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arxiv: 2402.13356 · v2 · pith:NYLY5U5Knew · submitted 2024-02-20 · 🧮 math.RT · math.RA

Fishing for complements

Lidia Angeleri H\"ugel , David Pauksztello , Jorge Vit\'oria This is my paper

Pith reviewed 2026-05-25 08:42 UTC · model grok-4.3

classification 🧮 math.RT math.RA
keywords presilting objectssilting complementsco-t-structurestriangulated categoriessilting-discrete algebraslarge silting complexesderived categories
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The pith

Finite-dimensional algebras are silting discrete if and only if every bounded large silting complex is equivalent to a compact one.

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

The paper determines necessary and sufficient conditions under which a presilting object in a triangulated category admits a complement that turns it into a silting object. These conditions are formulated for both ordinary presilting objects and their large counterparts by examining the co-t-structures they induce. The same criteria recover earlier existence theorems for complements in derived categories of hereditary abelian categories. A direct consequence is an if-and-only-if characterization of silting-discrete finite-dimensional algebras in terms of their bounded large silting complexes.

Core claim

Given a presilting object in a triangulated category, necessary and sufficient conditions for the existence of a complement are found by studying the associated co-t-structures. This is done for both classic and large presilting objects. As a result, some known cases of complement existence are recovered, and a finite-dimensional algebra is silting discrete precisely when every bounded large silting complex is equivalent to a compact one.

What carries the argument

Associated co-t-structures of a presilting object, which supply the necessary and sufficient conditions for a complement to exist.

If this is right

  • Complements exist precisely when the co-t-structures attached to a presilting object satisfy the stated criteria.
  • The same criteria apply verbatim to large presilting objects.
  • Existence of complements in derived categories of hereditary abelian categories follows immediately from the general conditions.
  • Silting discreteness of a finite-dimensional algebra is equivalent to every bounded large silting complex being equivalent to a compact one.

Where Pith is reading between the lines

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

  • The co-t-structure technique may yield explicit lists of silting-discrete algebras by checking only bounded complexes.
  • The same conditions could be tested in triangulated categories arising from geometric or topological settings beyond algebra.
  • A computational search for counterexamples could focus on small-dimensional algebras whose large silting complexes are explicitly describable.

Load-bearing premise

The triangulated category admits well-behaved associated co-t-structures for any given presilting object.

What would settle it

A finite-dimensional algebra that is silting discrete but admits a bounded large silting complex with no compact equivalent would refute the characterization.

Figures

Figures reproduced from arXiv: 2402.13356 by David Pauksztello, Jorge Vit\'oria, Lidia Angeleri H\"ugel.

Figure 1
Figure 1. Figure 1: Each figure shows a region of the Auslander–Reiten quiver of Db(kQ). Top: the coaisle (M[< 0])⊥ associated to the silting object M is marked in red, with the silting object M marked in deeper red. Middle: the (unbounded) coaisle [X[< 0])⊥ associated with the presilting object X = S2[2] ∈ (M[< 0])⊥ is marked in blue. Bottom: the intersection V := (X[< 0])⊥ ∩ (M[< 0])⊥ = (X ⊕ M)[< 0]⊥. One can see that no o… view at source ↗
read the original abstract

Given a presilting object in a triangulated category, we find necessary and sufficient conditions for the existence of a complement. This is done both for classic (pre)silting objects and for large (pre)silting objects. The key technique is the study of associated co-t-structures. As a consequence of our techniques we recover some known cases of the existence of complements, including for derived categories of some hereditary abelian categories and for silting-discrete algebras. Moreover, we also show that a finite-dimensional algebra is silting discrete if and only if every bounded large silting complex is equivalent to a compact one.

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.

Referee Report

0 major / 0 minor

Summary. The paper claims to establish necessary and sufficient conditions for the existence of complements to presilting objects (both classical and large) in triangulated categories by studying associated co-t-structures. As consequences, it recovers known results on complements for derived categories of certain hereditary abelian categories and for silting-discrete algebras, and proves that a finite-dimensional algebra is silting-discrete if and only if every bounded large silting complex is equivalent to a compact one.

Significance. If the central claims hold, the work provides a uniform technique via co-t-structures that characterizes silting discreteness and unifies classical and large settings for complements. The explicit recovery of prior results as sanity checks is a strength, as is the focus on falsifiable equivalences between bounded large and compact silting complexes.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report and for accurately summarizing the main results of the manuscript. We are pleased that the uniform approach via co-t-structures, the recovery of known results, and the characterization of silting discreteness are viewed as strengths. No specific major comments appear in the provided report, so we offer no point-by-point responses at this time. We remain available to address any concrete concerns should they be supplied.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives necessary and sufficient conditions for complements to presilting objects by analyzing associated co-t-structures in triangulated categories, then applies this uniformly to obtain the silting-discrete characterization as a consequence. This is a standard mathematical argument that recovers prior results as sanity checks rather than redefining inputs or fitting parameters to force predictions. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the logical chain; the central iff statement follows from the co-t-structure technique without reducing to its own assumptions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of triangulated categories and the definitions of presilting and silting objects from prior literature; no free parameters or invented entities are indicated.

axioms (2)
  • domain assumption The ambient category is triangulated.
    This is the setting in which presilting objects and co-t-structures are defined.
  • domain assumption Presilting objects admit associated co-t-structures whose properties control complement existence.
    This is the key technique stated in the abstract.

pith-pipeline@v0.9.0 · 5627 in / 1175 out tokens · 42488 ms · 2026-05-25T08:42:28.637426+00:00 · methodology

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Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages · 28 internal anchors

  1. [1]

    \tau-tilting theory

    T. Adachi, O. Iyama, I. Reiten, -tilting theory , Compos.\ Math.\ 150 (2014), no.\ 3, 415--452, also 1210.1036

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  2. [2]

    Discreteness of silting objects and t-structures in triangulated categories

    T. Adachi, Y. Mizuno, D. Yang, Discreteness of silting object and t-structures in triangulated categories , Proc.\ London Math.\ Soc.\ 118 (2019), no.\ 3, 1--42, also 1708.08168

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  3. [3]

    Silting mutation in triangulated categories

    T. Aihara, O. Iyama, Silting mutation in triangulated categories , J.\ Lond.\ Math.\ Soc.\ (2) 85 (2012), no.\ 3, 633--668, also 1009.3370

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  4. [4]

    Classifying tilting complexes over preprojective algebras of Dynkin type

    T. Aihara, Y. Mizuno, Classifying tilting complexes over preprojective algebras of Dynkin type , Algebra Number Theory 11 (2017), no.\ 6, 1287--1315, also 1509.07387

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  5. [5]

    Bousfield localization on formal schemes

    L. Alonso Tarr\'io, A. Jerem\'ias L\'opez, M. J. Souto Salorio, Construction of t-structures and equivalences of derived categories , Trans.\ Amer.\ Math.\ Soc.\ 355 (2003), no.\ 6, 2523--2543, also math/0307189

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  6. [6]

    Angeleri H\"ugel, F

    L. Angeleri H\"ugel, F. U. Coelho, Infinitely generated complements to partial tilting modules , Math.\ Proc.\ Cambridge Philos.\ Soc.\ 132 (2002), no.\ 1, 89--96

    work page 2002

  7. [7]

    Silting modules

    L. Angeleri H\"ugel, F. Marks, J. Vi\'oria, Silting modules , Int.\ Math.\ Res.\ Not.\ IMRN (2016), no.\ 4, 1251--1284, also 1405.2531

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  8. [8]

    A characterisation of $\tau$-tilting finite algebras

    L. Angeleri H\"ugel, F. Marks, J. Vi\'oria, A characterisation of -tilting finite algebras . in ``Model theory of modules, algebras and categories'', Contemp.\ Math.\ Vol.\ 19, pp. 75--89, Amer.\ Math.\ Soc., Providence RI, 2019, also 1801.04312

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  9. [9]

    Partial silting objects and smashing subcategories

    L. Angeleri H\"ugel, F. Marks, J. Vi\'oria, Partial silting objects and smashing subcategories , Math.\ Z.\ 296 (2020), no.\ 3-4, 887--900, also 1902.05817

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  10. [10]

    A. A. Beilinson, J. Bernstein, P. Deligne, O. Gabber, Faisceux Pervers , Ast\'erisque 100 (2018), Soc.\ Math.\ France, Second Edition

    work page 2018

  11. [11]

    A. I. Bondal, Operations on t-structures and perverse coherent sheaves , Izv.\ Ross.\ Akad.\ Nauk.\ Ser.\ Mat.\ 77 (2013), no.\ 4, 5--30, also 1308.2549

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  12. [12]

    M. V. Bondarko, Weight structures vs.\ t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general) , J.\ K-Theory 6 (2010), no.\ 3, 387--504, also 0704.4003

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  13. [13]

    Bongartz, Tilted algebras , in ``Representations of algebras (Puebla, 1980)'', Lecture Notes in Math.\ Vol

    K. Bongartz, Tilted algebras , in ``Representations of algebras (Puebla, 1980)'', Lecture Notes in Math.\ Vol. 903, pp. 26--38, Springer, Berlin-New York, 1981

    work page 1980

  14. [14]

    Breaz, On a characterization of (co)silting objects , 2303.06843

    S. Breaz, On a characterization of (co)silting objects , 2303.06843

    work page arXiv

  15. [15]

    Averaging t-structures and extension closure of aisles

    N. Broomhead, D. Pauksztello, D. Ploog, Averaging t-structures and extension closure of aisles , J.\ Algebra 394 (2013), 51--78, also 1208.5691

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  16. [16]

    ustle, D. Yang, Ordered exchange graphs , in ``Advances in representation theory of algebras'', EMS Ser.\ Congr.\ Rep.\, pp. 135--193, Eur.\ Math.\ Soc.\, Z\

    T. Br\"ustle, D. Yang, Ordered exchange graphs , in ``Advances in representation theory of algebras'', EMS Ser.\ Congr.\ Rep.\, pp. 135--193, Eur.\ Math.\ Soc.\, Z\"urich, 2013, also 1302.6045

    work page arXiv 2013

  17. [17]

    X.-W. Chen, C. M. Ringel, Hereditary triangulated categories , J.\ Noncommut.\ Geom.\ 12 (2018), no.\ 4, 1425--144, also 1606.08279

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  18. [18]

    W. Dai, C. Fu, A reduction approach to silting objects of derived categories of hereditary categories , Colloq.\ Math.\ 170 (2022), no.\ 2, 239--252, also 2011.10728

    work page arXiv 2022

  19. [19]

    $\tau$-tilting finite algebras, bricks and $g$-vectors

    L. Demonet, O. Iyama, G. Jasso, -tilting finite algebras, bricks, and g -vectors , Int.\ Math.\ Res.\ Not.\ IMRN (2019), no.\ 3, 852--892, also 1503.00285

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  20. [20]

    General Presentations of Algebras

    H. Derksen, J. Fei, General presentations of algebras , Adv.\ Math.\ 278 (2015), 210--237, also 0911.4913

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  21. [21]

    P. C. Eklof, J. Trlifaj, How to make Ext vanish , Bull.\ London Math.\ Soc.\ 33 (2001), no.\ 1, 41--51

    work page 2001

  22. [22]

    Happel, I

    D. Happel, I. Reiten, S. O. Smal \, Tilting in abelian categories and quasitilted algebras , Mem.\ Amer.\ Math.\ Soc.\ 120 (1996), no.\ 575, viii+88 pp

    work page 1996

  23. [23]

    On t-structures and Torsion Theories Induced by Compact Objects

    M. Hoshino, Y. Kato, J.-I. Miyachi, On t-structures and torsion theories induced by compact objects , J.\ Pure Appl.\ Algebra 167 (2002), no.\ 1, 15--35, also math/0005172

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  24. [24]

    Intermediate co-t-structures, two-term silting objects, tau-tilting modules, and torsion classes

    O. Iyama, P. J rgensen, D. Yang, Intermediate co-t-structures, two-term silting objects, -tilting modules, and torsion classes , Algebra Number Theory 8 (2014), no.\ 10, 2413--2431, also 1311.4891

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  25. [25]

    Silting reduction and Calabi--Yau reduction of triangulated categories

    O. Iyama, D. Yang, Silting reduction and Calabi-Yau reduction of triangulated categories , Trans.\ Amer.\ Math.\ Soc.\ 370 (2018), no.\ 11, 7861--7898, also 1408.2678

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  26. [26]

    H. Jin, S. Schroll, Z. Wang, A complete derived invariant and silting theory for graded gentle algebras , 2303.17474

    work page arXiv

  27. [27]

    Kalck, A finite-dimensional algebra with a phantom (a corollary of an example by J

    M. Kalck, A finite-dimensional algebra with a phantom (a corollary of an example by J. Krah) , 2304.08417

    work page arXiv

  28. [28]

    Keller, Deriving DG categories , Ann.\ Sci.\ \'Ecole Norm.\ Sup.\ 27 (1994), no.\ 1, 63--102

    B. Keller, Deriving DG categories , Ann.\ Sci.\ \'Ecole Norm.\ Sup.\ 27 (1994), no.\ 1, 63--102

    work page 1994

  29. [29]

    On triangulated orbit categories

    B. Keller, On triangulated orbit categories , Doc. Math. 10 (2005), 551--581, also math/0503240

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  30. [30]

    Keller, D

    B. Keller, D. Vossieck, Aisles in derived categories , Deuxi\`eme Contact Franco-Belge en Alg\`ebre (Faulx-les-Tombes, 1987), Bull.\ Soc.\ Math.\ Belg.\ S\'er.\ 40 (1988), no.\ 2, 239–253

    work page 1987

  31. [31]

    Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras

    S. Koenig, D. Yang, Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras , Doc.\ Math.\ 19 (2014), 403--438, also 1203.5657

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  32. [32]

    Y.-Z. Liu, Y. Zhou, A negative answer to the complement question for presilting complexes , 2302.12502

    work page arXiv

  33. [33]

    Auslander-Buchweitz context and co-t-structures

    O. Mendoza Hern\'andez, E. C. S\'aenz Valadez, V. Santiago Vargas, M. J. Souto Salorio, Auslander--Buchweitz context and co-t-structures , Appl.\ Categ.\ Structures 21 (2013), no. 5, 417--440, also 1002.4604

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  34. [34]

    The t-structures generated by objects

    A. Neeman, The t-structures generated by objects , Trans.\ Amer.\ Math.\ Soc.\ 374 (2021), no.\ 11, 8161--8175, also 1808.05267

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  35. [35]

    Silting Theory in triangulated categories with coproducts

    P. Nicol\'as, M. Saor\'in, A. Zvonareva, Silting theory in triangulated categories with coproducts , J.\ Pure Appl.\ Algebra 223 (2019), no.\ 6, 2273--2319, also 1512.04700

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  36. [36]

    Compact Corigid Objects in Triangulated Categories and Co-t-structures

    D. Pauksztello, Compact corigid objects in triangulated categories and co-t-structures , Cent.\ Eur.\ J.\ Math.\ 6 (2008), no.\ 1, 25--42, also 0705.0102

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  37. [37]

    Contractibility of the stability manifold for silting-discrete algebras

    D. Pauksztello, M. Saor\'in, A. Zvonareva, Contractibility of the stability manifold for silting-discrete algebras , Forum Math.\ 30 (2018), no.\ 5, 1255--1263, also 1705.10604

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  38. [38]

    Realisation functors in tilting theory

    C. Psaroudakis, J. Vit\'oria, Realisation functors in tilting theory , Math.\ Z.\ 288 (2018), no.\ 3-4, 965--1028, also 1511.02677

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  39. [39]

    Rickard, A

    J. Rickard, A. Schofield, Cocovers and tilting modules , Math.\ Proc.\ Cambridge Philos.\ Soc.\ 106 (1989), no.\ 1, 1--5

    work page 1989

  40. [40]

    On exact categories and applications to triangulated adjoints and model structures

    M. Saor\'in, J. , On exact categories and applications to triangulated adjoints and model structures , Adv.\ Math.\ 228 (2011), no.\ 2, 968--1007, also 1005.3248

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  41. [41]

    J. , D Posp\' i s il, On compactly generated torsion pairs and the classification of co-t-structures for commutative noetherian rings , Trans.\ Amer.\ Math.\ Soc.\ 368 (2016), no.\ 9, 6325--6361, also 1212.3122

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  42. [42]

    Wei, Semi-tilting complexes , Israel J.\ Math.\ 194 (2013), no.\ 2, 871--893

    J. Wei, Semi-tilting complexes , Israel J.\ Math.\ 194 (2013), no.\ 2, 871--893

    work page 2013

  43. [43]

    Stability conditions, torsion theories and tilting

    J. Woolf, Stability conditions, torsion theories and tilting , J.\ London Math.\ Soc.\ 82 (2010), no.\ 3, 663--682, also 0909.0552

    work page internal anchor Pith review Pith/arXiv arXiv 2010