{"id":"05a85ae9-fa3e-4fa3-9e2b-c54a63337fde","arxiv_id":"2502.05655","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper introduces folded gentle algebras and proves they have complete string/band module classifications, explicit Auslander-Reiten sequences, and closure under derived equivalence.","lead":"Folded gentle algebras are a new class of finite-dimensional algebras built by folding gentle algebras along a two-fold symmetry, with special 'crease loops' satisfying irreducible quadratic relations. The paper proves these algebras admit a complete classification of indecomposable modules, explicit Auslander-Reiten sequences, and are closed under derived equivalence.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"M3 symmetric band classification rests on an existence claim that Remark 6.14 itself leaves open; Theorem 6.21 is incomplete as written.","rationale":"The paper's unfolding strategy is coherent, and large parts of Sections 5 and 6 are carefully executed. The reader's conditional verdict is appropriate. However, the reader's identified weakest assumption, axiom (FG5), is a definitional hypothesis rather than a gap in the argument; the paper is explicit that reducible clans are excluded. The more load-bearing weakness is internal: the completeness of the symmetric band class M3 is asserted through Remark 6.18, which cites Remark 6.14, and Remark 6.14 itself concedes that the needed sufficiency criterion is not known in general. Since Theorem 6.21 is the central classification theorem, this is not a cosmetic issue. The proposed test is deliberately small: a single folded gentle algebra over Q with a known symmetric band, where the relevant quadratic extensions are both Q(i), so the representation theory of \\Lambda_w is concrete enough to check the predicted unique \\psi class. A negative result would require a revision of M3 and of Theorem 6.21; a positive result for this example would not prove the general case but would at least confirm that the gap is not an immediate counterexample. The secondary finiteness statement in Theorem 7.7 is also false over Q because irreducible quadratic polynomials give infinitely many non-isomorphic degree-2 field extensions, but this does not affect the classification theorem; it is a separate correction.","tokens_in":69175,"tokens_out":12357,"duration_ms":120894,"concrete_test":"Work over K = Q with the folded gentle algebra of Example 2.10, where the crease relations are \\eta_2^2 + \\varepsilon_2 = 0 and \\eta_3^2 - 2\\eta_3 + 5\\varepsilon_3 = 0, and take the symmetric band w = \\eta_2 \\beta \\eta_3 \\beta^{-1} from Example 4.6. For the irreducible polynomials p = x^2 + 1 and p = x^2 + x + 2, compute the sets \\Psi^{(1)}_{w,p} and \\Psi^{(2)}_{w,p} exactly as in Definition 6.19: enumerate conjugacy classes of automorphisms \\psi satisfying \\psi^2 - 2\\psi + 5 = 0 and compare the companions of p and gp under the Z2-action. Remark 6.18 predicts exactly one of the two sets is nonempty for each p. If either is empty or both are nonempty, Theorem 6.21 fails for this example; the computation is finite linear algebra and would settle whether the unresolved sufficiency in Remark 6.14 actually breaks the classification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 6.21's classification depends on the claim, in Definition 6.19 and Remark 6.18, that every symmetric band w contributes exactly one indecomposable module M(w, m_{\\psi_p}, \\psi_p) for each orbit p \\in \\Pi_w, i.e. that |\\Psi_{w,p}| = 1. This is used in the proof of Theorem 6.21 to write every Z2-invariant even-parity band summand of U(M) as the image of a unique M3-object. But Remark 6.14, immediately after Proposition 6.12, states that determining when the image under U of a symmetric band module is indecomposable 'appears to be quite a hard problem in general', that a necessary condition is that \\hat{\\varphi} be similar to \\mu_{\\hat{w}}^2 \\lambda_2 \\lambda'_2 \\hat{\\varphi}^{-1}, and that 'it is not at all obvious if this condition is sufficient in general.' The proof of Proposition 6.12(e)\\Rightarrow(f) proves indecomposability only under the hypothesis that the relevant direct summand already exists; it does not construct \\psi for every p. Remark 6.14's subsequent 'Consequently' asserts existence of the needed symmetric band module, but this is precisely the point flagged as unresolved. Thus the completeness, and the no-repetition clause, of M3 are not established as written: if for some K and p both \\Psi^{(1)}_{w,p} and \\Psi^{(2)}_{w,p} were empty, an indecomposable module would be missing from Theorem 6.21.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces folded gentle algebras, a class of finite-dimensional algebras defined by gentle quiver data together with crease loops satisfying irreducible quadratic relations. The main results are: folded gentle algebras are biserial (Theorem 3.4); a classification of their indecomposable modules as string modules and as asymmetric and symmetric band modules (Theorem 6.21); a classification of Auslander-Reiten sequences (Theorem 6.26); and closure of the class under derived equivalence, with a finiteness statement for the number of derived-equivalent algebras (Theorem 7.7). The proofs are built on an unfolding functor U : mod A → mod  to a gentle algebra  with a Z2-action, and on folding arguments that reduce statements to known results for gentle algebras.","tokens_in":69426,"tokens_out":11722,"duration_ms":121780,"significance":"If the main results hold, the paper makes a substantial contribution: it identifies and analyzes a natural gentle-like class corresponding to Dynkin/Euclidean types C and ∼C, extending the analogy between gentle algebras and type A, and skewed-gentle algebras and type D. The unfolding method is well motivated, the explicit examples are useful, and the reduction to known gentle-algebra theorems (Butler–Ringel, Schröer–Zimmermann, Rickard) is a clear strategy. The paper does not rely on circular reasoning, and its central construction is original. However, the completeness of the module classification and the finiteness clause of the derived-equivalence theorem currently require additional support.","major_comments":[{"comment":"The assertion |Ψ_{w,p}| = 1 is load-bearing for the completeness of M3 in Theorem 6.21, but it is not proved. Remark 6.14 explicitly states that the condition for U(M(w,m,ψ)) to be indecomposable is not known to be sufficient in general, and the existence claim in that remark produces only a symmetric quasi-band module whose image contains a given band module as a direct summand. Proposition 6.12 is conditional: it constructs a symmetric band module only when a band module M(ŵ,m̂,φ̂) is already a direct summand of U(M), and it does not show that for every p ∈ Π_w there exists ψ with H_{w,deg p}ψ similar to φ_p or H_{w,2deg p}ψ similar to φ_p ⊕ φ_{gp}. Since the proof of Theorem 6.21 uses Remark 6.18 to lift every even-parity Z2-invariant summand of U(M) to a unique object of M3, the completeness and no-repetition clauses of Theorem 6.21 are not established as written.","section":"§6.4 (Definition 6.19 and Remark 6.18)"},{"comment":"The proof of the finiteness assertion relies on the statement that for any n there are only finitely many folded gentle algebras up to Morita equivalence with n vertices. This is false for a general field K admitting a degree-2 extension. For example, take K = Q and, for each squarefree integer a, the folded gentle triple with one crease vertex and crease relation η² − a = 0; the corresponding algebras are the fields Q(√a), which are pairwise non-isomorphic and hence not Morita equivalent, so there are infinitely many Morita classes with one vertex. The finiteness clause therefore needs either a restriction on K (for example K finite or a fixed quadratic extension) or a more careful counting argument.","section":"§7.3 (Theorem 7.7, final paragraph)"}],"minor_comments":[{"comment":"The statement 'Then U(M) ∼= U(M) only if M ∼= N' should read 'U(M) ∼= U(N) only if M ∼= N'.","section":"Lemma 6.15"},{"comment":"The line 'Ψ_{w,p} = Φ^{(1)}_{w,p} ∪ Φ^{(2)}_{w,p}' should use Ψ on the right-hand side, and the convention Φ(2m+1)_w = ∅ should be stated before Φ(deg p) is used for odd p.","section":"Definition 6.19"},{"comment":"The symbol Q is used both for the quiver and for the field of rational numbers; consider renaming one of them to avoid confusion.","section":"Example 2.8"},{"comment":"The phrase 'in terms symmetric and asymmetric string and band modules' is missing 'of' before 'symmetric'.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The finiteness issue in Theorem 7.7 is easy to repair by restricting the ground field or the quadratic extension, but the M3 existence gap is more important: the paper's own Remark 6.14 signals that the sufficiency of the indecomposability condition is open, so Remark 6.18 should not be cited as though it resolved that point. If the author supplies the missing existence argument for the symmetric band modules, the central classification would be solid and the paper would be a valuable contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere is my read on Duffield's folded gentle algebras paper.\n\nThe genuinely new thing is the class itself: gentle algebras with crease loops satisfying irreducible quadratic relations, obtained by folding ordinary gentle algebras. That is a natural and useful construction, and the unfolding functor gives a clean route to the main theorems. The biseriality result (Theorem 3.4) is solid, the string module correspondence (Proposition 6.3) is carefully worked out, and the Auslander–Reiten classification is a real contribution. If the module classification is right, this is a valuable paper.\n\nThe soft spots are real, though. The M3 part of Theorem 6.21 is not proven as written. Definition 6.19 and Remark 6.18 assert that for each orbit p in Π_w there is exactly one ψ, so |Ψ_{w,p}| = 1. But Remark 6.14, immediately before, says it is \"not at all obvious\" whether the necessary condition for the image of a symmetric band module to be indecomposable is sufficient; the proof of Proposition 6.12(e)⇒(f) only shows indecomposability assuming the relevant direct summand already exists. Nothing constructs the ψ needed for completeness. The stress-test note is right: if for some band and field both Ψ^{(1)} and Ψ^{(2)} are empty, the classification misses an indecomposable. That is a load-bearing gap, not a cosmetic one.\n\nThe finiteness claim in Theorem 7.7 is false over fields such as Q. There are infinitely many irreducible quadratics over Q, so infinitely many non-Morita-equivalent folded gentle algebras on a single crease vertex. The closure under derived equivalence may survive, but the \"finitely many\" statement needs to be dropped or repaired.\n\nMinor issue: Corollary 6.4 invokes \"special biserial\" for algebras that are not special biserial in the usual sense; the biseriality theorem should be the justification instead.\n\nOverall: the paper deserves a serious referee. The unfolding framework is sound, and most of the arguments are careful. The M3 gap needs fixing—either by proving existence or by restricting the classification—and the finiteness claim needs correction. I would send it to review, expecting heavy revision.","headline":"New class with a genuinely useful unfolding approach, but the symmetric band classification has a real existence gap and the finiteness claim in Theorem 7.7 is false over fields like Q.","tokens_in":69977,"tokens_out":2701,"would_cite":true,"duration_ms":28686,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","16G70","16E35","16G60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper defines folded gentle algebras—gentle-like algebras whose crease loops satisfy irreducible quadratic relations—and proves that their indecomposable modules are exactly the symmetric and asymmetric string and band modules, with…","keywords":["folded gentle algebras","string modules","band modules","derived equivalence","folding","clannish algebras","biserial algebras","Auslander-Reiten sequences"],"falsifier":"Take the quiver of Example 2.7 with one crease loop η and one ordinary arrow α, but over the field R replace the relation $η^{2}$ + 1 = 0 by the reducible relation $η^{2}$ − η = 0. Then the local algebra at the crease vertex becomes K × K rather than a field extension, the simple module there is 1-dimensional, and the unfolding functor no longer identifies mod A with the Z2-fixed part of a gentle module category; checking whether Theorem 6.21 still holds in this example would settle whether irreducibility of the crease quadratics is necessary.","tokens_in":68918,"feed_emoji":"🔁","tokens_out":5050,"duration_ms":50802,"temperature":0.7,"pith_summary":"This paper introduces a new class of finite-dimensional algebras, called folded gentle algebras, built from gentle algebras by a folding construction that adds loops satisfying irreducible quadratic equations at special 'crease' vertices. These algebras generalize the iterated tilted algebras of Dynkin types C and ~C, filling a gap left by gentle and skewed-gentle algebras for the non-simply-laced infinite families. The main theorem classifies all indecomposable modules: they are exactly the string modules together with symmetric and asymmetric band modules, with no repetitions. The paper also classifies the Auslander-Reiten sequences and proves the class is closed under derived equivalence, with only finitely many Morita classes in each derived equivalence class.","feed_headline":"Folding gentle algebras preserves their string-and-band module story","feed_subtitle":"A symmetry trick extends gentle-algebra structure to types C and ~C, with derived equivalence closure.","key_machinery":"The load-bearing construction is the unfolding procedure: to a folded gentle triple one associates a gentle pair by doubling ordinary vertices and arrows while leaving crease vertices fixed, on which Z2 acts by swapping the two copies. The crease loops at fixed vertices are governed by an irreducible quadratic $x^{2}$ − λ1 x − λ2, so the local endomorphism ring at a crease vertex is a degree-2 field extension rather than a split algebra; this irreducibility is what makes the folded algebra behave like a genuine quotient of a gentle algebra. The unfolding functor U from mod A to mod  is exact and faithful, sending string modules to sums of two string modules (or one, for symmetric strings) and band modules to corresponding pairs, with the correspondence controlled by the maps ρ and θ between folded and unfolded words. The derived-equivalence closure is obtained by applying the same folding argument to the known gentle-algebra result via the repetitive algebra and tilting complexes.","core_discovery":"The central claim is that folded gentle algebras, defined by the usual gentle axioms on ordinary arrows plus crease loops whose quadratic relations are irreducible over the ground field, have a module category completely described by combinatorial words. Theorem 6.21 states that every indecomposable module is, up to isomorphism, exactly one of a string module, an asymmetric band module, or a symmetric band module, with the three collections disjoint and exhaustive. The proof proceeds by unfolding: every folded gentle algebra arises as a Z2-quotient of a gentle algebra, and the unfolding functor matches each folded string or band to a pair of gentle strings or bands swapped by the folding action, with symmetric bands arising from Z2-invariant bands of even parity. A second theorem asserts that any algebra derived equivalent to a folded gentle algebra is again folded gentle, and that only finitely many Morita equivalence classes arise in each derived equivalence class.","pith_inferences":["The same folding construction may extend to the other non-simply-laced infinite families, B and ~B, by choosing crease loops with appropriate irreducible quadratics; the paper treats only C and ~C.","The irreducibility of the crease quadratics means the classification is sensitive to the arithmetic of the ground field; testing whether the symmetric band indexing changes when the field is changed (e.g., from R to Q) could reveal field-dependent phenomena not visible over algebraically closed fields.","Since folded gentle algebras are clannish algebras with irreducible clans, the results suggest that many techniques developed for gentle algebras, such as geometric models via surface triangulations, might have folded analogues where the Z2-symmetry becomes an orbifold structure.","The finiteness of derived equivalence classes mirrors the gentle case and suggests that folded gentle algebras form a natural testbed for derived-tame classification beyond the simply-laced setting."],"forward_implications":["The module category of a folded gentle algebra is completely described by strings and bands, with symmetric bands indexed by orbits of irreducible polynomials under a Z2-action.","The class of folded gentle algebras is closed under derived equivalence, and there are only finitely many algebras up to Morita equivalence in each derived equivalence class.","Auslander-Reiten sequences are classified combinatorially: string modules follow the hook/cohook rules, band modules sit in homogeneous tubes, and symmetric strings give valued arrows of type (1,2) and (2,1).","Folded gentle algebras are biserial over fields that need not be algebraically closed, extending the structural theory of gentle algebras to the C and ~C Dynkin/Euclidean families.","The unfolding functor provides a constructive bridge: every folded gentle module corresponds to a pair of gentle modules permuted by the Z2 folding action, so existing gentle-algebra computations can be lifted and then folded back."],"supporting_citations":[{"why":"Supplies the string and band module classification for tame biserial algebras that the folded classification extends.","marker":"[39]"},{"why":"Provides the clannish algebra framework, including symmetric and asymmetric string/band modules, of which folded gentle algebras are the irreducible case.","marker":"[12]"},{"why":"Proves that gentle algebras are closed under derived equivalence; the paper's Theorem 7.7 is obtained by applying a folding argument to this result.","marker":"[33]"},{"why":"Gives the quiver with relations of the repetitive algebra of a gentle algebra, adapted here to folded gentle algebras in Lemma 7.2.","marker":"[32]"},{"why":"Classifies Auslander-Reiten sequences for string algebras, providing the hook/cohook description used in Theorem 6.26.","marker":"[10]"},{"why":"Characterizes derived equivalence via tilting complexes, used in the proof of Theorem 7.7 to identify derived equivalent algebras with stable endomorphism algebras.","marker":"[29]"},{"why":"Describes representations of species, the context in which the type C and ~C algebras were previously studied and which folded gentle algebras generalize.","marker":"[16]"},{"why":"Introduces iterated tilted algebras of types Bn and Cn, the class that folded gentle algebras generalize.","marker":"[3]"}],"fun_headline_variants":["Fold to unlock: string and band modules for gentle-type algebras","Folded gentle algebras: symmetric and asymmetric bands decide","Derived equivalence keeps folded gentle algebras in the family","String and band classification for folded gentle algebras"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Every crease loop must satisfy an irreducible quadratic relation over the ground field; if any such quadratic splits, the folded gentle structure collapses into the different reducible clannish case and the string/band classification and the Z2-quotient description no longer apply.","fun_headline_variants_meta":{"raw":{"variants":["Fold to unlock: string and band modules for gentle-type algebras","Folded gentle algebras: symmetric and asymmetric bands decide","Derived equivalence keeps folded gentle algebras in the family","String and band classification for folded gentle algebras"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00024,"raw_usage":{"total_tokens":1459,"prompt_tokens":830,"completion_tokens":629,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":446,"completion_tokens_details":{"reasoning_tokens":566}},"tokens_in":446,"tokens_out":629,"duration_ms":6476,"temperature":1.0,"reasoning_tokens":566,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T18:29:33.262518+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the quiver of Example 2.7 with one crease loop η and one ordinary arrow α, but over the field R replace the relation $η^{2}$ + 1 = 0 by the reducible relation $η^{2}$ − η = 0. Then the local algebra at the crease vertex becomes K × K rather than a field extension, the simple module there is 1-dimensional, and the unfolding functor no longer identifies mod A with the Z2-fixed part of a gentle module category; checking whether Theorem 6.21 still holds in this example would settle whether irreducibility of the crease quadratics is necessary.","supporting_citations":[{"cited_title":"Wald and J","cited_arxiv_id":null,"evidence_quote":"Supplies the string and band module classification for tame biserial algebras that the folded classification extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the clannish algebra framework, including symmetric and asymmetric string/band modules, of which folded gentle algebras are the irreducible case."},{"cited_title":"Schr¨ oer and A","cited_arxiv_id":null,"evidence_quote":"Proves that gentle algebras are closed under derived equivalence; the paper's Theorem 7.7 is obtained by applying a folding argument to this result."},{"cited_title":"Schr¨ oer","cited_arxiv_id":null,"evidence_quote":"Gives the quiver with relations of the repetitive algebra of a gentle algebra, adapted here to folded gentle algebras in Lemma 7.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classifies Auslander-Reiten sequences for string algebras, providing the hook/cohook description used in Theorem 6.26."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Characterizes derived equivalence via tilting complexes, used in the proof of Theorem 7.7 to identify derived equivalent algebras with stable endomorphism algebras."},{"cited_title":"Dlab and C","cited_arxiv_id":null,"evidence_quote":"Describes representations of species, the context in which the type C and ~C algebras were previously studied and which folded gentle algebras generalize."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces iterated tilted algebras of types Bn and Cn, the class that folded gentle algebras generalize."}],"review_version":1}