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arxiv: 2606.21424 · v1 · pith:M6LHIAGZnew · submitted 2026-06-19 · 🧮 math.RT · math.CO

Resolving subcategories for gentle algebras III: Tilting modules for gentle tree algebras

Benjamin Dequ\^ene , Micha\"el Schoonheere This is my paper

Pith reviewed 2026-06-26 12:52 UTC · model grok-4.3

classification 🧮 math.RT math.CO
keywords gentle algebrasresolving subcategoriestilting modulesAuslander-Reiten correspondencesurface modelscombinatorial modelstree algebras
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The pith

Gentle tree algebras admit a combinatorial realization of the Auslander-Reiten correspondence between resolving subcategories and tilting modules.

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

The paper establishes an explicit combinatorial matching between resolving subcategories of the module category and tilting modules for gentle tree algebras. It relies on a modified surface model and poset techniques developed earlier in the series to turn the abstract Auslander-Reiten bijection into a directly computable correspondence. A sympathetic reader would care because the construction makes it possible to list and generate these subcategories from any collection of modules using only combinatorial data on the quiver and relations. The result extends the earlier classification work to the full correspondence with tilting modules.

Core claim

Via the modified surface model for gentle algebras with finite global dimension, the authors provide a combinatorial realization of the Auslander-Reiten one-to-one correspondence between resolving subcategories and tilting modules in the module category of a gentle tree algebra KQ/<R>.

What carries the argument

The modified surface model combined with combinatorial and poset techniques that identify all resolving subcategories and pair them with tilting modules.

Load-bearing premise

The modified surface model and combinatorial techniques from the prior papers in the series correctly classify every resolving subcategory of the module category for gentle tree algebras.

What would settle it

A resolving subcategory or tilting module for a gentle tree algebra that fails to match under the proposed combinatorial correspondence.

Figures

Figures reproduced from arXiv: 2606.21424 by Benjamin Dequ\^ene, Micha\"el Schoonheere.

Figure 1
Figure 1. Figure 1: Illustration of a crossing corresponding to a basis element of Hom(M(δ), M(η)). The shaded part is where all the segments of δ and all the ones of η, given by the cutting of Σ with Γ(Gammaa◦ ), are homotopic. Convention 2.6. Whenever we say that two arcs cross, they cross in their relative interior. Proposition 2.7 ([DS25a]). Let (Σ,M, ∆◦ ) be a dualizable ◦-dissected marked surface with M◦ ⊂ ∂Σ. Consider … view at source ↗
Figure 2
Figure 2. Figure 2: The two types of accordions κi representing the inde￾composable summands of Ker(f). Proposition 2.8 ([DS25a]). Let (Σ,M, ∆◦ ) be a dualizable ◦-dissected marked surface with M◦ ⊂ ∂Σ. Let δ, η ∈ A . We distinguish betweenn two types of extensions from M(δ) to M(η) which are: 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of an overlap extension. • Arrow extension : whenever we have a non-split short exact sequence M(η) E M(δ) , where E ∈ ind(Q, R), then we can construct γ(E) from δ and η as described in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of an arrow extension. The gray area corresponds to one cell of Γ(∆◦ ). 2.2. Neighboring projective accordions. We will use the geometric model to compute resolving subcategories and consider minimal projective resolutions. Let (Q, R) be a gentle quiver with finite global dimension. This is equivalent to M ⊂ ∂Σ. The •-dissection Prj(∆◦ ) given by the projective accordions is homeomorphic to ∆◦… view at source ↗
Figure 5
Figure 5. Figure 5: In this example, we have wte(δ, η) = 3. We can therefore characterize combinatorially pairs of accordions that are Ext￾orthogonal. Proposition 2.18 ([Cha25]). Let δ, η ∈ A . Then δ ⊥ η if and only if the following assertions hold: (i) δ and η do not cross; and, (ii) for any common endpoint e, we have wte(δ, η) = 0. We can now provide a characterization of rigid and tilting subsets of accordions. Theorem 2.… view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of scell(δ) = (ηL, CL) and tcell(δ) = (ηR, CR) given δ ∈ A . Lemma 3.10 ([DS25a]). Let (Σ,M, ∆◦ ) be a ◦-dissected marked disc with M ⊂ ∂Σ. Let δ ∈ A ′ . If δ ⊈ C for some C ∈ Γ(Prj(∆◦ )), then we set scell(δ) = (ηL, CL) and tcell(δ) = (ηR, CR); otherwise, we set C = CL = CR the cell containing δ. The curve ς ∈ R ′ (δ) must satisfy all of the following assertions: (i) if ς crosses ηL and wL(δ)… view at source ↗
Figure 7
Figure 7. Figure 7: Example of the determination of the upper source and target of a pair (δ, η) ∈ (A ′ ) 2 . • there exists ρ ∈ Prj(∆◦ ) such that ρ ⊥̸ δ and ρ ⊥̸ η; and, • δ is above η. In such a case, the •-arc ξ such that s(ξ) = s(δ,η) and t(ξ) = t(δ,η) is an accordion, and we call it the Co-Z-completion of (δ, η). Lemma 3.19. Let (δ, η) ∈ (A ′ ) 2 . Assume that (δ, η) admits a Co-Z-completion ξ. Then ξ ∈ A . Definition 3… view at source ↗
Figure 8
Figure 8. Figure 8: Construction of the Co-Z-completion ξ ∈ A of (δ, η). Proposition 3.21. Let (δ, η) ∈ (A ′ ) 2 admit a Co-Z-completion ξ. Then ξ ∈ R ′ (δ, η). 4. Combinatorial realization of the Auslander–Reiten correspondence For Λ an Artin algebra of finite global dimension, M. Auslander and I. Reiten [AR91] establish a one-to-one correspondence from cotilting objects (of finite pro￾jective dimension) to contravariantly f… view at source ↗
Figure 9
Figure 9. Figure 9: All possible distinct behavior we can have between ρ ∈ (v|w)Prj and the accordions non Ext-orthogonal accordions to ρ contained in ∂C, for C ∈ L (Db). satisfying (1a). Let δ ∈ A ′ be contained in ∂C, sharing with ρ a common endpoint u ∈ M•, and satisfying ρ ≺u δ. Denote by e ̸= u the other endpoint of ρ in ∂C. Fix an orientation of Σ around δ and so a coloration colδ of Nproj(δ)0 such that e ∈ Nproj(δ) □ 0… view at source ↗
Figure 10
Figure 10. Figure 10: Construction of ν (the densely-dotted purple line) in the case where ρ ∈ (v|w)Prj (the continuous red line) crosses a unique δ ∈ D (the loosely-dotted blue line) contained in ∂C (with C the blue colored area). By assumption, OvExt(δ, ρ) ̸= ∅, and by construction, ν ∈ OvExt(δ, ρ). There￾fore ν ∈ R(δ) ⊂ R(D). Assume that ν ∈ Prj(∆◦ ). By construction, ν ⊂ C, and therefore ν does not cross any arc within D. … view at source ↗
Figure 11
Figure 11. Figure 11: Two pictures illustrating the construction of ν (the densely-dotted purple line) in the case where ρ ∈ (v|w)Prj (the con￾tinuous red line) satisfies condition (2c) (on the left) and condition (2a) (on the right): in both cases, δ, η ∈ D (the loosely-dotted blue line) are the two accordions contained in ∂C (with C the blue colored area) such that δ ⊥̸ ρ and η ⊥̸ ρ. Note that, in the (2a) configuration, ν /… view at source ↗
Figure 12
Figure 12. Figure 12: A gentle tree (Q, R) with the Res-relation order on its non-projective indecomposable representations, and the dualizable •-dissected marked surface associated to (Q, R). 4.6. An example. Let us consider (Q, R) the gentle tree shown in [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The geometric tilting collections T1 = {η, ς, ρ1, ρ4}, and T2 = {δ, µ, ρ1, ρ3} associated to the geometric resolving sub￾category R(η, ς) = R(η)∪R(ς) and R(δ, µ) = R(δ)∪R(µ) calcu￾lated in [DS25b, Section 6.3]. Example 4.29. See [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The construction of the geometric tilting collection T = {η, ς, δ, ρ1} associated to the geometric resolving subcategory given by taking the set of all accordions A . Acknowledgements B.D. thanks the Institut des Sciences Mathematiques (UQAM) and the Engi￾neering and Physical Sciences Research Council (EP/W007509/1) for their partial funding support. The authors acknowledge the CHARMS program grant (ANR-1… view at source ↗
read the original abstract

This paper is the third part of a series that intends to study the resolving subcategories for gentle algebras over an algebraically closed field $\mathbb{K}$. As in the previous two papers, we continue to focus on gentle trees $(Q,R)$. Via a modified surface model for gentle algebras with finite global dimension, we developed combinatorial, poset, and quiver representation techniques that allow one to calculate all the resolving subcategories of $\mathbb{K}Q/\langle R \rangle$-mod. Furthermore, they enable one to calculate the resolving subcategory generated by any collection of $\mathbb{K}Q/\langle R \rangle$-modules. In this paper, based on those techniques, we give a combinatorial realization of the Auslander--Reiten one-to-one correspondence between resolving subcategories and tilting modules in $\mathbb{K}Q/\langle R \rangle$-mod.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper is the third in a series on resolving subcategories of module categories over gentle algebras. Focusing on gentle tree algebras (Q,R), it employs the modified surface model, poset techniques, and quiver representation methods developed in the prior two papers to furnish an explicit combinatorial realization of the Auslander-Reiten bijection between resolving subcategories and tilting modules over KQ/<R>.

Significance. If the prior combinatorial identification of all resolving subcategories is correct, the explicit realization supplies a concrete, computable correspondence that strengthens the utility of the surface model for gentle tree algebras. The work credits the series' combinatorial and poset tools for enabling the bijection and extends them to tilting modules without introducing new free parameters or ad-hoc axioms.

minor comments (2)
  1. The introduction should include a short recap (one paragraph) of the key combinatorial objects (e.g., the specific poset or surface arcs) that label the resolving subcategories, to make the tilting-module correspondence readable without immediate consultation of Papers I and II.
  2. Notation for the modified surface model is used throughout; a single consolidated table or diagram in §2 comparing the original and modified models would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for recognizing its significance in providing a concrete combinatorial realization of the Auslander-Reiten correspondence, and for the recommendation of minor revision. No specific major comments were listed in the report.

Circularity Check

1 steps flagged

Moderate circularity from reliance on prior self-authored papers in the series

specific steps
  1. self citation load bearing [Abstract]
    "Via a modified surface model for gentle algebras with finite global dimension, we developed combinatorial, poset, and quiver representation techniques that allow one to calculate all the resolving subcategories of ℤQ/⟨R⟩-mod. ... In this paper, based on those techniques, we give a combinatorial realization of the Auslander--Reiten one-to-one correspondence between resolving subcategories and tilting modules in ℤQ/⟨R⟩-mod."

    The combinatorial realization depends on the modified surface model and techniques developed in the authors' previous two papers to identify resolving subcategories; the correspondence is thus built upon self-cited foundational work without external verification mentioned.

full rationale

The paper is the third in a series and explicitly bases its main result on techniques from the prior papers by the same authors. While the realization of the Auslander-Reiten correspondence is a new combinatorial contribution, the validity hinges on the correctness of the prior identification of resolving subcategories, introducing moderate circularity burden as per the self-citation load-bearing pattern. No other patterns like self-definitional or fitted predictions are present. The central claim retains independent content beyond the citation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the work relies on standard background from representation theory of algebras and the authors' prior papers.

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