arxiv: 2604.26199 · v1 · submitted 2026-04-29 · 🧮 math.CT · math.RA· math.RT

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Metrics on triangulated categories and restrictions of (co)-t-structures

Wei Hu, Ziheng Liu

Pith reviewed 2026-05-07 12:46 UTC · model grok-4.3

classification 🧮 math.CT math.RAmath.RT

keywords silting subcategoriest-structurestriangulated categoriesright coherent ringscontravariant finitenessprecompletionmetricsKoenig-Yang correspondences

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The pith

A silting subcategory is contravariantly finite in the precompletion exactly when the induced canonical t-structure restricts to it.

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

The paper proves that in compactly generated triangulated categories admitting small coproducts, a silting subcategory of compact objects is contravariantly finite in the precompletion or completion if and only if the canonical t-structure it induces restricts to that precompletion or completion. This equivalence supplies a purely categorical test for right coherence of rings. The work also lifts the correspondences among silting objects, bounded t-structures, co-t-structures, and simple-minded collections to triangulated categories equipped with metrics, while showing that the lifted correspondences continue to commute with mutations and to preserve partial orders.

Core claim

For compactly generated triangulated categories admitting small coproducts, silting subcategories of compact objects give rise to canonical t-structures. A silting subcategory being contravariantly finite in the precompletion (or completion) is equivalent to the canonical t-structure restricting to this precompletion (or completion). This yields a purely categorical characterization of right coherent rings: a ring R is right coherent if and only if the standard t-structure on D(Mod-R) restricts to a t-structure on K^{-,b}(proj-R). The correspondences between silting objects, bounded (co)-t-structures, and simple-minded collections extend to the metric framework and still commute withmutation

What carries the argument

The equivalence between contravariant finiteness of a silting subcategory in the precompletion and restriction of the induced canonical t-structure to that precompletion.

Load-bearing premise

The triangulated category is compactly generated, admits small coproducts, and the silting subcategories consist of compact objects that induce canonical t-structures under the given metric.

What would settle it

A single compactly generated triangulated category containing a silting subcategory of compact objects that is not contravariantly finite in the precompletion yet whose canonical t-structure still restricts to the precompletion.

read the original abstract

This paper explores the restriction behavior of silting-induced $t$-structures and co-$t$-structures on triangulated categories endowed with metrics. For compactly generated triangulated categories admitting small coproducts, silting subcategories of compact objects give rise to canonical $t$-structures. We establish that a silting subcategory being contravariantly finite in the precompletion (or completion) is equivalent to the canonical $t$-structure restricting to this precompletion (or completion). This result yields a purely categorical characterization of right coherent rings: a ring $R$ is right coherent if and only if the standard $t$-structure on $\mathcal{D}({\sf Mod}\text{-}R)$ restricts to a $t$-structure on $\mathcal{K}^{-,b}({\sf proj}\text{-}R)$. Furthermore, we show that the correspondences between silting objects, bounded (co)-$t$-structures, and simple-minded collections given by Koenig and Yang can be extended to the metric framework of triangulated categories, and still commute with mutation operations and preserve natural partial orders.

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 gives a clean equivalence between contravariant finiteness of compact silting subcategories in metric completions and restriction of the induced t-structure, which yields a categorical characterization of right coherent rings.

read the letter

The main advance is the equivalence: for compactly generated triangulated categories with coproducts, a silting subcategory of compact objects is contravariantly finite in the metric precompletion or completion exactly when the canonical t-structure restricts there. This produces the stated characterization of right coherent rings via the standard t-structure on D(Mod-R) restricting to K^{-,b}(proj-R). The authors also extend the Koenig-Yang correspondences between silting objects, bounded (co)-t-structures, and simple-minded collections to the metric setting, and check that the extensions still commute with mutation and preserve the partial orders. The metric framework organizes the restriction and extension behavior in a uniform way, and the ring example follows directly once the general equivalence is available. The arguments use standard properties of silting subcategories and the metric definitions without circularity or extra fitting parameters. The compact generation and coproduct assumptions are stated explicitly and align with the usual setup in this area. The only real limitation is that the metric on triangulated categories is a specialized tool, so the results stay mostly within the homological algebra community already working on silting and completions. Readers outside that circle will find the overhead noticeable but not prohibitive. This is for people who already know Koenig-Yang and want to see how the correspondences and restriction criteria behave under metrics. It is a solid, incremental piece that deserves a serious referee to check the technical steps in the restriction equivalence and the metric preliminaries. I would send it to peer review.

Referee Report

2 major / 3 minor

Summary. The paper investigates restriction properties of silting-induced t-structures and co-t-structures on triangulated categories equipped with metrics. For compactly generated triangulated categories with small coproducts, it shows that a silting subcategory of compact objects is contravariantly finite in the metric precompletion (resp. completion) if and only if the induced canonical t-structure restricts to that precompletion (resp. completion). This equivalence is applied to obtain a purely categorical characterization of right coherent rings: a ring R is right coherent precisely when the standard t-structure on D(Mod-R) restricts to a t-structure on K^{-,b}(proj-R). The paper further extends the Koenig-Yang correspondences among silting objects, bounded (co)-t-structures, and simple-minded collections to the metric setting, verifying that these correspondences commute with mutation and preserve the natural partial orders.

Significance. If the central equivalences hold, the work supplies a new, purely categorical criterion for right coherence of rings and broadens the scope of silting theory by incorporating metric structures on triangulated categories. The extension of the Koenig-Yang correspondences to this setting is a useful generalization that preserves key structural features such as mutation compatibility and order preservation, potentially aiding further research on derived categories and silting subcategories.

major comments (2)

[§3] §3 (equivalence theorem): the proof that contravariant finiteness in the precompletion is equivalent to restriction of the canonical t-structure relies on the metric allowing extension/restriction of the t-structure; the argument should explicitly verify that the compact generation and small-coproduct hypotheses suffice to preserve the necessary orthogonality conditions without additional finiteness assumptions on the silting subcategory.
[§4] §4 (ring-coherence characterization): the identification of K^{-,b}(proj-R) as the metric completion of the compact projective silting subcategory must be checked in detail, particularly that the boundedness condition K^{-,b} aligns exactly with the completion construction used in the general equivalence.

minor comments (3)

The precise definition of a 'metric' on a triangulated category (including the axioms for the distance function and its compatibility with the triangulated structure) should be stated explicitly in the preliminaries rather than deferred to later sections.
Notation for precompletion versus completion is used interchangeably in several statements; a short clarifying remark or diagram distinguishing the two constructions would improve readability.
[Abstract] The abstract and introduction refer to 'Koenig and Yang' without a full citation; ensure the reference list contains the precise bibliographic entry for their work on silting and t-structures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive recommendation. The comments highlight opportunities to strengthen the clarity of the proofs in §§3 and 4. We address each point below and will incorporate the suggested expansions in the revised version.

read point-by-point responses

Referee: [§3] §3 (equivalence theorem): the proof that contravariant finiteness in the precompletion is equivalent to restriction of the canonical t-structure relies on the metric allowing extension/restriction of the t-structure; the argument should explicitly verify that the compact generation and small-coproduct hypotheses suffice to preserve the necessary orthogonality conditions without additional finiteness assumptions on the silting subcategory.

Authors: We agree that an explicit verification of the orthogonality preservation would improve readability. In the revised manuscript we will insert a short auxiliary lemma (or expanded remark) right after the statement of the main equivalence in §3. The lemma will use the compact generation of the ambient triangulated category together with the existence of all small coproducts to show that the relevant Hom-vanishing conditions between the silting subcategory and its orthogonal complement continue to hold after passage to the metric precompletion, without imposing any further finiteness hypotheses on the silting subcategory itself. The argument relies only on the standard properties of compact objects and the definition of the metric structure already introduced in the paper. revision: yes
Referee: [§4] §4 (ring-coherence characterization): the identification of K^{-,b}(proj-R) as the metric completion of the compact projective silting subcategory must be checked in detail, particularly that the boundedness condition K^{-,b} aligns exactly with the completion construction used in the general equivalence.

Authors: We thank the referee for this observation. While the identification is implicit in the application of the general theorem, we concede that a fully explicit verification is desirable. In the revision we will enlarge the discussion in §4 by adding a dedicated paragraph (or short subsection) that directly compares the metric completion of the silting subcategory of compact projective R-modules with the category K^{-,b}(proj-R). We will verify step-by-step that the boundedness condition defining K^{-,b} coincides exactly with the objects obtained by the completion construction of §3, using the standard t-structure on D(Mod-R) and the fact that compact projectives generate under the given metric. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard categorical equivalences

full rationale

The paper proves an equivalence between a silting subcategory being contravariantly finite in the metric precompletion/completion and the canonical t-structure restricting to that completion, using the compactly generated hypothesis and small coproducts. This is applied to characterize right coherent rings via restriction of the standard t-structure to K^{-,b}(proj-R), identifying the latter as the relevant completion of the compact projective silting subcategory. The extension of Koenig-Yang correspondences to the metric setting commutes with mutation and preserves orders by direct verification. No step reduces by definition to its inputs, no parameters are fitted then renamed as predictions, and cited prior results (including silting theory) are external and not load-bearing self-citations. The argument is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides insufficient detail to list specific free parameters or invented entities; results rest on standard assumptions of triangulated category theory.

axioms (1)

domain assumption Compactly generated triangulated categories admit small coproducts and silting subcategories of compact objects induce canonical t-structures.
Invoked to establish the main equivalence and ring characterization.

pith-pipeline@v0.9.0 · 5492 in / 1259 out tokens · 51877 ms · 2026-05-07T12:46:51.488098+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 3 canonical work pages

[1]

Aihara and O

T. Aihara and O. Iyama. Silting mutation in triangulated categories.J. Lond. Math. Soc. (2), 85(3):633–668, 2012

2012
[2]

Alonso Tarr´ ıo, A

L. Alonso Tarr´ ıo, A. Jerem´ ıas L´ opez, and M. J. Souto Salorio. Construction oft-structures and equivalences of derived categories.Trans. Amer. Math. Soc., 355(6):2523–2543, 2003

2003
[3]

A. A. Beilinson, J. Bernstein, and P. Deligne. Faisceaux pervers. InAnalysis and topology on singular spaces, I (Luminy, 1981), volume 100 ofAst´ erisque, pages 5–171. Soc. Math. France, Paris, 1982

1981
[4]

Beligiannis and I

A. Beligiannis and I. Reiten. Homological and homotopical aspects of torsion theories. Mem. Amer. Math. Soc., 883, 07 2007

2007
[5]

Biswas, H

R. Biswas, H. Chen, K. M. Rahul, C. J. Parker, and J. Zheng. Boundedt-structures, fini- tistic dimensions, and singularity categories of triangulated categories, arXiv:2401.00130

work page arXiv
[6]

M. V. Bondarko. Weight structures vs.t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general).J. K-theory, 6:387–504, 2007

2007
[7]

R. Fushimi. The correspondence between silting objects andt-structures for non-positive dg algebras. InProceedings of the 56th Symposium on Ring Theory and Representation Theory, pages 22–25. Symp. Ring Theory Represent. Theory Organ. Comm., Koganei, 2025

2025
[8]

Goodbody, T

I. Goodbody, T. Raedschelders, and G. Stevenson. Approximable triangulated categories and reflexive dg-categories,Appl. Cat. Str.(2026) 34-19

2026
[9]

Hoshino, Y

M. Hoshino, Y. Kato, and J.-I. Miyachi. Ont-structures and torsion theories induced by compact objects.J. Pure Appl. Algebra, 167(1):15–35, 2002

2002
[10]

Keller and P

B. Keller and P. Nicol´ as. Weight structures and simple dg modules for positive dg algebras. Int. Math. Res. Not. IMRN, (5):1028–1078, 2013

2013
[11]

Koenig and D

S. Koenig and D. Yang. Silting objects, simple-minded collections,t-structures and co-t- structures for finite-dimensional algebras.Doc. Math., 19:403–438, 2014

2014
[12]

H. Krause. Completing perfect complexes.Math. Zeit., 296:1387–1427, 2020

2020
[13]

H. Krause. Completions of triangulated categories. InTriangulated categories in repre- sentation theory and beyond—the Abel Symposium 2022, volume 17 ofAbel Symp., pages 169–193. Springer, Cham, [2024]©2024

2022
[14]

Marks and A

F. Marks and A. Zvonareva. Lifting and restricting t-structures.Bull. Lond. Math. Soc., 55(2):640–657, 2023. 34

2023
[15]

Mendoza Hern´ andez, E

O. Mendoza Hern´ andez, E. C. S´ aenz Valadez, V. Santiago Vargas, and M. J. Souto Salorio. Auslander-Buchweitz context and co-t-structures.Appl. Categ. Structures, 21(5):417–440, 2013

2013
[16]

A. Neeman. The categoriesT c andT b c determine each other. arXiv:1806.06471v1

work page arXiv
[17]

A. Neeman. Thet-structures generated by objects.Trans. Amer. Math. Soc., 374(11):8161– 8175, 2021

2021
[18]

A. Neeman. Boundedt-structures on the category of perfect complexes.Acta Math., 233(2):239–284, 2024

2024
[19]

A. Neeman. Triangulated categories with a single compact generator and a brown repre- sentability theorem.Invent. Math., 2026

2026
[20]

Pauksztello

D. Pauksztello. Compact corigid objects in triangulated categories and co-t-structures. Central Eur. J. Math., 6:25–42, 2008

2008
[21]

K. M. Rahul. Representability theorems via metric techniques, arXiv:2504.11768

work page arXiv
[22]

J. Rickard. Equivalences of derived categories for symmetric algebras.J. Algebra, 257(2):460–481, 2002

2002
[23]

Saor´ ın and A

M. Saor´ ın and A. Zvonareva. Lifting of recollements and gluing of partial silting sets.Proc. Roy. Soc. Edinburgh Sect. A, 152(1):209–257, 2022

2022
[24]

Su and D

H. Su and D. Yang. From simple-minded collections to silting objects via Koszul duality. Algebr. Represent. Theory, 22(1):219–238, 2019. Wei Hu, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China Email:huwei@bnu.edu.cn Ziheng Liu, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China Email:alglzh@...

2019