{"id":"21960f52-9d6c-42fa-a6d3-c9fb91fba387","arxiv_id":"2508.19763","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under a strong-source or strong-sink condition on maximal forbidden paths, gentle algebras satisfy hb.dim A ≤ 2·gl.dim A − 1 (or 2·f.dim A − 1), yielding a new quasi-tilted criterion when gl.dim A = 2.","lead":"This paper finds conditions on the quiver of a gentle algebra under which the sum of projective and injective dimensions of every indecomposable module is strictly less than twice the global dimension. It also gives a new combinatorial test for whether a gentle algebra of global dimension two is quasi-tilted.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.2 is the keystone and is not proved; Corollary 3.6 misstates its consequence, and Theorem 3.7 silently uses a corrected four-term maximum. The hb.dim bound is not securely established until this is resolved.","rationale":"The reader identified Proposition 3.2 as the weakest assumption and Corollary 3.6 as a misstatement. My stress-test confirms this: the paper's main theorems reduce to the string-module dimension formula, which is unproved; the corollary is false as stated; and the proof of Theorem 3.7 implicitly uses the corrected four-term version. I found no independent counterexample to the main theorem's conclusion, and the proof's case analysis suggests the bound may be salvageable, but the paper as written is not fully sound. The conditional verdict is therefore appropriate: the result should be accepted only after Proposition 3.2 is proved and Corollary 3.6 is corrected. No change to the reader's verdict is needed.","tokens_in":19590,"tokens_out":14467,"duration_ms":157821,"concrete_test":"Use QPA/GAP on the gentle algebras of Examples 5.1, 5.4(2), and a large random set of gentle algebras with finite global dimension: compute proj.dim and inj.dim of every string module directly, and compare them against (i) the displayed Corollary 3.6 formula and (ii) the corrected four-term maximum max_{X,Y∈{L,R}}(u_X+d_Y). A mismatch in (ii) would refute Theorem 3.7; a mismatch only in (i) would confirm that the corollary is false but the main bound may survive. Independently, complete the induction proof of Proposition 3.2 from Lemma 3.1, tracking all cases where one, both, or neither endpoint maximal forbidden path exists.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central bound of Theorem 3.7 (and its infinite-gl.dim analogue Theorem 3.10, and the quasi-tilted criterion in §4) rests on Proposition 3.2, which claims pd M(s)=max{dL,dR} and id M(s)=max{uL,uR} for string modules. This proposition is stated without proof: the text only says 'By repeatedly using Lemma 3.1, we have the following result.' No induction is supplied, and the hypotheses of the four inequalities (the 'vertex without relation' conditions) are never checked in the applications. More concretely, Corollary 3.6 states pd+id = max{u_X+d_X | X∈{L,R}}, but this is not what Proposition 3.2 implies: since pd=max(dL,dR) and id=max(uL,uR), the sum is max over all four pairings u_i+d_j. The proof of Theorem 3.7, case (b), actually uses the four-term maximum (terms i+l−j and j+ℓ−i), so the stated corollary is false as written and the main proof silently relies on an unstated correction. Because every subsequent bound, including the quasi-tilted characterization, inherits this formula, the central claim is not securely established: a missing proof or a hidden failure of Proposition 3.2 would invalidate Theorem 3.7 regardless of the local strong-source condition.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the homological bound hb.dim A = sup_{M in ind(modA)} (proj.dim M + inj.dim M) for gentle algebras. The main results are Theorem 3.7 and Theorem 3.10: under the combinatorial condition that the starting vertices (or ending vertices) of all maximal forbidden paths of length at least 2 are strong sources (resp. strong sinks), the bound hb.dim A <= 2 gl.dim A - 1 holds when gl.dim A is finite, and a finitistic-dimensional analogue holds when gl.dim A is infinite. For gl.dim A = 2 this gives a sufficient condition for quasi-tiltedness, and Corollary 4.6 claims a necessary and sufficient condition in terms of four string conditions (Qt1)-(Qt4). The proofs are based on formulas for projective and injective dimensions of string modules in Proposition 3.2 and on known results on global and finitistic dimensions of gentle algebras.","tokens_in":20032,"tokens_out":14630,"duration_ms":150630,"significance":"If fully established, the paper would provide a clean combinatorial criterion ensuring hb.dim A <= 2 gl.dim A - 1 and a quasi-tilted characterization for gentle algebras. The approach is not circular: it builds on established results (Theorems 3.4 and 3.8) and on the authors' own Proposition 3.2, with no parameter fitting. The examples are explicit and appear consistent. However, the keystone Proposition 3.2 is not proved, and Corollary 3.6 contains a false equality that is used in the main arguments. The proof of Theorem 3.7 silently uses the correct four-term maximum, so the main claim may be recoverable, but the manuscript in its present form does not establish it. The contribution is potentially valuable but needs substantial verification.","major_comments":[{"comment":"Proposition 3.2 is the keystone of the paper: Theorems 3.7, 3.10, Lemma 4.5 and Corollary 4.6 all use the equalities proj.dim M(s)=max{d_L,d_R} and inj.dim M(s)=max{u_L,u_R}. The proposition is introduced by 'By repeatedly using Lemma 3.1, we have the following result' with no proof and no induction. Moreover, each of the four assertions has a 'vertex without relation' hypothesis that is never verified in the applications. For instance, in Theorem 3.7 case (a), an arrow alpha inside a maximal forbidden path is assigned u_L=u_R=0 although the adjacent forbidden subpaths end/start at the endpoints of alpha; whether those vertices are 'without relation' on the required sequences is not checked. If Proposition 3.2 is not proved, or if its hypotheses fail in these cases, the main bounds are unsupported.","section":"§3.1, Proposition 3.2"},{"comment":"The displayed equality in Corollary 3.6, proj.dim M + inj.dim M = max{u_X+d_X | X in {L,R}}, is false as a consequence of Proposition 3.2. Since proj.dim M=max(d_L,d_R) and inj.dim M=max(u_L,u_R), the sum is max_{X,Y in {L,R}}(u_X+d_Y). The two-term maximum can be strictly smaller (e.g., u_L=1, u_R=100, d_L=100, d_R=1 gives 101 vs 200). The proof of Theorem 3.7 case (b) uses the four-term maximum (i+ell-i, i+l-j, j+ell-i, j+l-j), so it silently relies on a corrected statement that is never stated or proved. Lemma 3.9 repeats the same incorrect max, and Theorem 3.10 inherits the problem. This is a load-bearing error in the written arguments.","section":"§3.3, Corollary 3.6"},{"comment":"The quasi-tilted characterization rests on Proposition 3.2. Lemma 4.5 asserts an exhaustive classification of strings into the four shapes of Figure 4.1 and then concludes length bounds from Proposition 3.2; the step for shapes (3) and (4) -- that all four constituent forbidden paths have length 1 'by gl.dimA=2 and Proposition 3.2' -- is not demonstrated. Corollary 4.6's 'if' direction applies Proposition 3.2 to strings satisfying (Qt1)-(Qt4) without checking the vertex-without-relation hypotheses, and the proof contains a jump from 'inj.dimM(s) >= 2' to a contradiction that relies on these unverified formulas. Thus the necessary-and-sufficient claim is not established in the present form.","section":"§4.2, Lemma 4.5 and Corollary 4.6"}],"minor_comments":[{"comment":"Notation is inconsistent: (2.1) writes F1=f1...fdL but then says f1,...,fdR; (2.2) says 's(F1)=t(s)' and 'g1,...,gm', which should likely be s(F2)=t(s) and g1,...,gdR.","section":"§3.1, Proposition 3.2(2)"},{"comment":"The sentence 'Lemma 3.5 follows proj.dimB(n,lambda)+inj.dimB(n,lambda)=2' should cite Proposition 3.3; Lemma 3.5 is about string modules.","section":"§3.3, proof of Theorem 3.7"},{"comment":"The sentence 'The following result provides another case such that (3.1) holds' is dangling: no result follows before the beginning of Section 3.4.","section":"§3.3, after Theorem 3.7"},{"comment":"The final inequality is written 'proj.dimM + inj.dimM < 2*f.dimA - 1', but the theorem proves 'less than or equal to'.","section":"§4.1, Remark 4.4"},{"comment":"The condition 'proj.dimM+inj.dimM<infinity' is displayed as a subscript on the supremum in Question 1.1; it should be part of the set over which the supremum is taken. Also, the Introduction cites 'Corollaries 4.2 and 4.2' but the second should be 4.3.","section":"§1 and §4.1"},{"comment":"The displayed projective and injective resolutions are hard to parse because of the direct-sum notation, e.g., 'P(3)^{oplus 2} oplus P(4)' and '(3/4)^{oplus 2} oplus 4'. Please clarify the notation.","section":"§5.3, Example 5.4(2)"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper up front: it gives genuinely new combinatorial conditions, the strong-source/strong-sink hypotheses, under which homological bound hb.dim A ≤ 2·gl.dim A − 1, and it adds a new if-and-only-if criterion for quasi-tilted gentle algebras of global dimension 2. That is real content, and the quasi-tilted characterization complements rather than repeats the earlier criteria of Huard–Liu and Coelho–Tosar.\n\nThe trouble is that the whole tower rests on Proposition 3.2, which gives formulas for proj.dim and inj.dim of string modules in terms of four lengths of adjacent maximal forbidden paths. The proposition is asserted after \"By repeatedly using Lemma 3.1,\" but no induction is given, and the hypotheses about vertices being \"without relation\" are never checked in the applications. That alone would make me demand a full proof or a precise citation.\n\nWorse, Corollary 3.6 states proj.dim M + inj.dim M = max{u_X + d_X | X ∈ {L,R}}. That is not what Proposition 3.2 implies: the sum of two maxima is the maximum over all four pairings u_i + d_j. The proof of Theorem 3.7 actually uses the four-term maximum (terms like i + l − j and j + ℓ − i), so the corollary is wrong as written and the main argument silently relies on an unstated correction. Lemma 3.9 repeats the same wrong expression for the infinite-global-dimension case.\n\nThere are also small signs of carelessness in the proofs. Theorem 3.10 says \"1 ≤ l ≤ gl.dim A\" even though gl.dim A = ∞, and Theorem 3.7 case (a) sets u_L = u_R = 0 for a single arrow inside a maximal forbidden path without justification — those values look inconsistent with the definition of u_L as the length of a left maximal forbidden path ending at s(s). These may be fixable, but they add to the sense that Proposition 3.2 is being applied loosely.\n\nThe main theorems may well survive correction: the four-term max is what is actually used, and the strong-source/sink idea looks robust. But as written, the central bound is not securely established because its key input is unproved and its stated corollary is false. The paper is worth serious referee attention, not a desk reject, but it needs major revision: prove or properly source Proposition 3.2, correct Corollary 3.6 and Lemma 3.9, and go through the applications to verify the technical hypotheses.\n\nWho is this for? Specialists in gentle algebras who care about homological dimensions and quasi-tilted algebras. It is not a paper that will change the field, but the combinatorial conditions and the quasi-tilted criterion are useful. I would send it back for revision rather than accept it in its current form.","headline":"New enough combinatorial conditions for homological bounds on gentle algebras, but the keystone Proposition 3.2 is unproved and Corollary 3.6 is false as stated, so the main theorems need repair before the results can be trusted.","tokens_in":20436,"tokens_out":5734,"would_cite":false,"duration_ms":58942,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G10","16G20","16E05","16E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For gentle algebras, a local quiver condition pushes the projective-plus-injective bound below twice the global dimension.","keywords":["gentle algebras","homological dimensions","global dimension","finitistic dimension","quasi-tilted algebras","string modules","forbidden paths"],"falsifier":"Compute the projective and injective resolutions of a string module whose two endpoints sit inside one or two maximal forbidden paths of length 2 in a gentle algebra satisfying the strong-source condition; if any such module has proj.dim + inj.dim = 2·gl.dim, Theorem 3.7 fails. More directly, exhibit a string s for which Proposition 3.2's formula gives a value different from the actual resolution lengths; that single counterexample would invalidate the chain of inequalities.","tokens_in":19513,"feed_emoji":"📐","tokens_out":6406,"duration_ms":62145,"temperature":0.7,"pith_summary":"This paper studies the homological bound hb.dim A, the largest value of proj.dim M + inj.dim M among indecomposable modules of a gentle algebra A. It tries to show that, under a purely local condition on the bound quiver, this sum is at most 2·gl.dim A − 1 whenever the global dimension is finite, and at most 2·f.dim A − 1 when it is infinite. The condition asks that every maximal forbidden path of length at least two starts at a strong source or ends at a strong sink. A direct corollary is that gentle algebras of global dimension 2 satisfying this condition are quasi-tilted, and a necessary and sufficient quiver condition for quasi-tiltedness of gentle algebras of global dimension 2 is given in terms of four string forms (Qt1)–(Qt4). If right, the paper tightens the trivial bound by one and turns a homological property into a combinatorial check.","feed_headline":"Forbidden-path condition trims homological bound to 2·gl.dim − 1","feed_subtitle":"For gentle algebras, a local quiver condition cuts the naive bound by one and yields a quasi-tilted test.","key_machinery":"The machinery is the combinatorics of maximal forbidden paths anchored to endpoints of strings. A maximal forbidden path is a path of arrows with each consecutive pair lying in the ideal, not extendable at either end. A strong source or sink is a vertex incident to at most one arrow. For a string module M(s), the paper sets uL, uR, dL, dR to be the lengths of the left/right maximal forbidden paths adjacent to the two endpoints of the string and claims proj.dim M(s) = max(dL, dR) and inj.dim M(s) = max(uL, uR). The strong-source/sink condition ensures the four sums uX + dY stay one below 2·gl.dim, so the global bound follows by checking these local path lengths.","core_discovery":"The central claim is Theorem 3.7: for a non-hereditary gentle algebra A = kQ/I with finite global dimension, if all maximal forbidden paths of length ≥ 2 have either starting vertices that are strong sources or ending vertices that are strong sinks, then hb.dim A ≤ 2·gl.dim A − 1. Theorem 3.10 gives the analogous finitistic-dimensional bound when gl.dim A = ∞. The proof runs through string modules: Proposition 3.2 identifies proj.dim M(s) and inj.dim M(s) with the maxima of lengths of right/left maximal forbidden paths adjacent to the string endpoints; Lemma 3.5 shows these paths are unique in the finite-global-dimension setting; the strong-source/sink hypothesis then bounds the four sums uL","pith_inferences":["The equality stated in Corollary 3.6, as printed, appears to select max{uX + dX} over only two items; the intended bound is the maximum over the four cross-sums uL + dL, uL + dR, uR + dL, uR + dR, and the surrounding inequalities still work with that reading.","The same endpoint-length formulas could be tested on other string algebras, suggesting that a class larger than gentle algebras might satisfy an analogous 2·gl.dim − 1 bound whenever a uniqueness lemma like Lemma 3.5 holds.","The strong-source/sink condition can be read as requiring each long maximal forbidden path to have one free end; algebras where both ends are already occupied escape the bound, so a full characterization may depend on counting how many forbidden paths meet at a vertex.","Supplying a complete proof of Proposition 3.2 would turn the paper's conditional bound into explicit formulas for hb.dim on a wider family of gentle algebras and for related invariants such as the shod property."],"forward_implications":["For every gentle algebra in the class of Theorem 3.7, the sum proj.dim + inj.dim never reaches 2·gl.dim; the naive ceiling is improved by exactly one.","Gentle algebras with gl.dim = 2 and the strong-source/sink condition are quasi-tilted, so quasi-tiltedness can be certified by scanning the bound quiver.","In infinite global dimension, modules with finite projective and injective dimension obey a finitistic analogue 2·f.dim − 1, linking the homological bound to the finitistic dimension.","Corollary 4.6 gives a necessary and sufficient condition for quasi-tilted gentle algebras of global dimension 2 purely in terms of four string shapes.","The counterexamples in Section 5 show the strong-source/sink condition is sufficient, not necessary."],"supporting_citations":[{"why":"Supplies Theorem 3.4: global dimension equals the supremum of forbidden path lengths, used throughout the proofs of the bounds.","marker":"[LGH23]"},{"why":"Supplies Theorem 3.8: the finitistic dimension of a gentle algebra equals the supremum of maximal forbidden path lengths, underpinning Theorem 3.10.","marker":"[GR05]"},{"why":"Classifies indecomposable modules of gentle algebras as string and band modules, giving the objects whose dimensions are bounded.","marker":"[BR87]"},{"why":"Introduces the string and band classification that the module-theoretic analysis relies on.","marker":"[WW85]"},{"why":"Provides the result that quasi-simple band modules have projective dimension 1, used in Proposition 3.3.","marker":"[cPS20]"},{"why":"Gives an earlier quasi-tilted characterization via cohomological widths, providing the comparison point for Corollary 4.6.","marker":"[CT09]"},{"why":"Defines quasi-tilted algebras and states the homological condition used in Section 4.","marker":"[HRS94]"},{"why":"Introduces the forbidden thread terminology that the paper's maximal forbidden paths are built on.","marker":"[AAG08]"}],"fun_headline_variants":["Forbidden paths cut gentle algebra bound to 2·gl.dim−1","Strong sources/sinks shrink gentle algebra's homological bound","New condition yields quasi-tilted gentle algebras","Bound sharpened to 2·gl.dim−1 for gentle algebras","Gentle algebra bound tightened via forbidden paths"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is Proposition 3.2, stated without proof: for every string module M(s), proj.dim M(s) = max(dL, dR) and inj.dim M(s) = max(uL, uR), where the four numbers are lengths of maximal forbidden paths adjacent to the string's endpoints. If this formula is off for some string, every subsequent bound—including Theorems 3.7 and 3.10 and Corollary 4.6—loses its footing.","fun_headline_variants_meta":{"raw":{"variants":["Forbidden paths cut gentle algebra bound to 2·gl.dim−1","Strong sources/sinks shrink gentle algebra's homological bound","New condition yields quasi-tilted gentle algebras","Bound sharpened to 2·gl.dim−1 for gentle algebras","Gentle algebra bound tightened via forbidden paths"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000803,"raw_usage":{"total_tokens":3294,"prompt_tokens":600,"completion_tokens":2694,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":344,"completion_tokens_details":{"reasoning_tokens":2609}},"tokens_in":344,"tokens_out":2694,"duration_ms":20740,"temperature":1.0,"reasoning_tokens":2609,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T15:29:26.067850+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the projective and injective resolutions of a string module whose two endpoints sit inside one or two maximal forbidden paths of length 2 in a gentle algebra satisfying the strong-source condition; if any such module has proj.dim + inj.dim = 2·gl.dim, Theorem 3.7 fails. More directly, exhibit a string s for which Proposition 3.2's formula gives a value different from the actual resolution lengths; that single counterexample would invalidate the chain of inequalities.","supporting_citations":[],"review_version":1}