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Tilting-completion for gentle algebras

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Pith's one-line read Every almost-tilting module over a gentle algebra can be completed to a tilting module, with at most 2n complements.

desk verdict A real advance for gentle algebras with a reusable surface-cutting method, but the proof of the main bound rests on a 'straightforward check' that a referee should demand to see in full. read the letter →

arxiv 2412.13971 v3 pith:3G6P2C5J submitted 2024-12-18 math.RT

Wen Chang This is my paper · ORCID
classification math.RT MSC 16D9016E3557M50
keywords gentlealgebrastiltingmodulesalmost-tiltingpartial-tiltingcomplementsmarkedsurfacessurfacecutsstring
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

The paper establishes that over any gentle algebra of rank $n$, every almost-tilting module—a self-orthogonal module of finite projective dimension with exactly $n-1$ indecomposable direct summands—can be completed to a tilting module, and that the number of non-isomorphic complements is at most $2n$. This gives a positive answer to the completion question $(C_{n-1})$ for gentle algebras and confirms a modified complement conjecture in this class. The proof is geometric: modules are viewed as zigzag arcs on a marked surface, and completing a module becomes finding an arc that finishes a dissection. By cutting the surface along the given arcs, the problem reduces to three small base surfaces, where the possible completing arcs can be counted. The paper also constructs, for every $n \geq 3$ and $1 \leq m \leq n-2$, a connected gentle algebra with a pre-tilting module of rank $m$ that is not partial-tilting, showing that completion can fail when more than one summand is missing.

What carries the argument

The proof works on the marked surface attached to a gentle algebra: indecomposable string modules are represented by zigzag arcs, extensions and projective dimensions are read from weighted intersections of arcs, and a pre-tilting module becomes a collection of non-intersecting zigzag arcs whose oriented intersections all have weight zero. Cutting the surface along such an arc produces a new marked surface and simple coordinate whose associated gentle algebra has rank $n+1$ but the same number of arrows. Cutting along all $n-1$ arcs of an almost-tilting module leaves a single exceptional subsurface of rank one, which Lemma 3.19 classifies as a disk, a once-punctured disk, or an annulus; counting the possible completing arcs on that small surface gives between $1$ and $n+1$ choices, and lifting these choices back through the cuts yields the bound $2n$ complements.

What would settle it

Find a gentle algebra of rank $n$ with an almost-tilting module having more than $2n$ complements, or exhibit a marked surface whose exceptional subsurface after cutting $n-1$ pre-tilting arcs is not one of the three shapes in Lemma 3.19. Either would break the chain from Lemma 3.19 to Theorem 3.20.

Watch

Extended reading notes

Core claim

The central claim, Theorem 3.20, is that an almost-tilting module over a gentle algebra is always partial-tilting: it has at least one complement, and the total number of complements is bounded by $2n$, where $n$ is the rank of the algebra, meaning the number of non-isomorphic indecomposable projective modules. The same statement verifies the modified complement conjecture for gentle algebras: complements are finite in number, and the bound $2n$ replaces the originally proposed $2n-1$ because a rank-two example already admits four complements. A complementary construction, Theorem 3.22, shows that for any $n \geq 3$ and $1 \leq m \leq n-2$, some connected gentle algebra of rank $n$ has a pre-tilting module of rank $m$ that cannot be completed; in particular, the positive result for almost-tilting modules is close to optimal.

Load-bearing premise

The load-bearing premise is Lemma 3.19's assertion that after cutting, the single exceptional subsurface is a disk, a once-punctured disk, or an annulus with between $1$ and $n+1$ completing arcs; this classification is recorded as a 'straightforward check' rather than a fully enumerated case analysis.

Editorial extensions

If this is right

  • The completion question $(C_{n-1})$ has a positive answer for every gentle algebra of rank $n$: every almost-tilting module is partial-tilting.
  • A maximal partial-tilting module over a gentle algebra has finitely many complements, at most $2n$, so the finiteness and bounded-complement conjectures hold in this class.
  • An orthogonal module of full rank $n$ over a gentle algebra is automatically tilting, so the tilting condition (T3) can be replaced by a rank count in this setting.
  • For ranks $m \leq n-2$, completion can fail, so the positive result for $n-1$ summands is the strongest possible statement that close to full rank.
  • The cutting construction gives a new gentle algebra of rank $n+1$ with the same number of arrows, providing a concrete reduction tool for further module-theoretic questions.

Reading between the lines

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

  • The paper leaves open whether the $2n$ bound is sharp: it constructs an annulus example attaining $2n-1$ complements and sketches a gluing route toward $2n$, but does not exhibit an algebra reaching the bound.
  • The same surface-cutting induction could be applied to silting theory in the derived category of a gentle algebra, where completion questions for pre-silting objects are known to behave differently; the cutting picture may identify exactly where module-level and derived-level completability diverge.
  • The results suggest an extremal dichotomy for gentle algebras: completion always works for full rank and rank $n-1$, while every rank $\leq n-2$ admits a failure. One could test whether a similar dichotomy holds for other tame algebras.
  • The proof reduces complement counting to the topology of a single exceptional subsurface, so a combinatorial model for complements of a maximal partial-tilting module might be extracted purely from the shape of that subsurface.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proves that every almost-tilting module over a gentle algebra of rank n is partial-tilting and has at most 2n complements, confirming a modified Happel conjecture for this class. The proof uses the surface model for module categories of gentle algebras, introduces a cutting construction for marked surfaces and simple coordinates, and reduces the problem to three base cases: disks, once-punctured disks, and annuli. The paper also constructs, for every n ≥ 3 and 1 ≤ m ≤ n−2, a connected gentle algebra with a pre-tilting module of rank m that is not partial-tilting, extending the Rickard–Schofield counterexample.

Significance. If the main theorem is correct, it affirmatively answers question (C_{n−1}) for all gentle algebras and supplies a finite, explicit bound 2n for the number of complements, matching the asymptotic bound suggested by Mantese's example. The surface-cutting reduction is a promising technique with potential applications beyond tilting completion. The paper is well-structured and carefully integrates existing geometric models from [BC21, OPS18, C23]. However, the quantitative bound 2n depends on a case count in Lemma 3.19 that is delegated to a 'straightforward check' with illustrative pictures rather than a complete proof, and this is the central load-bearing step of the paper.

major comments (2)
  1. [§3.3, Lemma 3.19] The bound 1 ≤ r ≤ n+1 in Lemma 3.19 is the exact input that Theorem 3.20 converts into the final bound m ≤ 2n. The proof of this lemma is not complete: the disk case is dismissed with 'The proof of the claim is a straightforward check' and four representative pictures in Figure 15, while the once-punctured disk and annulus are treated by 'a similar argument' with four representatives in Figure 16 plus one exception in Figure 17. The manuscript does not prove that these pictures exhaust all simple coordinates satisfying the bigon conditions, nor does it show that the dashed ◦-arcs are the only zigzag arcs with zero-weight intersections. A missed configuration with r > n+1 would leave the statement 'almost-tilting implies partial-tilting' intact but would invalidate the quantitative claim that gives the paper its main theorem. This is a missing verification, not a disagreement with the geometric framework; the proof should supply a complete, exhaustive case analysis or a verification procedure that covers all possible configurations.
  2. [§3.2, Proposition 3.15] The rank increase rank(Sγ, Mγ) = rank(S, M) + 1 is used in Theorem 3.20 to conclude that the induced algebra AΓ has rank 2n−1, and the equality |Q̂1| = |Q1| is used implicitly in the same reduction. The proof of Proposition 3.15 derives the rank formula from the assertions that the number of marked points increases by two and the Euler characteristic changes by one, with the marked-point count said to be 'proved case-by-case, seeing the pictures in Figure 21'. Since this numerical formula is load-bearing for the main theorem, the proof should include the explicit five-case verification (or a uniform argument) rather than relying solely on a figure. The statement also contains a typographical error: the equality should read |Q̂1| = |Q1|, not |Q1| = |Q1|.
minor comments (5)
  1. [§3.2, Proposition 3.15] The sentence 'the Euler character of S is the same as the Euler character of the topological quotient of S, that is, equals χ − 1' is confusing and appears to contain a typo; χ was defined as the Euler characteristic of the original surface S, so the statement should refer to the Euler characteristic of Sγ, not of S.
  2. [§3.4, Theorem 3.22] In the proof of the claim for the torus example, the phrase 'if the endpoints of γ are pi, 2 ≤ i ≤ p − 1' uses p without definition; it should presumably be 2 ≤ i ≤ n−1, with n the number of marked ◦-points on the boundary.
  3. [§3.3, Lemma 3.19] The three possibilities for the exceptional subsurface (disk, once-punctured disk, annulus) are said to be 'depicted in Figure 13', but the text also notes that the annulus is homotopic to a once-punctured disk when the boundary formed by ◦-arcs is viewed as a puncture; the criterion for distinguishing these cases in the subsequent count should be stated explicitly.
  4. [Appendix] The appendix explicitly constructs the algebra associated with the cutting surface only under the assumption that the cutting arc γ intersects each •-arc at most once; it would be helpful to state whether the cases appearing in Lemma 3.19 satisfy this assumption, or to what extent the general construction remains open.
  5. [References] The URL for [S23] ends in 'FD-Atlas.htmpl', which looks like a typo for 'FD-Atlas.html'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the 2n complement bound is produced by a surface-cutting induction and a local case count, not by assuming the conclusion.

full rationale

The central claim is not circular. Theorem 3.20 is proved by cutting the surface along the n-1 arcs of an almost-tilting module, isolating one exceptional component, and applying Lemma 3.19, which counts possible completing arcs in three local shapes. That count is a case analysis on disks, once-punctured disks, and annuli; it does not presuppose the existence of a completion or the bound 2n. The lifting correspondence (Corollary 3.13) and the geometric characterization of tilting modules (Proposition 3.2) are proved in the paper. The main external input is the author's own [C23] dictionary identifying weighted intersections with Ext degrees (Proposition 2.15), but this is a prior theorem with stated assumptions that do not include tilting completion, and it is anchored in independent models [BC21, OPS18]. Thus the self-citation is not a circular reduction. The proof of Lemma 3.19 does contain an abbreviated case check ('The proof of the claim is a straightforward check'), which is an omitted-detail risk that could affect the numerical bound if a configuration were missed, but that is a correctness concern, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities: the proof rests on the established surface model for gentle algebras and background results cited from prior work. The cutting surface construction is a defined tool, not a postulated entity with independent evidence requirements.

assumptions (4)
  • domain assumption The marked surface model of [C23], building on [BC21] and [OPS18], gives a bijection between zigzag circles and indecomposable string modules, and identifies Ext groups with weighted oriented intersections.
    The proof of Proposition 3.2 and all subsequent geometric arguments rest on this correspondence; it is cited rather than rederived in this paper.
  • domain assumption [APS23, Proposition 5.7]: a pre-silting complex over a gentle algebra is silting if and only if its rank equals the algebra's rank.
    Used in Proposition 3.2 to identify tilting dissections with tilting modules via the projective resolution.
  • standard math Rank-one marked surfaces are exactly a disk with two boundary circle points or a once-punctured disk with one boundary circle point (Lemma 2.4).
    Used to classify the exceptional sub-surface in Lemma 3.19; the annulus case is treated as homotopic to a once-punctured disk.
  • domain assumption Gentle algebras are Iwanaga-Gorenstein [GR02], so self-orthogonal modules over gentle algebras have finite projective dimension.
    Used before Corollary 3.3 to justify that self-orthogonal modules over gentle algebras have finite projective dimension, and hence are pre-tilting or tilting according to rank.

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Pith. "Pith review of Tilting-completion for gentle algebras." pith.science (2026). https://pith.science/paper/3G6P2C5J

@misc{pith2026241213971,
  author       = {Pith},
  title        = {Pith review of: Tilting-completion for gentle algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3G6P2C5J}},
  note         = {Machine review of arXiv:2412.13971}
}
abstract

It is demonstrated that any almost-tilting module over a gentle algebra is indeed partial-tilting, meaning it can be completed as a tilting module. Furthermore, such a module has at most $2n$ possible complements, thereby confirming a (modified) conjecture of Happel for the case of gentle algebras. Additionally, for any $n\geq 3$ and $1\leq m \leq n-2$, there always exists a (connected) gentle algebra with rank $n$ and a pre-tilting module of rank $m$ which is not partial-tilting. The tool we use is the surface model associated with the module category of a gentle algebra. The main technique is an induction process involving surface cuts, which is hoped to be beneficial for other applications as well.

Figures

Figures reproduced from arXiv: 2412.13971 by the authors.

Figure 1
Figure 1. The left picture is a marked surface with a simple coordinate ∆∗ , where the arcs in ∆∗ are the •-arcs. The right picture shows the gentle algebra associated with it, where the dotted lines represent the (quadratic) relations in the algebra. 2.5. Zigzag arcs and string modules. Now we recall the geometric model of the module category of a gentle algebra, see [BC21, C23] for details. Let’s start with con￾structing a … view at source ↗
Figure 2
Figure 2. Two types of polygons formed by arcs in a simple coordinate and boundary segments, where a zigzag arc α passes through the polygon along an oriented intersection ai+1. p ℓ ∗ t ℓ ∗ t+ω α β ℓ ∗ 1 ℓ ∗ n P p [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. For a zigzag arc α with endpoint p which intersects ℓ ∗ t , the weight wp(α) of α at p equals n − t. The weight ω(p) of an oriented intersection p from α to β is defined as wp(α) − wp(β), which equals ω. Definition 2.9. Let (S,M, ∆∗ ) be a marked surface with a simple coordinate. For a zigzag ◦-arc α on the surface, we call the string module Mα of A arising from ω(α) the string module of α. Proposition 2.10. Let (S,… view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: The top picture depicts a simple ◦-arc γ on (S,M) cuts an •-arc ℓ ∗ into several segments ςi , with endpoints ℘i−1 and ℘i , which natu￾rally induce segments ςi on the surface Sγ, whose endpoints are ℘ ′ i−1 /℘′′ i−1 and ℘ ′ i /℘′′ i , that is the red crossed points in …
Figure 5
Figure 5. Figure 5: An example of the cutting surface. •-point q ′/q′′ on γ ′/γ′′, and up to homotopy (concerning the arcs γ ′ , γ′′ ), we can always assume that these points are at εj ′s. See the polygon Pb γ in the right picture of [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: The leftmost is a polygon Pγ formed by •-arcs or segments in the set ∆∗ \L ∪ {ς1, · · · , ςt , ℓ∗ ∈ L}∪ {ε1, · · · , εr}, which contains no ◦- point. Then it must be of the form as depicted in the right two pictures, where in both cases the polygon can be contracted to…
Figure 7
Figure 7. Figure 7: The zigzag simple ◦-arc γ goes through a boundary polygon P of ∆∗ several times, which cuts some •-arcs in ∆∗ as segments, denoted by ςi ′ . At the same time, the •-arcs in ∆∗ cut γ as segments εj ′ . By replacing the •-arcs in P that intersects γ, we get a polygon Pγ,…
Figure 8
Figure 8. Figure 8: When cutting the surface along a simple zigzag ◦-arc γ with two endpoints p coincide, the polygon P of ∆∗ which contains p becomes several polygons. The left-above picture is when p is a puncture, where we get two polygons of ∆∗ γ each which contains the new ◦-point p …
Figure 9
Figure 9. Figure 9: Consider an oriented intersection a from an •-arc ℓ ∗ 1 to an •- arc ℓ ∗ 2 in a polygon of ∆∗ . The red-dashed arcs are the arcs in ∆∗ γ which are induced from ℓ ∗ 1 and ℓ ∗ 2 . Then a induces an oriented intersection ba in a polygon of ∆∗ γ based at the newly added •-…
Figure 10
Figure 10. Figure 10: An example to show the zigzag arcs γ1 and γ2 that form a bigon with new boundary segments γ ′ and γ ′′ respectively. This is a special case when the simple zigzag ◦-arc γ factors through the •-arcs in ∆∗ at most once. with γ. Then α and γ share common endpoints or com…
Figure 11
Figure 11. Figure 11: The algebras A and Aγ associated with the surface and the cutting surface appearing in Example 3.5, see the [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: The three kinds of surfaces (S,M, ∆∗ ) satisfies conditions in Lemma 3.19, where the arcs in ∆∗ can be partially determined by using the condition that Ln−1 i=1 Mγi is pre-tilting in mod-A, see the •-arcs in the pictures. Proposition 3.17. If the algebra AΓ is represe…
Figure 13
Figure 13. Figure 13: Three possibilities of the sub-surface formed by ◦-arcs γi and boundary segments, whose rank is one, that is, it only needs one more ◦-arc to cut it into polygons each which contains exactly one •-point. γ1 γ2 γn−1 ℓ ∗ 1,2 ℓ ∗ 2,3 qm qn qn+1 q1 pm pn+1 p1 p2 [PITH_FU…
Figure 14
Figure 14. Figure 14: We label the •-points qi , the ◦-points pi , the ◦-arc γi and the •-arcs ℓ ∗ ij for the case of a disk appearing in Lemma 3.19. Since each ◦-arc γi and the boundary segment γ ′ i form a bigon on (S,M, ∆∗ ) which has exactly one •-point, the shape of the surface (S,M) …
Figure 15
Figure 15. Figure 15: By adding the dashed •-arcs, we complete the •-arcs in the disk of [PITH_FULL_IMAGE:figures/full_fig_p023_15.png]
Figure 16
Figure 16. Figure 16: Complete the •-arcs in the once-punctured disk and the annulus of [PITH_FULL_IMAGE:figures/full_fig_p025_16.png]
Figure 17
Figure 17. Figure 17: A gentle algebra from an annulus with rank n, and an almost-tilting module which has n + 1 complements, where n = 8 in the picture. The almost-tilting module is given by the solid ◦-arcs, and its complements are given by the dashed arcs. γ [PITH_FULL_IMAGE:figures/fu…
Figure 18
Figure 18. Figure 18: An example of a gentle algebra with rank n, and an almost￾tilting on it which has 2n − 1 complements, where n = 2. The almost￾tilting module is given by the solid ◦-arc γ, and its complements are given by the dashed arcs. pre-tilting module over A(n), which we claim n…
Figure 19
Figure 19. Figure 19: The surface model and the associated gentle algebra such that the pre-tilting module Lm i=1 Mγi is not partial-tilting for any 1 ≤ m ≤ n−2, as a generalization of the example given in [RS89] when n = 3. Now we prove the claim. At first, if one endpoint of γ is p1, the…
Figure 20
Figure 20. Figure 20: A counterexample from an annulus such that a pre-tilting module is not partial-tilting. 4. Appendix Here we list all the possibilities of the cutting surfaces, depending on the endpoints of the arc γ, see [PITH_FULL_IMAGE:figures/full_fig_p028_20.png]
Figure 21
Figure 21. Figure 21: For a simple ◦-arc γ on (S,M), the surface Sγ is obtained by cutting S along γ. The set Mγ is defined as Mγ = M \ {p1, p2} ∪ {p ′ 1 , p′ 2 , p′′ 1 , p′′ 2 } ∪ {q ′ , q′′}, where some points may coincide. There are five cases, depending on the positions of the endpoint…
Figure 22
Figure 22. Figure 22: A marked surface (S,M, ∆∗ ) with a simple zigzag ◦-arc γ. {bb : x → xbi+2 | b : x → xi+1 with ui+1 = u ′ i+1b}∪ {bb : x → xbi+1 | b : x → xi+1 with ui+2 = u ′ i+2b}, where i = 2j − 1 for 1 ≤ j ≤ m/2. (3) the relation set Ib is a union of the following four sets: • Ib′…
Figure 23
Figure 23. Figure 23: The gentle algebra associated with the marked surface (S,M, ∆∗ ) given in [PITH_FULL_IMAGE:figures/full_fig_p031_23.png]
Figure 24
Figure 24. Figure 24: The cutting surface (Sγ,Mγ, ∆∗ γ ) obtained by cutting (S,M, ∆∗ ) along γ in [PITH_FULL_IMAGE:figures/full_fig_p032_24.png]
Figure 25
Figure 25. Figure 25: The gentle algebra associated with the cutting surface (Sγ,Mγ, ∆∗ γ ) given in [PITH_FULL_IMAGE:figures/full_fig_p033_25.png]

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Works this paper leans on

50 extracted references · 44 canonical work pages · cited by 1 Pith paper

  1. [1]

    Angeleri-Hügel and F

    L. Angeleri-Hügel and F. U. Coelho, Infinitely generated complements to partial-tilting modules. Math. Proc. Cambridge Philos. Soc. 132(2002), no.1, 89–96

    work page 2002

  2. [2]

    Assem and D

    I. Assem and D. Happel, Generalized tilted algebras of type A_ n . Comm. Algebra 9 (1981), no. 20, 2101-2125

    work page 1981

  3. [3]

    Auslander, M

    M. Auslander, M. Platzeck, and I. Reiten, Coxeter functors without diagrams. Trans. Amer. Math. Soc. 250 (1979), 1–46

    work page 1979

  4. [4]

    Amiot, P-G

    C. Amiot, P-G. Plamondon and S. Schroll, A complete derived invariant for gentle algebras via winding numbers and Arf invariants. Selecta Math. (N.S.) 29 (2023), no. 2, Paper No. 30, 36 pp

    work page 2023

  5. [5]

    Amiot, P-G

    C. Amiot, P-G. Plamondon and S. Schroll, A complete derived invariant for gentle algebras via winding numbers and Arf invariants. (2019) Preprint https://arxiv.org/abs/1904.02555 arXiv:1904.02555

    arXiv 2019

  6. [6]

    Assem and A

    I. Assem and A. Skowro\' n ski, Iterated tilted algebras of type affine A_n . Math. Z. 195 (1987), no. 2, 269-290

    work page 1987

  7. [7]

    Bongartz, Tilted algebras

    K. Bongartz, Tilted algebras. Lecture Notes in Math. , 903 Springer-Verlag, Berlin-New York, 1981, pp. 26-38

    work page 1981

  8. [8]

    Brenner and M

    S. Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gel’fand-Ponomarev reflection functors, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832, Springer, Berlin, 1980, pp. 103-169

    work page 1979

Show all 50 references
  1. [9]

    Baur and R

    K. Baur and R. Coelho Sim\ oes, A geometric model for the module category of a gentle algebra. Int. Math. Res. Not. IMRN (2021), no. 15, 11357-11392

    2021

  2. [10]

    I. N. Bernstein, I. M. Gelfand, and V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk 28 (1973), no. 2(170), 19–33

    1973

  3. [11]

    M. C. R. Butler and C. M. Ringel, Auslander-Reiten sequences with few middle terms and applications to string algebras. Comm. Algebra 15 (1987), no. 1-2, 145-179

    1987

  4. [12]

    A. B. Buan and O. Solberg, Relative cotilting theory and almost complete cotilting modules, Algebras and modules, II (Geiranger, 1996) , CMS Conf. Proc., vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 77-92

    1996

  5. [13]

    W. W. Crawley-Boevey, Maps between representations of zero-relation algebras. J. Algebra 126, no. 2 (1989): 259-63

    1989

  6. [14]

    Chang, Geometric models for the algebraic hearts in the derived category of a gentle algebra

    W. Chang, Geometric models for the algebraic hearts in the derived category of a gentle algebra. (2023) Preprint https://arxiv.org/abs/2305.03998 arXiv:2305.03998 , to appear Isr. J. Math

    2023 arXiv

  7. [15]

    Chang, Maximal rigid modules over a gentle algebra and applications to higher Auslander-Reiten theory

    W. Chang, Maximal rigid modules over a gentle algebra and applications to higher Auslander-Reiten theory. (2025) Preprint https://arxiv.org/abs/2503.06819 arXiv:2503.06819

    2025

  8. [16]

    Chang, F

    W. Chang, F. Haiden and S. Schroll, Braid group actions on branched coverings and full exceptional sequences. Adv. Math. 472 (2025), Paper No. 110284

    2025

  9. [17]

    Coelho, D

    F. Coelho, D. Happel and L. Unger, Complements to partial-tilting modules. J. Algebra , 170 (1994), no. 1, 184-205

    1994

  10. [18]

    Chang, H

    W. Chang, H. B. Jin, and S. Schroll, Recollements of derived categories of graded gentle algebras and surface cuts, 2022 Preprint arXiv:2206.11196

    2022 arXiv

  11. [19]

    Chang and S

    W. Chang and S. Schroll, A geometric realization of silting theory for gentle algebras. Math. Z. 303 (2023), no. 3, Paper No. 67, 37 pp

    2023

  12. [20]

    Chang and S

    W. Chang and S. Schroll, Exceptional sequences in the derived category of a gentle algebra. Selecta Math. (N.S.) 29 (2023), no. 3, Paper No. 33, 41 pp

    2023

  13. [21]

    Colpi and J

    R. Colpi and J. Trlifaj, Tilting modules and tilting torsion theories. J. Algebra 178 (1995), no.2, 614–634

    1995

  14. [22]

    Enomoto, Maximal self-orthogonal modules and a new generalization of tilting modules

    H. Enomoto, Maximal self-orthogonal modules and a new generalization of tilting modules. (2023) Preprint https://arxiv.org/ abs/2301.13498 arXiv:2301.13498

    2023 arXiv

  15. [23]

    Geiss and I

    C. Geiss and I. Reiten, Gentle algebras are Gorenstein. In Representations of algebras and related topics. Proceedings from the 10th international conference, ICRA X, Toronto, Canada, 2002. , American Mathematical Society (AMS), Fields Institute Communications 45, 129-133 (2005)

    2005

  16. [24]

    Happel, On the derived category of a finite-dimensional algebra, Comment

    D. Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339–389

    1987

  17. [25]

    Happel, partial-tilting modules and recollement

    D. Happel, partial-tilting modules and recollement. Contemp. Math. , 131, Part 2 American Mathematical Society, Providence, RI, 1992, 345-361

    1992

  18. [26]

    Happel, Self-orthogonal Modules

    D. Happel, Self-orthogonal Modules. Abelian Groups and Modules. Math. Appl. 343. Dordrecht: Kluwer Acad. Plub. (1995), pp. 257-276

    1995

  19. [27]

    Haiden, L

    F. Haiden, L. Katzarkov and M. Kontsevich, Flat surfaces and stability structures. Publ. Math. Inst. Hautes \'Etudes Sci. 126 (2017), 247-318

    2017

  20. [28]

    Happel and L

    D. Happel and L. Unger, Almost complete tilting modules. Proc. Amer. Math. Soc. , 107 (1989), no. 3, 603-610

    1989

  21. [29]

    Happel and L

    D. Happel and L. Unger, Complements and the generalized Nakayama conjecture, Algebras and modules, II (Geiranger, 1996) , CMS Conf. Proc., vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 293–310

    1996

  22. [30]

    Happel and C.M

    D. Happel and C.M. Ringel, Tilted algebras. Trans. Amer. Math. Soc. , 274(1982), no.2, 399-443

    1982

  23. [31]

    H. B. Jin, S. Schroll and Z. F. Wang, A complete derived invariant and silting theory for graded gentle algebras, 2023 Preprint arXiv:2303.17474

    2023 arXiv

  24. [32]

    Kopřiva and J

    J. Kopřiva and J. Šťovíček, Semiorthogonal decompositions for bounded derived categories of gentle algebras, 2022 Preprint arXiv:2209.14496

    2022 arXiv

  25. [33]

    Lekili and A

    Y. Lekili and A. Polishchuk, Derived equivalences of gentle algebras via Fukaya categories. Math. Ann. 376 (2020), no. 1-2, 187-225

    2020

  26. [34]

    Zhou, A negative answer to Complement Question for presilting complexes, 2023 Preprint arXiv:2302.12502

    Y-Z Liu and Y. Zhou, A negative answer to Complement Question for presilting complexes, 2023 Preprint arXiv:2302.12502

    2023 arXiv

  27. [35]

    Miyashita, Tilting modules of finite projective dimension, Math

    Y. Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 (1986), 113–146

    1986

  28. [36]

    Mantese, Complements to projective almost complete tilting modules, Comm

    F. Mantese, Complements to projective almost complete tilting modules, Comm. Algebra 33 (2005), no. 9, 2921-2940

    2005

  29. [37]

    Marczinzik, On self-orthogonal modules in Iwanaga-Gorenstein rings

    R. Marczinzik, On self-orthogonal modules in Iwanaga-Gorenstein rings. (2023) Preprint https://arxiv.org/ abs/2303.10433 arXiv:2303.10433

    2023 arXiv

  30. [38]

    Opper, P-G Plamondon S

    S. Opper, P-G Plamondon S. Schroll A geometric model for the derived category of gentle algebras 2018 Preprint arXiv:1801.09659

    2018 arXiv

  31. [39]

    Y. Palu, V. Pilaud and P-G Plamondon, Non-kissing and non-crossing complexes for locally gentle algebras. J. Comb. Algebra 3 (2019), no. 4, 401-438

    2019

  32. [40]

    Y. Palu, V. Pilaud and P-G Plamondon, Non-kissing complexes and tau-tilting for gentle algebras. Mem. Amer. Math. Soc. 274 (2021), no. 1343, vii+110 pp

    2021

  33. [41]

    Rickard, Morita theory for derived categories, J

    J. Rickard, Morita theory for derived categories, J. London Math. Soc. 39 (1989), 436–456

    1989

  34. [42]

    Rickard and A

    J. Rickard and A. Schofield, Cocovers and tilting modules. Math. Proc. Cambridge Philos. Soc. , 106 (1989), no. 1, 1-5

    1989

  35. [43]

    Riedtmann, and A

    C. Riedtmann, and A. Schofield, On open orbits and their complements. J. Algebra , 130 (1990), no. 2, 388-411

    1990

  36. [44]

    Riedtmann, and A

    C. Riedtmann, and A. Schofield, On a simplicial complex associated with tilting modules. Comment. Math. Helv. , 66 (1991), no. 1, 70-78

    1991

  37. [45]

    Schröer, Atlas of finite-dimensional algebras

    J. Schröer, Atlas of finite-dimensional algebras. http://www.math.uni-bonn.de/people/schroer/FD-Atlas.htmpl

  38. [46]

    Unger, On the combinatorics of the set of tilting modules, Contemp

    L. Unger, On the combinatorics of the set of tilting modules, Contemp. Math. , 673 American Mathematical Society, Providence, RI, 2016, 227-249

    2016

  39. [47]

    Unger, Schur modules over wild, finite-dimensional path algebras with three simple modules

    L. Unger, Schur modules over wild, finite-dimensional path algebras with three simple modules. J. Pure Appl. Algebra , 64 (1990), no. 2, 205–222

    1990

  40. [48]

    J. Q. Wei, Semi-tilting complexes. Israel J. Math. 194 (2013), no.2, 871-893

    2013

  41. [49]

    Wald and J

    B. Wald and J. Waschbsch, Tame biserial algebras. J. Algebra 95 (1985), no. 2, 480-500

    1985

  42. [50]

    X. J. Zhang Self-orthogonal tau-tilting modules and tilting modules 2019 Preprint arXiv:1911.07667

    2019 arXiv

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