Mutating ordered $\tau$-rigid modules with applications to Nakayama algebras

Aslak B. Buan; H{\aa}vard U. Terland; Maximilian Kaipel

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Mutating ordered $\tau$-rigid modules with applications to Nakayama algebras

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Pith's one-line read For Nakayama algebras, mutation connects every complete τ-exceptional sequence to every other.

desk verdict A genuinely new transitivity theorem for Nakayama algebras with a mostly sound proof, but Lemma 5.1 has a gap that needs a real fix before the main theorem is fully established. read the letter →

arxiv 2501.13694 v1 pith:AQSLY5LR submitted 2025-01-23 math.RT

Aslak B. Buan , Maximilian Kaipel , H{aa}vard U. Terland This is my paper · ORCID

classification math.RT MSC 16G2016G7018E40

keywords τ-tiltingtheoryτ-exceptionalsequencesTF-orderedτ-rigidmodulesNakayamaalgebrasuniserialmutationtransitivityBongartzcompletionτ-perpendicularcategories

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 proves that over any Nakayama algebra, mutation is transitive: any complete τ-exceptional sequence can be turned into any other by a finite chain of left mutations. The proof works by translating mutation into an operation on TF-ordered τ-rigid modules, which are ordered collections of τ-rigid modules that encode such sequences, and then giving an explicit four-case formula for the operation. A key step is showing that the relevant τ-perpendicular subcategories are again Nakayama algebras, so the two-term mutation rules can be applied inside reductions. If the result stands, it extends the previously known rank-two case to all uniserial algebras and provides a concrete combinatorial algorithm for moving between complete sequences.

What carries the argument

The central objects are TF-ordered τ-rigid modules, ordered τ-rigid modules M1 ⊕ ... ⊕ Mt with Mi not generated by the later summands, which are in bijection with τ-exceptional sequences. The argument is carried by the V-map and E-map associated to a τ-rigid pair: the V-map bijects summands of the co-Bongartz completion to summands of the Bongartz completion, and the E-map reduces τ-rigid pairs to relative τ-rigid pairs in the τ-perpendicular subcategory J(M,P). For Nakayama algebras the paper computes these maps explicitly using the maximal and minimal completions described in Propositions 4.9 and 4.11, then proves the four-case mutation formula of Theorem 1.3. The reduction step is justified by Lemma 4.7, which says J(M,P) is again Morita equivalent to a Nakayama algebra, allowing longer mutations to be handled inside smaller categories of the same type.

What would settle it

Build the directed graph whose vertices are the TF-ordered τ-tilting modules of the radical-cube-zero Nakayama algebra with cyclic quiver on four vertices and whose edges are the four-case left mutations of Theorem 1.3; if any two vertices in the same connected component of the τ-tilting exchange graph lie in different components of this directed graph, transitivity fails.

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Extended reading notes

Core claim

The central claim is Theorem 5.15: for a Nakayama algebra, mutation of TF-ordered τ-tilting modules, and therefore of complete τ-exceptional sequences, is transitive. The paper establishes this by proving Theorem 1.3, which describes left mutation of a TF-ordered module B ⊕ C in four exhaustive and mutually exclusive cases: if C is projective with no homomorphisms to B, the pair simply swaps; if C is projective and Hom(C,B) ≠ 0, the second entry becomes the torsion-free functor f_C(B); if C is generated by B, the first entry becomes a radical of B (or its projective cover when B is projective); and in the remaining irregular case, the second entry becomes the quotient B/C. Transitivity follows by combining a lemma that any two TF-orders of the same τ-tilting module are connected by mutations with a proposition that lifts each ordinary τ-tilting mutation to a sequence of TF-ordered mutations; connectivity of the τ-tilting exchange graph for τ-tilting finite algebras then connects everything.

Load-bearing premise

The proof assumes that every τ-perpendicular subcategory J(M,P) of a Nakayama algebra is again Morita equivalent to a Nakayama algebra, so that the same two-term mutation formulas apply inside each reduction step.

Editorial extensions

If this is right

Any two TF-orders of the same τ-tilting module over a Nakayama algebra are connected by a sequence of left mutations (Lemma 5.2).
Each ordinary τ-tilting mutation between τ-tilting modules can be lifted to a sequence of mutations of TF-ordered modules, so the graph of TF-orders lies over the connected τ-tilting exchange graph (Proposition 5.14).
Combining these, the mutation graph of complete τ-exceptional sequences over a Nakayama algebra is connected (Theorem 5.15).
The four-case formula gives an explicit algorithm for left mutation in terms of radicals, projective covers, and torsion-free functors, so the transitivity proof is constructive.
Transitivity holds even though the mutation does not satisfy braid relations in this setting, as shown by Example 6.9.

Reading between the lines

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

Editorial inference: the only Nakayama-specific input in the transitivity argument is Lemma 4.7, so an analogous statement may hold for any class of τ-tilting finite algebras whose τ-perpendicular reductions stay inside the class.
Editorial inference: the explicit V-map computations could yield a direct count of φ-orbits and mutation graph components in the geometric disk model for Nakayama algebras, refining the comparison with rank-two wide subcategories that Example 6.8 shows is nontrivial.
Editorial inference: one could test whether the four-case formulas of Theorem 1.3 extend to all ordered τ-rigid pairs, not just TF-ordered ones, by inserting the E-map corrections that the paper uses for longer sequences.
Editorial inference: the failure of braid relations combined with transitivity suggests that the mutation graph is a natural combinatorial invariant for Nakayama algebras, and comparing its diameter across the two quiver shapes (linear and cyclic) would be a concrete next question.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First, what you should know: this paper gives explicit combinatorial formulas for mutating TF-ordered τ-rigid modules over Nakayama algebras (Theorem 1.3) and uses them to prove that mutation of complete τ-exceptional sequences is transitive (Theorem 5.15). That is a real, new result; previous work only covered rank-2 algebras. The general framework in Section 3, translating mutation into V- and E-maps, is genuinely useful and looks correct.

The proofs are mostly careful. Section 4's explicit Bongartz and co-Bongartz completions for Nakayama algebras are a nice contribution, and the case analysis for mutation, including the irregular case, is concrete and checkable. The V-map computations in Corollary 4.16 are elegant. The overall strategy for transitivity—connect TF-orders of the same module, then connect neighboring τ-tilting modules via a single TF-mutation—is sound in principle.

Now the soft spots. Lemma 5.1 is load-bearing and under-proved. The proof says: if φ(C⊕B) = B⊕C, then there must be an i with φ^i(B⊕C) = C⊕B because Λ is τ-tilting finite. Finiteness alone does not rule out tails in a non-injective map. You need either to prove φ is injective (hence a permutation) on the finite set of TF-ordered pairs, or to exhibit the sequence explicitly. The later case analysis does this for the projective and irregular cases, but the generic non-projective regular case is left to that finiteness assertion. This is not a fatal flaw—I expect it can be repaired—but it is a genuine gap in the proof of Theorem 5.15.

Two smaller things. Lemma 4.7 (τ-perpendicular categories are again Nakayama) is a one-sentence uniseriality argument; it is plausible but deserves a real proof, since the induction in Proposition 5.14 depends on it. And the homological arguments in Lemmas 5.9, 5.12, and 5.13 are dense enough that a referee should ask for expansion; I did not find a contradiction, but they are not easy to verify.

Bottom line: the main theorem is likely true, the paper is a solid contribution to τ-tilting theory, and the explicit formulas will be cited by people working on mutation and exceptional sequences. But it is not ready as is. A serious referee should be sent it, with instructions to require a fix for Lemma 5.1 and more detail on Lemma 4.7. My own verdict: conditional accept, if those are addressed.

Referee Report

2 major / 4 minor

Summary. The paper studies the mutation of τ-exceptional sequences introduced in [BHM24] through the equivalent language of TF-ordered τ-rigid modules. The authors translate the general mutation rules into operations on TF-ordered modules using the V-map and E-map, then specialize to Nakayama algebras, where they obtain explicit formulas for left mutation in four mutually exclusive cases (TF-1a, TF-1b, TF-2a, TF-2b, TF-3, TF-4). The main application is Theorem 5.15, which states that mutation of TF-ordered τ-tilting modules, and hence of complete τ-exceptional sequences, is transitive for Nakayama algebras. The proof strategy is to show (Lemma 5.2) that any two TF-orders of the same τ-tilting module are connected by mutations, and (Proposition 5.14) that a single τ-tilting mutation lifts to a mutation between suitable TF-orders; connectivity of the τ-tilting mutation graph then gives transitivity.

Significance. If the main theorem is correct, it provides the first known non-hereditary class of algebras of arbitrary rank for which mutation of complete τ-exceptional sequences is transitive, complementing the rank-2 result of [BHM24]. The paper is also valuable for its explicit combinatorial description of mutation for Nakayama algebras, including the irregular case, and for its use of Adachi's disk model to visualize the formulas. The authors build on published results rather than introducing ad-hoc assumptions, and the paper contains many worked examples that illustrate all six mutation cases. The central derivation is detailed and the overall strategy is coherent; however, one load-bearing proof step in Lemma 5.1 is under-justified, and one structural lemma (Lemma 4.7) is stated with a proof that is too terse for the role it plays in the induction.

major comments (2)

[§5, Lemma 5.1] The proof of the first case relies on the assertion: 'If the second equality is true then there must be an i such that φ^i(B⊕C)=C⊕B as wanted, since Λ is τ-tilting finite.' This is not justified by finiteness alone. From φ(C⊕B)=B⊕C, one cannot conclude that the forward orbit of B⊕C under φ ever reaches C⊕B unless φ is known to be injective (or bijective) on a finite invariant set containing both elements. The paper does not cite or prove such injectivity at this point. Since Lemma 5.2 and hence Theorem 5.15 depend on this step, the argument must be repaired. One possible repair is to cite the invertibility of the mutation operation from [BHM24] and explain that φ is a bijection on the relevant finite set. Alternatively, the step can be avoided entirely: because C⊕B is TF-ordered, C∉GenB by Definition 2.13, so in the stated left-regular case Theorem 3.26 already yields φ(B⊕C)=C⊕B directly, with the projective subcase handled by Lemma 3.4(c).
[§4, Lemma 4.7] Lemma 4.7 is load-bearing because Proposition 5.14 applies Lemma 5.1 inside τ-perpendicular categories J(M,P), which must themselves be module categories of Nakayama algebras. The proof is a single sentence: since every indecomposable Λ-module is uniserial, so is every indecomposable object of J(M,P), hence the Morita equivalent algebra Γ is Nakayama. This needs expansion: J(M,P) is a wide subcategory, and it is not immediate that the subobject lattice of an object of J(M,P) in the ambient category is inherited by the wide subcategory in a way that preserves uniseriality. The claim is true, but the proof should spell out that the subobjects of an object in the wide subcategory are exactly the ambient submodules that also lie in the subcategory, so the submodule lattice of each object remains a chain.

minor comments (4)

[§5, first paragraph] There is a typo: 'Nakayama algebas' should read 'Nakayama algebras'.
[§5, proof of Lemma 5.11] In the paragraph beginning 'Next we claim that it follows that Hom(X, Z) ≠ 0', the same statement appears twice; the second occurrence should likely refer to Hom(X, U1) ≠ 0, based on the surrounding argument.
[§4, Proposition 4.9] The notation B(M) for the Bongartz completion is used before its formal definition in Section 2.1; while the definition appears earlier, it would help the reader to include a forward reference at first use in the introduction of the Nakayama-specific results.
[§6, Example 6.4] There is a typo: 'Nakayma algebra' should be 'Nakayama algebra'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 5.15 is derived from independent τ-tilting and τ-exceptional-sequence machinery; no fitted parameter is renamed as a prediction.

full rationale

The paper's central claim is the transitivity of mutation of TF-ordered τ-rigid modules (and hence of complete τ-exceptional sequences) over Nakayama algebras. The derivation chain is: (i) translate the mutation defined in [BHM24] into TF-ordered language (Theorem 3.26); (ii) specialize Bongartz/co-Bongartz completions and V-maps to Nakayama algebras (Propositions 4.9, 4.11, 4.14, 4.15, Theorem 4.17); (iii) show that any two TF-orders of the same τ-rigid module are mutation-connected (Lemma 5.2, using Lemma 5.1 and Proposition 3.18); and (iv) lift τ-tilting mutations to TF-order mutations (Proposition 5.14). Each step is either proved from stated assumptions or cited to published theorems with proofs ([AIR14], [MT20], [BH23], [BHM24]); the cited theorems do not assume or contain the Nakayama transitivity conclusion. The one-sentence proof of Lemma 4.7 is terse, but that is a completeness concern, not circularity: the lemma's assertion is not an input to itself. Similarly, the skeptical concern about Lemma 5.1 (τ-tilting finiteness alone does not obviously force the forward φ-orbit to hit the swapped order) is a possible missing argument, not an equivalence-by-construction. No fitted parameters are renamed as predictions, no central definition is self-referential, and no uniqueness theorem from the authors is invoked to forbid alternatives. The argument is therefore self-contained in the relevant sense; the only reservations are about proof details, not circularity.

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

The paper introduces no empirical fitting and no invented entities. Its central claims rest on established theorems from τ-tilting theory, on the classification of τ-rigid modules over Nakayama algebras, and on the mutation framework of [BHM24]. These are domain assumptions with published proofs, not ad hoc postulates.

assumptions (5)

domain assumption Bijection between functorially finite torsion classes and τ-tilting pairs, and the existence of Bongartz/co-Bongartz completions.
Taken from [AIR14, Thm. 2.7, Thm. 2.10] and [Dem+23]; used throughout Sections 2-4 to define V-maps and mutation.
domain assumption For a Nakayama algebra, an indecomposable non-projective module M is τ-rigid iff its Loewy length ℓ(M) is less than the rank n (Proposition 4.2).
Cited from [Ada16, Prop. 2.5]; this is the backbone of the explicit Bongartz completion formulas and the case distinction in Theorem 1.3.
domain assumption Uniseriality facts collected in Proposition 4.6, including the AR-translation formula τ M_{s,t} = M_{(s-1)n,(t-1)n} and the Hom-vanishing criteria.
Used pervasively in Section 4 to compute V-maps and in Section 5 for the exchange sequence analysis; derives from [ASS06].
domain assumption The mutation of τ-exceptional pairs and the left irregular mutation procedure as defined in [BHM24, Def.-Prop. 4.3, Sec. 4].
The paper's Theorem 3.26 is a translation of this mutation; Theorem 4.17 for the irregular case imports [BHM24, Prop. 3.13g].
domain assumption Nakayama algebras are τ-tilting finite, and the mutation graph of τ-tilting modules is connected.
Used in Theorem 5.15 to connect any two τ-tilting modules; follows from representation finiteness and [AIR14].

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Pith. "Pith review of Mutating ordered $\tau$-rigid modules with applications to Nakayama algebras." pith.science (2026). https://pith.science/paper/AQSLY5LR

@misc{pith2026250113694,
  author       = {Pith},
  title        = {Pith review of: Mutating ordered $\tau$-rigid modules with applications to Nakayama algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AQSLY5LR}},
  note         = {Machine review of arXiv:2501.13694}
}

abstract

A mutation operation for $\tau$-exceptional sequences of modules over any finite-dimensional algebra was recently introduced, generalising the mutation for exceptional sequences of modules over hereditary algebras. We interpret this mutation in terms of TF-ordered $\tau$-rigid modules, which are in bijection with $\tau$-exceptional sequences. As an application we show that the mutation is transitive for Nakayama algebras, by providing an explicit combinatorial description of mutation over this class of algebras.

Figures

Figures reproduced from arXiv: 2501.13694 by the authors.

**Figure 1.** Bongartz completion (left) and co-Bongartz completion (right) of an indecomposable τ -rigid M ∼= Ms,t on Adachi’s disk model In fact, we can also use the disk model to visualise the quiver of the τ -tilting reduction of an indecomposable module of a Nakayama algebra. This description improves Lemma 4.7 and generalises a description given in the proof of [Msa22, Thm. 6.5] for a special class of self-injective Nakayam… view at source ↗

**Figure 2.** The module Bb of Theorem 4.17 (left) and the quiver (in orange) of End(B(M))/⟨M⟩ embedded into Adachi’s disk model (right) The following example demonstrates the absence of the three arrows a, b, c in Proposition 6.5. Example 6.6. Let Λ ∼= K(5 a−→ 4 b −→ 3 c−→ 2 d −→ 1)/⟨dc, cba⟩. Let M ∼= 4 3 , then one can check easily that J(M) ∼= A1 × A1 × A1 × A1 ∼= K4 is the direct product of four Nakayama algebras of rank 1. … view at source ↗

**Figure 3.** The φ-orbit of a TF-ordered τ -rigid module B ⊕ C illustrating three cases of Theorem 1.3. The left direct summand corresponds to the orange arc and the right direct summand corresponds to the blue arc. T F − 4 T F − 2a T F − 3 φ φ φ 1 0 2 3 1 0 2 3 1 0 2 3 S(0) P(1)/S(3) S(1) P(1)/S(3) P(1)/S(3) S(0) 1 2 1 2 2 1 [PITH_FULL_IMAGE:figures/full_fig_p040_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The φ-orbit of a TF-ordered τ -rigid module B ⊕ C illustrating the remaining three cases of Theorem 1.3. The left direct summand corresponds to the orange arc and the right direct summand corresponds to the blue arc. and introducing two new “diagonal” arrows, from the …

**Figure 5.** Figure 5: A φ-orbit of length 4, which cannot come from applying [BHM24, Thm. 7.8] to any functorially finite wide subcategory of rank 2. The left direct summand corresponds to the orange arc and the right direct summands corresponds to the blue arc. group relations, we show the…

**Figure 6.** Figure 6: An example of a cyclic Nakayama algebra for which the mutation of TF-ordered τ -tilting modules does not satisfy the Braid relations. The solid arc corresponds to the left-most direct summand, the dashed one to the middle one and the dotted one to the right-most one. R…

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Transitivity of mutation of $\tau$-exceptional sequences in the $\tau$-tilting finite case
math.RT 2025-06 accept novelty 7.0 of 10

Mutation of complete τ-exceptional sequences is transitive for every τ-tilting finite algebra.
Mutation of $\tau$-exceptional sequences for acyclic quivers over local algebras
math.RT 2025-05 conditional novelty 6.0 of 10

For algebras R⊗kQ with R local and Q acyclic, BHM mutation of complete τ-exceptional sequences coincides with classical mutation, and the braid group acts transitively.

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Works this paper leans on

2 extracted references · 2 canonical work pages · cited by 2 Pith papers

[2021]

Exceptional collections on $\Sigma _{ 2 }$

arXiv: 2107.03051. [Jas15] Gustavo Jasso. “Reduction of τ -tilting Modules and Torsion Pairs”. In: Int. Math. Res. Not. 2015.16 (2015), pp. 7190–7237. [Kai24] Maximilian Kaipel. τ -cluster morphism categories of factor algebras. 2024. arXiv: 2408. 03818. [Kai23] Maximilian Kaipel. The category of a partitioned fan . To be published in J. Lond. Math. Soc.,...

work page Pith review arXiv 2015
[2022]

Which cluster morphism categories are CAT(0)

arXiv: 2203.16679. [IOU21] Akira Ishii, Shinnosuke Okawa, and Hokuto Uehara. Exceptional collections on Σ2

work page Pith review arXiv

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Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [2021]

Exceptional collections on $\Sigma _{ 2 }$

arXiv: 2107.03051. [Jas15] Gustavo Jasso. “Reduction of τ -tilting Modules and Torsion Pairs”. In: Int. Math. Res. Not. 2015.16 (2015), pp. 7190–7237. [Kai24] Maximilian Kaipel. τ -cluster morphism categories of factor algebras. 2024. arXiv: 2408. 03818. [Kai23] Maximilian Kaipel. The category of a partitioned fan . To be published in J. Lond. Math. Soc.,...

work page Pith review arXiv 2015

[2] [2022]

Which cluster morphism categories are CAT(0)

arXiv: 2203.16679. [IOU21] Akira Ishii, Shinnosuke Okawa, and Hokuto Uehara. Exceptional collections on Σ2

work page Pith review arXiv