Localization theorems for weakly approximable triangulated categories

Jinbi Zhang; Yaohua Zhang; Yongliang Sun

arxiv: 2402.04954 · v3 · submitted 2024-02-07 · 🧮 math.RT · math.CT· math.RA

Localization theorems for weakly approximable triangulated categories

Yongliang Sun , Jinbi Zhang , Yaohua Zhang This is my paper

Pith reviewed 2026-05-24 03:49 UTC · model grok-4.3

classification 🧮 math.RT math.CTmath.RA

keywords weakly approximable triangulated categoriesrecollementlocalization theoremssingularity categoriesderived categoriestriangulated categoriesexact sequences

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The pith

Under mild assumptions, a recollement of weakly approximable triangulated categories induces short exact sequences on natural subcategories and big singularity categories.

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

The paper establishes localization theorems for weakly approximable triangulated categories. It proves that recollements between such categories produce short exact sequences on several triangulated subcategories and on the associated big singularity categories. These results apply directly to derived categories of rings, DG algebras, and schemes. A reader would care because the exact sequences relate invariants across different categories, making localization phenomena more computable in algebraic settings.

Core claim

Under mild assumptions, a recollement of weakly approximable triangulated categories induces short exact sequences on several natural triangulated subcategories as well as on the associated (big) singularity categories, with the theorems illustrated through applications in the derived categories of rings, DG algebras, and schemes.

What carries the argument

The recollement of weakly approximable triangulated categories, which produces short exact sequences on subcategories and big singularity categories.

If this is right

Recollements in derived categories of rings produce exact sequences relating singularity categories.
The same exact sequences hold for recollements of DG algebras.
Derived categories of schemes inherit the exact sequences on their subcategories and singularity categories.
Invariants of singularity categories can be compared across recollements using the induced sequences.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The results may allow direct comparison of singularity categories without explicit computation of the categories themselves.
Exact sequences could simplify proofs of equivalences or vanishing results in K-theory of singularity categories.
The framework might apply to recollements arising from geometric morphisms between schemes.

Load-bearing premise

The categories must be weakly approximable and the recollement must satisfy the mild assumptions required for the sequences to be exact.

What would settle it

A concrete recollement of weakly approximable categories in which the induced maps on subcategories or big singularity categories fail to form short exact sequences would disprove the main theorems.

read the original abstract

Weakly approximable triangulated categories, introduced by Neeman, provide a powerful framework for studying localization phenomena in triangulated categories. In this paper, we establish new localization theorems showing that, under mild assumptions, a recollement of weakly approximable triangulated categories induces short exact sequences on several natural triangulated subcategories as well as on the associated (big) singularity categories. As applications, we illustrate our results in the derived categories of rings, DG algebras, and schemes.

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 supplies a general localization theorem for recollements of weakly approximable triangulated categories that produces short exact sequences on subcategories and big singularity categories under mild assumptions.

read the letter

The main new content is a set of localization theorems that start from a recollement of weakly approximable triangulated categories and, under mild assumptions, deliver short exact sequences on the natural triangulated subcategories plus the associated big singularity categories. This extends Neeman's framework in a direct way and aims to replace scattered case-by-case arguments with one reusable statement. The applications to derived categories of rings, DG algebras, and schemes are the right kind of check: they show the general result applies in the settings people actually compute in. That part is done cleanly and gives the theorems concrete reach. The soft spots are limited. Everything turns on the mild assumptions being satisfied in the target examples, so the paper needs to state those assumptions explicitly and verify them without extra hidden conditions; if the assumptions are more restrictive than advertised, the scope narrows. The proofs are not visible in the abstract, but the framing shows no circularity or self-referential fitting. The citation pattern to Neeman is appropriate and the work stays within the existing definition. This is for people already using recollements and singularity categories in homological algebra. A reader who needs a systematic way to produce exact sequences in these contexts will get direct value. The paper is coherent on its own terms and engages the literature honestly. It deserves a serious referee because the claim is specific, the framework is already standard, and the applications are standard enough that referees can check the details without starting from zero.

Referee Report

1 major / 2 minor

Summary. The paper establishes localization theorems showing that, under mild assumptions, a recollement of weakly approximable triangulated categories induces short exact sequences on several natural triangulated subcategories as well as on the associated big singularity categories. Applications are given to derived categories of rings, DG algebras, and schemes, building directly on Neeman's framework for weakly approximable categories.

Significance. If the theorems hold, the work provides a useful extension of localization techniques in triangulated categories to the weakly approximable setting, with direct applicability to standard objects in homological algebra and algebraic geometry. The explicit framing around recollements and singularity categories strengthens the potential utility for researchers working with derived categories.

major comments (1)

[§3] The main theorems are stated under 'mild assumptions' whose precise content is not listed in the general formulation (see the statement following the abstract and the opening of §3); this makes it hard to check whether the short exact sequences on subcategories and singularity categories follow without additional hidden conditions on the recollement functors.

minor comments (2)

Notation for the natural subcategories (e.g., the 'approximable' and 'compact' parts) should be introduced with a single consistent diagram or table early in the paper to aid readability across the applications.
[applications section] The applications section would benefit from a brief reminder of how the mild assumptions specialize in each concrete case (rings, DG algebras, schemes) rather than leaving the verification entirely to the reader.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation of minor revision. The single major comment can be addressed by improving the clarity of the general statements without altering the mathematical content.

read point-by-point responses

Referee: [§3] The main theorems are stated under 'mild assumptions' whose precise content is not listed in the general formulation (see the statement following the abstract and the opening of §3); this makes it hard to check whether the short exact sequences on subcategories and singularity categories follow without additional hidden conditions on the recollement functors.

Authors: We agree that the phrasing 'mild assumptions' in the abstract and the opening paragraph of §3 could be made more precise for the reader. The assumptions in question are those explicitly enumerated in the statements of the main results (Theorems 3.1, 3.4 and 3.7): weak approximability of the three categories in the recollement, the existence of the required adjoints, and the compactness or generation conditions needed for the singularity categories. No further hidden conditions on the recollement functors are imposed. In the revised version we will insert, immediately after the opening sentence of §3, a short enumerated list that collects these hypotheses verbatim from the theorem statements. This change will make the logical structure transparent while leaving the proofs unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from external definition

full rationale

The paper's central claims consist of localization theorems derived from the externally introduced definition of weakly approximable triangulated categories (Neeman) and standard recollement assumptions. The abstract and reader's summary frame the results as extensions to derived categories of rings, DG algebras, and schemes, with no equations, parameters, or premises that reduce by construction to fitted inputs or self-citations. No load-bearing self-citation chains, ansatzes smuggled via prior author work, or renamings of known results are indicated. This is the normal case of a self-contained mathematical derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the definition of weakly approximable triangulated categories and the standard axioms of recollements; no free parameters or invented entities appear in the abstract.

axioms (2)

domain assumption Weakly approximable triangulated categories satisfy the properties introduced by Neeman that enable localization phenomena.
The framework is invoked at the outset of the abstract.
standard math A recollement consists of adjoint functors between three triangulated categories satisfying the usual orthogonality and exactness conditions.
Standard background invoked when the authors speak of recollements inducing exact sequences.

pith-pipeline@v0.9.0 · 5600 in / 1416 out tokens · 41731 ms · 2026-05-24T03:49:56.278212+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

Finiteness of homological dimensions in triangulated categories
math.RT 2026-04 unverdicted novelty 5.0

Inequalities bound homological dimensions across recollements in triangulated categories, extending ring-theoretic finiteness results.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · cited by 1 Pith paper

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[1] [1]

Alonso Tarro, A

L. Alonso Tarro, A. Jeremias Lopez, M. J. Souto Salorio, Construction of t-structures and equivalences of derived categories, Trans. Amer. Math. Soc. 355(6) (2003), 2523-2543

work page 2003

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Amiot, Cluster categories for algebras of global dimension 2 and qu ivers with potential, Annal

C. Amiot, Cluster categories for algebras of global dimension 2 and qu ivers with potential, Annal. Institut Fourier 59(6) (2009), 2525-2590

work page 2009

[3] [3]

Angeleri H¨ ugel, S

L. Angeleri H¨ ugel, S. Koenig, Q. H. Liu, D. Yang, Ladders and simplicity of derived module categories, J. Algebra 472 (2017), 15–66

work page 2017

[4] [4]

A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux perverse (French), Analysis and topology on singular spaces, Asterisque, vol. 100 (1982), Soc. Math. France, Paris, 5-171

work page 1982

[5] [5]

Biswas, H

R. Biswas, H. X. Chen, C. Parker, K. M. Rahul, J. H. Zheng, Bounded t-structures, ﬁnitistic dimensions, and singularity categories of triangulated ca tegories, preprint (2024), arXiv: 2401.00130

work page arXiv 2024

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M. V. Bondarko, S. V. Vostokov, On Weakly Negative Subcategories, Weight Structures, and (Weakly) Approximable Triangulated Categories, Lobachevskii J. Math. 41 (2020), 151–159

work page 2020

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Burke, A

J. Burke, A. Neeman, B. Pauwels, Gluing approximable triangulated categories, Forum of Math., Sigma 11 (2023), 1–18

work page 2023

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Canonako, C

A. Canonako, C. Haesemeyer, A. Neeman, P. Stellari, The passage among the subcategories of weakly approximable triangulated categories, preprint (2024), arXiv:2402.04605

work page arXiv 2024

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X. W. Chen, Unifying two results of Orlov on singularity categories, Abh. Math. Semin. Univ. Hambg. 80 (2010), 207-212

work page 2010

[10] [10]

Guo, Cluster tilting objects in generalized higher cluster cate gories, J

L. Guo, Cluster tilting objects in generalized higher cluster cate gories, J. Pure Appl. Alg. 215(9) (2011), 2055-2071

work page 2011

[11] [11]

H. Jin, D. Yang, G. D. Zhou, A localisation theorem for singularity categories of prope r DG algebras, preprint (2023), arXiv:2302.05054

work page arXiv 2023

[12] [12]

Jorgensen, A new recollement for schemes, Houston J

P. Jorgensen, A new recollement for schemes, Houston J. Math. 35 (2009), 1071-1077

work page 2009

[13] [13]

Kalck, D

M. Kalck, D. Yang, Relative singularity categories I: Auslander resolutions , Adv. Math. 301 (2016), 973–1021

work page 2016

[14] [14]

Keller, Deriving DG categories, Ann

B. Keller, Deriving DG categories, Ann. Sci. ´Ecole Norm. Sup. (4) 27(1) (1994), 63-102

work page 1994

[15] [15]

Keller, Derived categories and their uses, in: Handbook of Algebra, vol

B. Keller, Derived categories and their uses, in: Handbook of Algebra, vol. 1, in: Handb. Algebr., vol. 1, Elsevier/North-Holland, Amsterdam, 1996 , pp. 671–701. LOCALIZATION THEOREMS 20

work page 1996

[16] [16]

Krause, Localization theory for triangulated categories, In T

H. Krause, Localization theory for triangulated categories, In T. Holm, P. Jørgensen, R. Rouquier (Eds.), Triangulated Categories (London Mathema tical Society Lecture Note Series, pp. 161-235). Cambridge: Cambridge University Press. (201 0)

work page

[17] [17]

Neeman, The connection between the K–theory localisation theorem o f Thomason, Trobaugh and Yao, and the smashing subcategories of Bousﬁel d and Ravenel, Ann

A. Neeman, The connection between the K–theory localisation theorem o f Thomason, Trobaugh and Yao, and the smashing subcategories of Bousﬁel d and Ravenel, Ann. Sci. ´Ec. Norm. Sup´ er.25 (1992), 547-566

work page 1992

[18] [18]

Neeman, Triangulated categories with a single compact generator an d a Brown repre- sentability theorem, preprint (2018), arXiv:1804.02240

A. Neeman, Triangulated categories with a single compact generator an d a Brown repre- sentability theorem, preprint (2018), arXiv:1804.02240

work page arXiv 2018

[19] [19]

Neeman, The categories T c and T b c determine each other, preprint (2018), arXiv:1806.06471

A. Neeman, The categories T c and T b c determine each other, preprint (2018), arXiv:1806.06471

work page arXiv 2018

[20] [20]

Neeman, The t-structures generated by objects, Trans

A. Neeman, The t-structures generated by objects, Trans. Amer. Math. Soc. 374 (2021), 8161-8175

work page 2021

[21] [21]

Neeman, Strong generators in Dperf (X) and Db coh(X), Ann

A. Neeman, Strong generators in Dperf (X) and Db coh(X), Ann. Math. 193 (2021), 689-732

work page 2021

[22] [22]

Neeman, Approximable triangulated categories, Cont

A. Neeman, Approximable triangulated categories, Cont. Math. 769 (2021), 111-155

work page 2021

[23] [23]

Neeman, Bounded t-structures on the category of perfect complexes, Acta Math

A. Neeman, Bounded t-structures on the category of perfect complexes, Acta Math. (to ap- pear), arXiv:2202.08861

work page arXiv

[24] [24]

Neeman, Triangulated categories, volume 148 of Anna ls of Mathematics Studies

A. Neeman, Triangulated categories, volume 148 of Anna ls of Mathematics Studies. Princeton University Press, Princeton, NJ, 2001

work page 2001

[25] [25]

D. O. Orlov, Formal completions and idempotent completions of triangul ated categories of singularities, Adv. Math. 226(1) (2011), 206-217

work page 2011

[26] [26]

Rickard, Morita theory of derived categories, J

J. Rickard, Morita theory of derived categories, J. Lond. Math. Soc. 39(2) (1988), 436-456. Yongliang Sun, School of Mathematics and Physics, Yancheng Institute of Technology, 224003 Jiangsu, People’s Republic of China Email address : syl13536@126.com Yaohua Zhang, Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei U...

work page 1988