Pith. sign in

REVIEW 3 major objections 6 minor 17 references

Homological Bounds of Gentle algebras

T0 review · 3 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For gentle algebras, a local quiver condition pushes the projective-plus-injective bound below twice the global dimension.

desk verdict 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. read the letter →

arxiv 2508.19763 v1 pith:RJMSOWWQ submitted 2025-08-27 math.RT math.RA

Yu-Zhe Liu , Xin Ma , Jiacheng Xu , Chao Zhang This is my paper · ORCID
classification math.RTmath.RA MSC 16G1016G2016E0516E10
keywords gentlealgebrashomologicaldimensionsglobaldimensionfinitisticquasi-tiltedstringmodulesforbiddenpaths
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

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.

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 (3)
  1. [§3.1, Proposition 3.2] 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.
  2. [§3.3, Corollary 3.6] 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.
  3. [§4.2, Lemma 4.5 and Corollary 4.6] 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.
minor comments (6)
  1. [§3.1, Proposition 3.2(2)] 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.
  2. [§3.3, proof of Theorem 3.7] 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.
  3. [§3.3, after Theorem 3.7] 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.
  4. [§4.1, Remark 4.4] The final inequality is written 'proj.dimM + inj.dimM < 2*f.dimA - 1', but the theorem proves 'less than or equal to'.
  5. [§1 and §4.1] 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.
  6. [§5.3, Example 5.4(2)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the missing proof of Proposition 3.2 and the misstatement in Corollary 3.6 are correctness issues, not circularity.

full rationale

The derivation chain is not circular. The main bounds (Theorem 3.7 and Theorem 3.10) are assembled from Lemma 3.1, which is proved in the paper, Proposition 3.2, which is asserted as a repeated application of Lemma 3.1, Lemma 3.5, which is proved from Theorem 3.4, and external global/finitistic dimension results. Proposition 3.2 is not fitted to the target bound and is not defined in terms of hb.dim; it is a general formula for projective and injective dimensions of string modules. Its lack of proof is a serious correctness gap, and Corollary 3.6 misstates its consequence: from pd = max{dL,dR} and id = max{uL,uR} one gets max over four pairings, not max{uL+dL,uR+dR}. The proof of Theorem 3.7 silently uses the correct four-term maximum. These are mathematical omissions/errors, not circularity: no equation is assumed in order to prove itself, and no output quantity is used as an input. The only overlapping-author citation that is load-bearing, Theorem 3.4 from [LGH23], is independently supported: the introduction attributes the same global-dimension characterization to [cPS20, FOZ24, Cha25] as well. Other cited results, such as [GR05], are external. Thus no 'prediction' reduces to a fitted parameter or to a self-citation chain; the paper's central claim, if correct, rests on an independent combinatorial derivation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper rests on standard representation theory of gentle algebras plus its own Proposition 3.2, which is asserted rather than proved. There are no fitted parameters or invented entities.

assumptions (5)
  • domain assumption The classification of indecomposable modules over gentle algebras as string and band modules (Theorem 2.3, from [WW85, BR87]).
    Invoked throughout, e.g., in Propositions 3.2 and 3.3, to reduce computations to strings and bands.
  • domain assumption Theorem 3.4: global dimension of a gentle algebra equals the supremum of lengths of all forbidden paths ([LGH23, Thms 5.10 and 6.9]).
    Used to relate gl.dim to forbidden path lengths in Lemma 3.5 and Theorem 3.7.
  • domain assumption Theorem 3.8: the finitistic dimension equals the supremum of lengths of all maximal forbidden paths ([GR05]).
    Used in the proof of Lemma 3.9 and Theorem 3.10.
  • ad hoc to paper Proposition 3.2: for a string module M(s), inj.dim M(s) = max{uL,uR} and proj.dim M(s) = max{dL,dR}, where uL,uR,dL,dR are lengths of maximal forbidden paths adjacent to the string endpoints.
    Stated without proof in Section 3.1 ('By repeatedly using Lemma 3.1, we have the following result'). This is the central computational tool for all subsequent bounds.
  • domain assumption Proposition 3.3: every band module over a gentle algebra has projective and injective dimension 1.
    Proved in the paper via a sketchy induction and the existence of Auslander-Reiten sequences for band modules; also known for quasi-simple band modules from [cPS20, Corollary 2.13].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Homological Bounds of Gentle algebras." pith.science (2026). https://pith.science/paper/RJMSOWWQ

@misc{pith2026250819763,
  author       = {Pith},
  title        = {Pith review of: Homological Bounds of Gentle algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RJMSOWWQ}},
  note         = {Machine review of arXiv:2508.19763}
}
read the original abstract

This paper studies the homological bounds of gentle algebras, i.e., the upper bounds for the sum of the projective and injective dimensions of indecomposable modules over gentle algebras. We provide conditions under which this sum is strictly less than twice the global dimension, and as an application, we give a characterization of quasi-tilted gentle algebras.

Figures

Figures reproduced from arXiv: 2508.19763 by the authors.

Figure 3.1
Figure 3.1. The projective cover of M(s) in the case for s = sI and its kernel Case 2 if there exists an arrow β with s(β) = su, then Ω1(M(sI)) has a direct summand R which is a direct string module with topR = S(t(β)). Thus, we have Ω1(M(sI)) = L ⊕ ÅMu−1 i=1 P(i) ã ⊕ R as required, where L and R may be zero, and if u = 1 then P is zero. A vertex v in gentle pair (Q, I) is called a vertex with relation(s) if there are two arrow… view at source ↗
Figure 3.2
Figure 3.2. The projective resolution of a band module B(1, λ) Next, we show the following key lemma by using Theorem 3.4. Lemma 3.5. Let M ∈ Im(M|Str(A) ) be a string module over gentle algebra A, and sM = M−1 (M) = s1 · · · st (s1, . . . , st ∈ Q1 ∪ Q−1 1 ) be the string corresponded by it. If gl.dimA < ∞, then the following statements hold. (1) There is at most one left maximal forbidden path F = f1 · · · fuL (resp., F ′ = f… view at source ↗
Figure 4.1
Figure 4.1. In (1) (resp., (2)), F1 and F2 are either maximal forbidden paths of length > 0 or forbidden paths of length 0, and v and w are vertices without relation on F −1 1 sF2 (resp., F1sF −1 2 ); in (3) and (4), F ′ 1 , F ′ 2 , F ′′ 1 and F ′′ 2 are forbidden paths of length ⩾ 1 such that F ′ 1F ′′ 1 and F ′ 2F ′′ 2 are maximal forbidden paths (2) shown in [PITH_FULL_IMAGE:figures/full_fig_p018_4_1.png] view at source ↗
Figures from the paper (1 more)
Figure 4.2
Figure 4.2. Figure 4.2: a −1 1 α ∈ I, where s = a1 · · · am Therefore, we have proj.dimM(s) + inj.dimM(s) ⩽ 3 for any string module M(s). It follows that A is quasi-tilted if gl.dimA = 2. Moreover, Lemma 4.5 implies that the inverse of the above result holds; therefore, we complete this pro…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 8 canonical work pages

  1. [5]

    [CE56] Henry Cartan and Samuel Eilenberg

    DOI:10.1080/00927878708823416. [CE56] Henry Cartan and Samuel Eilenberg. Homological algebra. Princeton University Press,

    doi:10.1080/00927878708823416

  2. [1979]

    [WW85] Burkhard Wald and Josef Waschb¨ usch

    DOI:10.1007/b98977. [WW85] Burkhard Wald and Josef Waschb¨ usch. Tame biserial algebras. J. Algebra, 1(95):480– 500,

    doi:10.1007/b98977

  3. [1985]

    [ZH16] Chao Zhang and Yang Han

    DOI:10.1016/0021-8693(85)90119-X. [ZH16] Chao Zhang and Yang Han. Brauer-Thrall type theorems for derived module categories. Algebr. Represent. Theor., 19(6):1369–1386,

    doi:10.1016/0021-8693(85)90119-x

  4. [1987]

    [ASS06] Ibrahim Assem, Daniel Simson, and Andrzej Skowro´ nski

    DOI:10.1007/bf01166463. [ASS06] Ibrahim Assem, Daniel Simson, and Andrzej Skowro´ nski. Elements of the Representa- tion Theory of Associative Algebras, Volume 1 Techniques of Representation Theory . Cambridge University Press, Cambridge,

    work page doi:10.1007/bf01166463

  5. [1994]

    DOI:10.1007/978-94-017-1556-0

    work page doi:10.1007/978-94-017-1556-0

  6. [2000]

    [HRS94] Dieter Happel, Idun Reiten, and Sverre O

    DOI:10.1016/S0022-4049(99)00101-2. [HRS94] Dieter Happel, Idun Reiten, and Sverre O. Smalø. Quasitilted Algebras, pages 163–181. Springer Netherlands, Dordrecht,

    work page doi:10.1016/s0022-4049(99)00101-2

  7. [2005]

    [HL00] Fran¸ cois Huard and Shiping Liu

    DOI:10.1090/fic/045. [HL00] Fran¸ cois Huard and Shiping Liu. Tilted string algebras. Journal of Pure and Applied Algebra, 153(2):151–164,

    work page doi:10.1090/fic/045

  8. [2006]

    [BCS21] Karin Baur and Raquel Coelho-Sim˜ oes

    DOI:10.1017/CBO9780511614309. [BCS21] Karin Baur and Raquel Coelho-Sim˜ oes. A geometric model for the module cate- gory of a gentle algebra. Int. Math. Res. Notices , 2021(15):11357–11392,

    doi:10.1017/cbo9780511614309 2021

Show all 17 references
  1. [2008]

    [AS87] Ibrahim Assem and Andrzej Skowro´ nski

    DOI:10.1016/j.jpaa.2007.05.014. [AS87] Ibrahim Assem and Andrzej Skowro´ nski. Iterated tilted algebras of type eAn. Math. Z., 195(2):269–290,

    2007 doi

  2. [2009]

    0 25 2025-08-24 [FOZ24] S

    10.1007/s10468-008-9105-6. 0 25 2025-08-24 [FOZ24] S. Ford, A. Oswald, and J. J. Zhang. Homological conditions on locally gengle algebras. arXiv:2409.08333,

    2025 arXiv

  3. [2016]

    [Zha19] Chao Zhang

    DOI:10.1007/s10468-016-9622-7. [Zha19] Chao Zhang. Indecomposables with smaller cohomological length in the derived cate- gory of gentle algebras. Sci. China Math. , 62:891–900,

    doi

  4. [2019]

    [ZL24] Houjun Zhang and Yu-Zhe Liu

    DOI:10.1007/s11425-017- 9270-x. [ZL24] Houjun Zhang and Yu-Zhe Liu. There are no strictly shod algebras in hereditary gentle algebras. Comm. Algebra, 0(0):1–14,

    doi

  5. [2020]

    [CT09] Fl´ avio U

    DOI:10.4153/S0008414X2000005X. [CT09] Fl´ avio U. Coelho and Cecilia Tosar. On the derived categories and quasitilted algebras. Algebr. Represent. Theor., 12(1):77–92,

    doi

  6. [2021]

    [BR87] Michael C.R

    DOI: 10.1093/imrn/rnz150 (Published:31 July 2019). [BR87] Michael C.R. Butler and Claus Michael Ringel. Auslander-Reiten sequences with few middle terms and applications to string algebras. Commun. Algebra, 15(1-2):145–179,

    2019 doi

  7. [2023]

    [OPS18] Sebastian Opper, Pierre-Guy Plamondon, and Sibylle Schroll

    DOI:10.1007/s11425-022-2120-8. [OPS18] Sebastian Opper, Pierre-Guy Plamondon, and Sibylle Schroll. A geometric model for the derived category of gentle algebras. arXiv:1801.09659,

    arXiv

  8. [2024]

    DOI:10.1080/00927872.2024.2400214. 0 26

    2024

  9. [2025]

    [cPS20] ˙Ilke C ¸ anak¸ cı, David Pauksztello, and Sibylle Schroll

    DOI:10.1007/s11856-025-2801-7. [cPS20] ˙Ilke C ¸ anak¸ cı, David Pauksztello, and Sibylle Schroll. On Extensions for Gentle Algebras. Canad. J. Math. , 73(1):1–44,

    doi

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.