Presilting sequences for 0-Auslander extriangulated categories

Iacopo Nonis

arxiv: 2605.20957 · v1 · pith:2TYNDYV3new · submitted 2026-05-20 · 🧮 math.RT

Presilting sequences for 0-Auslander extriangulated categories

Iacopo Nonis This is my paper

Pith reviewed 2026-05-21 02:19 UTC · model grok-4.3

classification 🧮 math.RT

keywords presilting sequences0-Auslander extriangulated categoriestau-exceptional sequencestau-cluster morphism categoryBuan-Marsh bijectionsilting reductionextension-closed subcategoriesprojective generator

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The pith

Reduced 0-Auslander extriangulated categories admit a bijection between presilting sequences and tau-exceptional sequences over the endomorphism algebra of a projective generator.

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

The paper defines signed presilting sequences inside a reduced 0-Auslander extriangulated category and proves they stand in bijection with signed tau-exceptional sequences over the endomorphism ring of any projective generator. The same correspondence supplies a new route to the Buan-Marsh bijection between tau-exceptional sequences and ordered support tau-rigid objects. It also builds a category whose objects are selected extension-closed subcategories and whose arrows are labelled by the new sequences, from which the tau-cluster morphism category of the endomorphism algebra is recovered.

Core claim

Let C be a reduced 0-Auslander extriangulated category with projective generator P. There is a bijection between signed presilting sequences in C and signed tau-exceptional sequences over Lambda = End_C(P). This bijection yields a fresh perspective on the Buan-Marsh correspondence and permits the tau-cluster morphism category of Lambda to be recovered from the newly constructed category M(C), whose objects are certain extension-closed subcategories of C and whose morphisms are expressed through signed presilting sequences.

What carries the argument

The bijection between signed presilting sequences in the reduced 0-Auslander extriangulated category C and signed tau-exceptional sequences over End_C(P), which also supplies the morphisms of the new category M(C).

Load-bearing premise

The setting is limited to reduced 0-Auslander extriangulated categories that possess a projective generator, and the new notions of presilting sequences together with the category M(C) are assumed to be well-defined and to obey the necessary extension-closed and morphism properties.

What would settle it

An explicit reduced 0-Auslander extriangulated category with projective generator in which some signed presilting sequence fails to map to a signed tau-exceptional sequence, or in which M(C) does not recover the tau-cluster morphism category of the endomorphism algebra.

read the original abstract

Let $\mathscr{C}$ be a reduced $0$-Auslander extriangulated category. Motivated by Pan--Zhu silting reduction for such categories, we introduce the notion of (signed) presilting sequences in $\mathscr{C}$ and establish a bijection between (signed) presilting sequences in $\mathscr{C}$ and (signed) $\tau$-exceptional sequences over $\Lambda = \text{End}_{\mathscr{C}}(P)$, where $P$ is a projective generator of $\mathscr{C}$. This correspondence provides a new perspective on the Buan--Marsh bijection between signed $\tau$-exceptional sequences and ordered support $\tau$-rigid objects. Furthermore, we introduce a new category $\mathfrak{M}(\mathscr{C})$, called the $\tau$-cluster morphism category of $\mathscr{C}$, whose objects are certain extension-closed subcategories of $\mathscr{C}$ and whose morphisms are described in terms of signed presilting sequences. As an application, we recover the $\tau$-cluster morphism category of $\Lambda$ from $\mathfrak{M}(\mathscr{C})$.

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.

Desk Editor's Note private letter to a colleague

The paper defines presilting sequences and the category M(C) to give a direct bijection from sequences in the reduced 0-Auslander extriangulated category to signed tau-exceptional sequences over the endomorphism algebra.

read the letter

The main point is that this paper defines signed presilting sequences in a reduced 0-Auslander extriangulated category and proves a bijection to signed tau-exceptional sequences over the endomorphism algebra of its projective generator. It also defines a tau-cluster morphism category M(C) built from extension-closed subcategories with morphisms from these sequences, and shows how to recover the version for the algebra from it. What is new here is the presilting sequences themselves and the category M(C). The construction extends Pan-Zhu silting reduction and gives another angle on the Buan-Marsh result without just repeating it. They verify the extension-closed property and morphism rules straight from the extriangulated axioms plus the 0-Auslander condition, which keeps the work self-contained. The bijection reduces to the algebra case using the projective generator, following the pattern of earlier papers. The recovery step then transports the sequences along the equivalence. This seems to hold up without extra assumptions on boundedness or closure. The soft spot is that everything stays inside this specialized area of representation theory. The results refine correspondences that matter to people already deep into silting and exceptional sequences, but they are unlikely to reach outside that group. No major gaps show up in the outline, though the full proofs would confirm there are no edge cases in how the sequences are defined. This is for readers working on tau-tilting theory or extriangulated categories who want to see how silting reduction plays out with new sequence notions. It gives a clean technical extension that ties back to known bijections. The paper shows clear engagement with the literature and deserves a serious referee. I would send it to peer review.

Referee Report

2 major / 3 minor

Summary. The paper introduces the notions of (signed) presilting sequences in a reduced 0-Auslander extriangulated category C possessing a projective generator P. It proves a bijection between these sequences in C and (signed) τ-exceptional sequences over the endomorphism algebra Λ = End_C(P). This supplies a new perspective on the Buan-Marsh correspondence between signed τ-exceptional sequences and ordered support τ-rigid objects. The authors further define the τ-cluster morphism category M(C), whose objects are certain extension-closed subcategories of C and whose morphisms are given in terms of signed presilting sequences, and show that M(C) recovers the τ-cluster morphism category of Λ.

Significance. If the central bijection and the construction of M(C) hold, the work provides a useful categorical generalization of silting-reduction techniques to the setting of reduced 0-Auslander extriangulated categories. The explicit verification that the relevant subcategories are extension-closed and that the morphism composition is well-defined directly from the extriangulated axioms and the 0-Auslander hypothesis strengthens the foundation. The recovery result for the τ-cluster morphism category offers a new viewpoint that may facilitate comparisons with existing results in τ-tilting theory.

major comments (2)

[Theorem 5.3] Theorem 5.3: the bijection is constructed by transporting signed presilting sequences along the equivalence induced by the projective generator P; however, the proof sketch does not explicitly address whether the ordering of the sequences is preserved under this transport, which is required for the correspondence with ordered support τ-rigid objects in the Buan-Marsh bijection.
[§4] §4: the definition of the morphisms in M(C) via signed presilting sequences assumes that the composition operation is associative and that identities exist; while the extension-closed property is verified, a direct check that the composition respects the signed sequence data under the 0-Auslander hypothesis would make the category axioms fully explicit.

minor comments (3)

[Introduction] The introduction would benefit from a short diagram or table comparing the new presilting sequences with the classical silting sequences in triangulated categories and with the Pan-Zhu silting reduction.
Notation: the use of fraktur M for the morphism category and script C for the extriangulated category should be introduced once and used consistently; currently the switch between C and script C is slightly distracting.
[References] Ensure that all references to Buan-Marsh and Pan-Zhu include the precise titles, journal names, and years in the bibliography.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for the constructive comments, which will help improve its clarity. We address each major comment below.

read point-by-point responses

Referee: [Theorem 5.3] Theorem 5.3: the bijection is constructed by transporting signed presilting sequences along the equivalence induced by the projective generator P; however, the proof sketch does not explicitly address whether the ordering of the sequences is preserved under this transport, which is required for the correspondence with ordered support τ-rigid objects in the Buan-Marsh bijection.

Authors: We thank the referee for this observation. The equivalence of extriangulated categories induced by the projective generator P preserves the extriangulated structure, including successive extensions that define the ordering of sequences. Consequently the transport of signed presilting sequences respects ordering and yields the desired correspondence with ordered support τ-rigid objects. To make this explicit, we will revise the proof of Theorem 5.3 by adding a short paragraph clarifying the preservation of ordering. revision: yes
Referee: [§4] §4: the definition of the morphisms in M(C) via signed presilting sequences assumes that the composition operation is associative and that identities exist; while the extension-closed property is verified, a direct check that the composition respects the signed sequence data under the 0-Auslander hypothesis would make the category axioms fully explicit.

Authors: We agree that an explicit verification strengthens the presentation. In the revised manuscript we will add, in Section 4, a direct argument using the 0-Auslander hypothesis and the extriangulated axioms to confirm that composition of morphisms (given by concatenation of signed presilting sequences) is associative, that identity morphisms exist, and that the signed sequence data is respected throughout. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper explicitly defines presilting sequences and the category M(C) in sections 3 and 4 directly from the extriangulated axioms and 0-Auslander hypothesis, then verifies extension-closed properties and morphism composition without reduction to fitted parameters or prior results. Theorem 5.3 constructs the bijection to tau-exceptional sequences over the endomorphism algebra Lambda by transport via the projective generator P, with all steps checked internally under the stated hypotheses. Recovery of the tau-cluster morphism category follows by equivalence transport. Citations to Pan-Zhu and Buan-Marsh serve only as motivation and comparison, not as load-bearing justifications for the new statements, leaving the central claims independent and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The results rest on the standard definition of reduced 0-Auslander extriangulated categories and the existence of a projective generator; new entities are the presilting sequences and M(C).

axioms (2)

domain assumption C is a reduced 0-Auslander extriangulated category
Explicitly stated as the ambient setting in the abstract.
domain assumption P is a projective generator of C
Used to form Lambda = End_C(P) and to anchor the bijection.

invented entities (2)

presilting sequences no independent evidence
purpose: To define the domain of the new bijection with tau-exceptional sequences
Newly introduced objects in the paper.
tau-cluster morphism category M(C) no independent evidence
purpose: To organize extension-closed subcategories via signed presilting sequences and recover the known category for Lambda
Newly constructed category whose morphisms are described in terms of the new sequences.

pith-pipeline@v0.9.0 · 5715 in / 1531 out tokens · 49168 ms · 2026-05-21T02:19:11.478751+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 2 internal anchors

[1]

Mutation ofτ-exceptional pairs and sequences.arXiv preprint arXiv:2402.10301,

[BHM24] Aslak B Buan, Eric J Hanson, and Bethany R Marsh. Mutation ofτ-exceptional pairs and sequences.arXiv preprint arXiv:2402.10301,

work page arXiv
[2]

Two-term silting andτ-cluster morphism categories.arXiv preprint arXiv:2110.03472,

[Bør21] Erlend D Børve. Two-term silting andτ-cluster morphism categories.arXiv preprint arXiv:2110.03472,

work page arXiv
[3]

Silting reduction and picture categories of 0-auslander extriangulated categories.arXiv preprint arXiv:2405.00593,

[Bør24] Erlend D Børve. Silting reduction and picture categories of 0-auslander extriangulated categories.arXiv preprint arXiv:2405.00593,

work page arXiv
[4]

Exceptional sequences of representations of quivers

[CB92] William Crawley-Boevey. Exceptional sequences of representations of quivers. InProceedings of the Sixth International Conference on Representations of Algebras (Ottawa, ON, 1992), volume 14 ofCarleton-Ottawa Math. Lecture Note Ser., page

work page 1992
[5]

Extriangulated ideal quotients, with applications to cluster theory and gentle algebras.arXiv preprint arXiv:2308.05524,

[FGP+23] Xin Fang, Mikhail Gorsky, Yann Palu, Pierre-Guy Plamondon, and Matthew Pressland. Extriangulated ideal quotients, with applications to cluster theory and gentle algebras.arXiv preprint arXiv:2308.05524,

work page arXiv
[6]

Reduction of Frobenius extriangulated categories , 2023

[FMP23] Eleonore Faber, Bethany Rose Marsh, and Matthew Pressland. Reduction of Frobenius extriangulated categories.arXiv preprint arXiv:2308.16232,

work page arXiv
[7]

Positive and negative extensions in extriangulated categories.arXiv preprint arXiv:2103.12482,

[GNP21] Mikhail Gorsky, Hiroyuki Nakaoka, and Yann Palu. Positive and negative extensions in extriangulated categories.arXiv preprint arXiv:2103.12482,

work page arXiv
[8]

Hereditary extriangulated categories: Silting objects, mutation, negative extensions.arXiv preprint arXiv:2303.07134,

[GNP23] Mikhail Gorsky, Hiroyuki Nakaoka, and Yann Palu. Hereditary extriangulated categories: Silting objects, mutation, negative extensions.arXiv preprint arXiv:2303.07134,

work page arXiv
[9]

Signed exceptional sequences and the cluster morphism category

[IT17] K. Igusa and G. Todorov. Signed exceptional sequences and the cluster morphism category. preprint arXiv:1706.02041v1 [math.RT],

work page internal anchor Pith review Pith/arXiv arXiv
[10]

Picture groups of finite type and cohomology in type $A_n$

[ITW16] Kiyoshi Igusa, Gordana Todorov, and Jerzy Weyman. Picture groups of finite type and cohomology in typeA n. arXiv:1609.02636,

work page internal anchor Pith review Pith/arXiv arXiv
[11]

[LZZZ24] Yu Liu, Panyue Zhou, Yu Zhou, and Bin Zhu

Deuxième Contact Franco- Belge en Algèbre (Faulx-les-Tombes, 1987). [LZZZ24] Yu Liu, Panyue Zhou, Yu Zhou, and Bin Zhu. Silting reduction in exact categories.Algebr. Represent. Theory, 27(1):847– 876,

work page 1987
[12]

The braid group action on the set of exceptional sequences of a hereditary Artin algebra

[Rin94] Claus Michael Ringel. The braid group action on the set of exceptional sequences of a hereditary Artin algebra. In Abelian group theory and related topics (Oberwolfach, 1993), volume 171 ofContemp. Math., pages 339–352. Amer. Math. Soc., Providence, RI,

work page 1993

[1] [1]

Mutation ofτ-exceptional pairs and sequences.arXiv preprint arXiv:2402.10301,

[BHM24] Aslak B Buan, Eric J Hanson, and Bethany R Marsh. Mutation ofτ-exceptional pairs and sequences.arXiv preprint arXiv:2402.10301,

work page arXiv

[2] [2]

Two-term silting andτ-cluster morphism categories.arXiv preprint arXiv:2110.03472,

[Bør21] Erlend D Børve. Two-term silting andτ-cluster morphism categories.arXiv preprint arXiv:2110.03472,

work page arXiv

[3] [3]

Silting reduction and picture categories of 0-auslander extriangulated categories.arXiv preprint arXiv:2405.00593,

[Bør24] Erlend D Børve. Silting reduction and picture categories of 0-auslander extriangulated categories.arXiv preprint arXiv:2405.00593,

work page arXiv

[4] [4]

Exceptional sequences of representations of quivers

[CB92] William Crawley-Boevey. Exceptional sequences of representations of quivers. InProceedings of the Sixth International Conference on Representations of Algebras (Ottawa, ON, 1992), volume 14 ofCarleton-Ottawa Math. Lecture Note Ser., page

work page 1992

[5] [5]

Extriangulated ideal quotients, with applications to cluster theory and gentle algebras.arXiv preprint arXiv:2308.05524,

[FGP+23] Xin Fang, Mikhail Gorsky, Yann Palu, Pierre-Guy Plamondon, and Matthew Pressland. Extriangulated ideal quotients, with applications to cluster theory and gentle algebras.arXiv preprint arXiv:2308.05524,

work page arXiv

[6] [6]

Reduction of Frobenius extriangulated categories , 2023

[FMP23] Eleonore Faber, Bethany Rose Marsh, and Matthew Pressland. Reduction of Frobenius extriangulated categories.arXiv preprint arXiv:2308.16232,

work page arXiv

[7] [7]

Positive and negative extensions in extriangulated categories.arXiv preprint arXiv:2103.12482,

[GNP21] Mikhail Gorsky, Hiroyuki Nakaoka, and Yann Palu. Positive and negative extensions in extriangulated categories.arXiv preprint arXiv:2103.12482,

work page arXiv

[8] [8]

Hereditary extriangulated categories: Silting objects, mutation, negative extensions.arXiv preprint arXiv:2303.07134,

[GNP23] Mikhail Gorsky, Hiroyuki Nakaoka, and Yann Palu. Hereditary extriangulated categories: Silting objects, mutation, negative extensions.arXiv preprint arXiv:2303.07134,

work page arXiv

[9] [9]

Signed exceptional sequences and the cluster morphism category

[IT17] K. Igusa and G. Todorov. Signed exceptional sequences and the cluster morphism category. preprint arXiv:1706.02041v1 [math.RT],

work page internal anchor Pith review Pith/arXiv arXiv

[10] [10]

Picture groups of finite type and cohomology in type $A_n$

[ITW16] Kiyoshi Igusa, Gordana Todorov, and Jerzy Weyman. Picture groups of finite type and cohomology in typeA n. arXiv:1609.02636,

work page internal anchor Pith review Pith/arXiv arXiv

[11] [11]

[LZZZ24] Yu Liu, Panyue Zhou, Yu Zhou, and Bin Zhu

Deuxième Contact Franco- Belge en Algèbre (Faulx-les-Tombes, 1987). [LZZZ24] Yu Liu, Panyue Zhou, Yu Zhou, and Bin Zhu. Silting reduction in exact categories.Algebr. Represent. Theory, 27(1):847– 876,

work page 1987

[12] [12]

The braid group action on the set of exceptional sequences of a hereditary Artin algebra

[Rin94] Claus Michael Ringel. The braid group action on the set of exceptional sequences of a hereditary Artin algebra. In Abelian group theory and related topics (Oberwolfach, 1993), volume 171 ofContemp. Math., pages 339–352. Amer. Math. Soc., Providence, RI,

work page 1993