander-Reiten quiver for an arrow whose source and target have no projective summands. The theorem predicts that this arrow is stably smonic, stably sepic, or stably sirreducible and that the surrounding triangle is induced by an Auslander-Reiten sequence with at most one indecomposable projective summand; an arrow of any other shape, or a triangle whose end terms are not $\\operatorname{rad}\\hat P$ and $\\hat P/\\operatorname{soc}\\hat P$ when a projective summand appears, would disprove the claim.","supporting_citations":[{"cited_title":"Giraldo, Irreducible morphisms between modules over a repetitive algebras , Algebr","cited_arxiv_id":null,"evidence_quote":"Supplies the transfer theorem and the smonic/sepic/sirreducible classification of irreducible morphisms in the module category of the repetitive algebra."},{"cited_title":"Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras , London Mathematical Society Lecture Notes Series, no","cited_arxiv_id":null,"evidence_quote":"Provides the triangulated structure of the stable category, the definition and formal properties of Auslander-Reiten triangles, and the fact that the two non-translation maps in such a triangle are irreducible."},{"cited_title":"Hughes and J","cited_arxiv_id":null,"evidence_quote":"Establishes the existence of Auslander-Reiten sequences in the category of modules over the repetitive algebra and introduces the repetitive-algebra setting."},{"cited_title":"Bautista and M","cited_arxiv_id":null,"evidence_quote":"Provides the general additive-category facts about irreducible morphisms under split projections used in Lemma 2.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the embedding of the bounded derived category into the stable category of the repetitive algebra, the motivation for studying Auslander-Reiten triangles in this setting."},{"cited_title":"Shapes of Auslander-Reiten Triangles","cited_arxiv_id":"1610.08457","evidence_quote":"Gives the companion shape classification of Auslander-Reiten triangles in the bounded derived category that this paper recovers and extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the standard definitions and radical formalism for Auslander-Reiten sequences and irreducible morphisms."}],"review_version":1}