Silting subcategories and (co)torsion pairs associated to extended hearts

Haicheng Zhang; Liangwei Huang

REVIEW 1 major objections 2 minor 1 cited by

Silting subcategories and (co)torsion pairs associated to extended hearts

T0 review · 1 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read Poset isomorphisms connect (d+1)-term silting subcategories to functorially finite s-torsion pairs in d-extended hearts and to hereditary complete cotorsion pairs.

desk verdict The paper sets up poset isomorphisms between (d+1)-term silting subcategories and torsion/cotorsion pairs in d-extended hearts, plus dg versions. read the letter →

arxiv 2606.13508 v1 pith:IDALL4L2 submitted 2026-06-11 math.RT math.CTmath.RA

Liangwei Huang , Haicheng Zhang This is my paper · ORCID

classification math.RTmath.CTmath.RA

keywords siltingsubcategoriesextendedheartss-torsionpairscotorsiontau-tiltingdgalgebrasposetisomorphismstriangulatedcategories

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 these three collections of objects stand in order-preserving bijection with one another inside a triangulated category equipped with a d-extended heart. The same pattern is shown to hold, after passage to dg algebras, between tau-tilting pairs, (d+1)-term silting complexes, and s-torsion pairs. A reader following the argument sees that questions about one of these structures can be translated directly into questions about the others while preserving the partial order. The correspondences therefore supply a uniform language for several variants of tilting and silting that appear in representation theory.

What carries the argument

The d-extended heart, which supplies the ambient abelian category in which s-torsion pairs and hereditary cotorsion pairs are defined and compared with silting subcategories via the stated poset isomorphisms.

What would settle it

An explicit triangulated category possessing a d-extended heart in which the map from (d+1)-term silting subcategories to functorially finite s-torsion pairs fails to be bijective or order-preserving.

Watch

Extended reading notes

Core claim

The central claim is that the poset of (d+1)-term silting subcategories is isomorphic to the poset of functorially finite s-torsion pairs inside the d-extended heart, which is in turn isomorphic to the poset of hereditary complete cotorsion pairs in an associated subcategory; the dg-algebra versions replace the first two posets by the poset of tau-tilting pairs and the poset of (d+1)-term silting complexes, respectively.

Load-bearing premise

The d-extended heart of the triangulated category exists and satisfies the technical conditions that let functorially finite s-torsion pairs and hereditary complete cotorsion pairs be defined inside it.

Editorial extensions

If this is right

Any classification of (d+1)-term silting subcategories immediately yields a classification of the corresponding s-torsion pairs and cotorsion pairs.
Properties preserved by the poset isomorphisms, such as finiteness or heredity, transfer between the three collections.
In the dg setting the same transfer applies between tau-tilting pairs and silting complexes.
The bijections are compatible with the natural partial orders on each side.

Reading between the lines

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

The isomorphisms may be used to transport mutation operations or approximation properties from one structure to the others.
The pattern suggests analogous correspondences could exist for other notions of extended hearts or for n-torsion pairs with n not equal to d.
Concrete computations of these posets for derived categories of gentle algebras or cluster-tilted algebras become interchangeable across the three descriptions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main takeaway is that the authors establish poset isomorphisms linking (d+1)-term silting subcategories, functorially finite s-torsion pairs inside the d-extended heart, and hereditary complete cotorsion pairs in a suitable subcategory. They also record the corresponding statements for dg algebras, relating tau-tilting pairs, (d+1)-term silting complexes, and torsion pairs.

This extends earlier correspondences between silting objects and torsion pairs by moving to the d-extended setting. The bijections look like a direct, incremental step from the cited literature rather than a wholesale reorganization.

The work does a reasonable job of collecting these objects into a single framework and stating the isomorphisms cleanly. If the proofs in the body are complete and the definitions of the extended hearts are handled without extra hidden assumptions, the results should be usable by people already working in this corner of representation theory.

The main limitation visible from the abstract is the lack of any proof outline or concrete example, which makes it hard to judge how much new technical work is required to make the functorially finite conditions hold. The central assumption—that the d-extended heart carries an abelian structure rich enough for these pairs—needs to be checked carefully in the full text, but nothing in the statement itself looks circular or contradictory.

This is aimed at specialists who already know silting theory and t-structures. A reader who wants explicit bijections for the d-case could extract something usable here.

I would send it to peer review. The claims are specific enough that referees can test them against the existing literature.

Referee Report

1 major / 2 minor

Summary. The manuscript establishes poset isomorphisms between (d+1)-term silting subcategories, functorially finite s-torsion pairs in the d-extended heart, and hereditary complete cotorsion pairs in a suitable subcategory of a triangulated category. As an application, it provides dg-algebra versions of these bijections relating τ-tilting pairs, (d+1)-term silting complexes, and functorially finite s-torsion pairs.

Significance. If the isomorphisms hold, the results unify aspects of silting theory with torsion and cotorsion pairs in the setting of extended hearts, extending classical correspondences (such as those for 2-term silting and torsion pairs) to higher d. The dg-algebra versions may facilitate applications in derived categories and representation theory of algebras. The paper ships explicit bijections that are functorial in the stated sense, which strengthens the contribution if the technical conditions on extended hearts are verified.

major comments (1)

[Introduction / Setup of extended hearts] The central claims rely on the existence and properties of the d-extended heart (including that it is abelian and admits functorially finite s-torsion pairs and hereditary complete cotorsion pairs). The abstract and setup assume these without an explicit verification or reference to a prior result establishing the required abelian structure and finiteness conditions for general d; this is load-bearing for all stated isomorphisms.

minor comments (2)

[Main theorems] Notation for s-torsion pairs and the precise definition of 'hereditary complete cotorsion pairs in a suitable subcategory' should be recalled or cross-referenced in the statement of the main theorems to improve readability.
[Application section] The dg-algebra versions are presented as an application; a brief comparison table or diagram relating the classical and dg cases would clarify the functoriality claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for recognizing the potential of our results to unify aspects of silting theory with torsion and cotorsion pairs via extended hearts. We address the single major comment below.

read point-by-point responses

Referee: [Introduction / Setup of extended hearts] The central claims rely on the existence and properties of the d-extended heart (including that it is abelian and admits functorially finite s-torsion pairs and hereditary complete cotorsion pairs). The abstract and setup assume these without an explicit verification or reference to a prior result establishing the required abelian structure and finiteness conditions for general d; this is load-bearing for all stated isomorphisms.

Authors: We agree that the properties of the d-extended heart are foundational to all stated isomorphisms. The construction of the d-extended heart appears in Section 2, where it is introduced as an abelian category following the standard extension procedure from the original heart. However, the introduction and setup do not contain an explicit reference or short verification confirming the abelian structure together with the existence of functorially finite s-torsion pairs and hereditary complete cotorsion pairs for arbitrary d. We will revise the manuscript by adding a reference to the prior result that establishes the abelian structure of the d-extended heart for general d, together with a brief paragraph in the setup section recalling why the required finiteness conditions hold. This change will clarify the load-bearing assumptions without affecting the main theorems or proofs. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper states and proves poset isomorphisms between (d+1)-term silting subcategories, functorially finite s-torsion pairs in the d-extended heart, and hereditary complete cotorsion pairs, plus dg-algebra versions involving τ-tilting pairs and silting complexes. These are direct theorem statements derived from the definitions of silting subcategories, extended hearts, torsion pairs, and cotorsion pairs in triangulated and dg categories. No equations reduce a claimed result to its own inputs by construction, no parameters are fitted and relabeled as predictions, and no load-bearing steps rely on self-citations whose content is itself unverified or defined circularly. The derivation is self-contained against standard external results in the field.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Relies on background axioms of triangulated categories and definitions of silting/torsion objects; no free parameters or new entities introduced in the abstract.

assumptions (2)

domain assumption Existence and basic properties of d-extended hearts in triangulated categories
Required to even state the torsion pairs and isomorphisms.
standard math Standard definitions and finiteness conditions for silting subcategories and cotorsion pairs
Drawn from prior literature in representation theory.

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Cite this review

Pith. "Pith review of Silting subcategories and (co)torsion pairs associated to extended hearts." pith.science (2026). https://pith.science/paper/IDALL4L2

@misc{pith2026260613508,
  author       = {Pith},
  title        = {Pith review of: Silting subcategories and (co)torsion pairs associated to extended hearts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IDALL4L2}},
  note         = {Machine review of arXiv:2606.13508}
}

abstract

We establish the poset isomorphisms between $(d+1)$-term silting subcategories, functorially finite $s$-torsion pairs in the $d$-extended heart, and hereditary complete cotorsion pairs in a suitable subcategory. As an application, we also give dg algebra versions of these bijections, which establish the poset isomorphisms between $\tau$-tilting pairs, $(d+1)$-term silting complexes, and functorially finite $s$-torsion pairs.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Extended heart construction (I): The heart of $n$-cotorsion pairs on triangulated categories
math.CT 2026-08 accept novelty 8.0 of 10

Every n-cotorsion pair on a triangulated category has a heart that is an abelian n-truncated category, carrying compatible pretriangulated and extriangulated structures.

Reference graph

Works this paper leans on

14 extracted references · 4 canonical work pages · cited by 1 Pith paper

[1]

Adachi, H

T. Adachi, H. Enomoto, M. Tsukamoto, Intervals ofs-torsion pairs in extriangulated categories with negative first extension, Math. Proc. Cambridge Philos. Soc. 174 (2023), 451–469. SILTING SUBCATEGORIES AND (CO)TORSION PAIRS 19

2023
[2]

Adachi, O

T. Adachi, O. Iyama, I. Reiten,τ-tilting theory, Compos. Math. 150 (2014), 415–452

2014
[3]

Adachi, Y

T. Adachi, Y. Mizuno, D. Yang, Discreteness of silting objects andt-structures in triangulated categories, Proc. London Math. Soc. (3) 118 (2019), 1–42

2019
[4]

Adachi, M

T. Adachi, M. Tsukamoto, Hereditary cotorsion pairs and silting subcategories in extriangulated categories, J. Algebra 594 (2022), 109–137

2022
[5]

Aihara, O

T. Aihara, O. Iyama, Silting mutation in triangulated categories, J. Lond. Math. Soc. (2) 85 (2012), 633–668

2012
[6]

Gupta, Ond-term silting objects and (co) torsion pairs in truncated derived categories, arXiv:2407.10562

E. Gupta, Ond-term silting objects and (co) torsion pairs in truncated derived categories, arXiv:2407.10562

work page arXiv
[7]

Mendoza Hern´ andez, E., S´ aenz Valadez, V

O. Mendoza Hern´ andez, E., S´ aenz Valadez, V. Santiago Vargas, M. Souto Salorio, Auslander- Buchweitz context and co-t-structures, Appl. Categ. Structures 21 (2013), 417–440

2013
[8]

Iyama, H

O. Iyama, H. Nakaoka, Y. Palu, Auslander-Reiten theory in extriangulated categories, Trans. Amer. Math. Soc. Ser. B 11 (2024), 248–305

2024

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Y. Liu, P. Zhou, Cotorsion pairs onm-term subcategories, Acta. Math. Sin.-English Ser. (3) 41 (2025), 1023–1036

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Mochizuki, M

N. Mochizuki, M. Plogmann, On the Auslander-Reiten theory for extended hearts of proper connec- tive dg algebras, arXiv:2505.16560
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Nakaoka, Y

H. Nakaoka, Y. Palu, Extriangulated categories, hovey twin cotorsion pairs and model structures, Cah. Topol. G´ eom. Diff´ er. Cat´ eg. 60 (2019), 117–193

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D. Pauksztello, A. Zvonareva, Co-t-structures, cotilting and cotorsion pairs, Math. Proc. Camb. Philos. Soc. (1) 175 (2023), 89–106

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J. Wei, Y. Zhou, AIR tilting subcategories of extended hearts, arXiv:2512.13218
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Zhou, Tilting theory for extended module categories, arXiv:2411.15473

Y. Zhou, Tilting theory for extended module categories, arXiv:2411.15473. Ministry of Education Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R.China Email address:2524416777@qq.com (L. Huang) Email address:zhanghc@njnu...

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.

[1] [1]

Adachi, H

T. Adachi, H. Enomoto, M. Tsukamoto, Intervals ofs-torsion pairs in extriangulated categories with negative first extension, Math. Proc. Cambridge Philos. Soc. 174 (2023), 451–469. SILTING SUBCATEGORIES AND (CO)TORSION PAIRS 19

2023

[2] [2]

Adachi, O

T. Adachi, O. Iyama, I. Reiten,τ-tilting theory, Compos. Math. 150 (2014), 415–452

2014

[3] [3]

Adachi, Y

T. Adachi, Y. Mizuno, D. Yang, Discreteness of silting objects andt-structures in triangulated categories, Proc. London Math. Soc. (3) 118 (2019), 1–42

2019

[4] [4]

Adachi, M

T. Adachi, M. Tsukamoto, Hereditary cotorsion pairs and silting subcategories in extriangulated categories, J. Algebra 594 (2022), 109–137

2022

[5] [5]

Aihara, O

T. Aihara, O. Iyama, Silting mutation in triangulated categories, J. Lond. Math. Soc. (2) 85 (2012), 633–668

2012

[6] [6]

Gupta, Ond-term silting objects and (co) torsion pairs in truncated derived categories, arXiv:2407.10562

E. Gupta, Ond-term silting objects and (co) torsion pairs in truncated derived categories, arXiv:2407.10562

work page arXiv

[7] [7]

Mendoza Hern´ andez, E., S´ aenz Valadez, V

O. Mendoza Hern´ andez, E., S´ aenz Valadez, V. Santiago Vargas, M. Souto Salorio, Auslander- Buchweitz context and co-t-structures, Appl. Categ. Structures 21 (2013), 417–440

2013

[8] [8]

Iyama, H

O. Iyama, H. Nakaoka, Y. Palu, Auslander-Reiten theory in extriangulated categories, Trans. Amer. Math. Soc. Ser. B 11 (2024), 248–305

2024