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arxiv: 2604.17962 · v1 · submitted 2026-04-20 · 🧮 math.RT · math.CO

The interval neighborhoods in the real Grothendieck groups

Sota Asai This is my paper

Pith reviewed 2026-05-10 04:01 UTC · model grok-4.3

classification 🧮 math.RT math.CO
keywords TF equivalencereal Grothendieck groupsilting conetau-tilting reductiong-fanpresilting complexinterval neighborhoodsimple-minded collection
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The pith

The closed interval neighborhood D(U) of each silting cone has a 2^{|U|}:1 correspondence with the TF equivalence classes of the tau-tilting reduced algebra B.

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

The paper shows that TF equivalence on the real Grothendieck group completes the g-fan, with the basic pieces being the open silting cones. It then focuses on the closed interval neighborhood D(U) around each such cone for a 2-term presilting complex U. The main result is a 2^{|U|}:1 correspondence between the TF equivalence classes inside D(U) and those on the real Grothendieck group of the reduced algebra B obtained from tau-tilting reduction at U. This is achieved by explicitly describing D(U) as a polyhedral cone whose inequalities and faces are given in terms of 2-term simple-minded collections and M-TF equivalences. A reader would care because the result allows reducing questions about the global structure of TF classes to the corresponding questions on smaller algebras.

Core claim

For a finite dimensional algebra A and a 2-term presilting complex U, there is a 2^{|U|}:1 correspondence between the TF equivalence classes in the closed interval neighborhood D(U) of the silting cone C°(U) in K_0(proj A)_R and the TF equivalence classes in K_0(proj B)_R, where B is the algebra from the tau-tilting reduction at U. The defining inequalities and faces of the polyhedral cone D(U) are described explicitly using 2-term simple-minded collections and M-TF equivalences.

What carries the argument

The closed interval neighborhood D(U) of the silting cone C°(U) in the real Grothendieck group, constructed via the tau-tilting reduction to B and described using 2-term simple-minded collections.

If this is right

  • The TF equivalence classes can be understood recursively by applying tau-tilting reductions at each silting complex.
  • The faces of each interval neighborhood D(U) are in bijection with certain M-TF equivalence classes for 2-term simple-minded collections.
  • The completion of the g-fan is built by gluing these interval neighborhoods around the silting cones.
  • The structure around each cone depends only on the reduced algebra B obtained at that step.

Where Pith is reading between the lines

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

  • Successive applications of this reduction could yield a full classification of TF classes for algebras that reduce to semisimple ones.
  • This might relate the TF fan to other fans appearing in tilting theory or cluster algebras, though the paper does not explore such links.
  • Computational verification on small quivers could confirm the multiplicity 2^{|U|} for specific examples.

Load-bearing premise

The TF equivalence is assumed to complete the g-fan, and the tau-tilting reduction at U must be defined for the finite dimensional algebra A with the given 2-term presilting complex U.

What would settle it

An explicit finite-dimensional algebra A together with a 2-term presilting complex U such that the number or structure of TF equivalence classes inside D(U) does not match 2 to the power of the number of summands in U times the number of classes for the reduced algebra B would falsify the correspondence.

read the original abstract

For a finite dimensional algebra $A$, the TF equivalence on the real Grothendieck group $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ can be regarded as a completion of the $g$-fan. For example, the silting cones $C^\circ(U)$ of 2-term presilting complexes $U$ give the most fundamental family of TF equivalence classes. The next step is studying the TF equivalence classes around each silting cone $C^\circ(U)$. Thus, in this paper, we investigate the closed interval neighborhood $D(U)$ of $C^\circ(U)$. As our main result, we give a $2^{|U|}:1$ correspondence between the TF equivalence classes in $D(U)$ and those in $K_0(\operatorname{\mathsf{proj}} B)_\mathbb{R}$, where $B$ is the algebra appearing in the $\tau$-tilting reduction at $U$. For this purpose, we give an explicit description of defining inequalities and the faces of $D(U)$ as a polyhedral cone, by using 2-term simple-minded collections and $M$-TF equivalences.

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 / 2 minor

Summary. For a finite-dimensional algebra A, the paper treats TF equivalence on the real Grothendieck group K_0(proj A)_R as a completion of the g-fan. It studies the closed interval neighborhood D(U) of the silting cone C°(U) associated to a 2-term presilting complex U. The central result is an explicit description of D(U) as a polyhedral cone via its defining inequalities and faces, obtained using 2-term simple-minded collections and M-TF equivalences, together with a 2^{|U|}:1 correspondence between the TF equivalence classes inside D(U) and the TF equivalence classes in K_0(proj B)_R, where B is the algebra obtained by τ-tilting reduction at U.

Significance. If the stated correspondence and the polyhedral description of D(U) are correct, the work supplies a recursive reduction that relates the local structure of TF equivalence classes around each silting cone to the corresponding classes on a smaller algebra. This strengthens the g-fan completion perspective by furnishing an explicit combinatorial bridge between the original and reduced settings, using only standard objects from silting and τ-tilting theory.

minor comments (2)
  1. The abstract and introduction would benefit from a short sentence recalling the precise definition of an M-TF equivalence (or a reference to the section where it is introduced), since the correspondence relies on this notion.
  2. In the description of the faces of D(U), it would be helpful to include a small concrete example (e.g., a hereditary algebra of small rank) that illustrates how the 2^{|U|}:1 map acts on the inequalities.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report, so we have no points requiring detailed rebuttal or clarification. We will incorporate any minor editorial or typographical adjustments in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity in the claimed correspondence or cone description

full rationale

The paper's main result establishes a 2^{|U|}:1 correspondence between TF equivalence classes in the interval neighborhood D(U) and those in the reduced algebra B via the standard τ-tilting reduction. This is supported by an explicit description of the polyhedral cone D(U) using 2-term simple-minded collections and M-TF equivalences, which are defined independently in the literature. The construction treats the g-fan completion as established prior work without reducing the central claim to a self-referential definition or fitted input. All load-bearing steps rely on recalled standard definitions rather than circular self-citations or ansatzes smuggled in. The derivation chain remains self-contained against external benchmarks in representation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information is available from the abstract to list free parameters, axioms, or invented entities. The work presupposes standard notions from representation theory (Grothendieck groups, silting complexes, τ-tilting reduction) whose precise axiomatic status cannot be audited here.

pith-pipeline@v0.9.0 · 5500 in / 1333 out tokens · 55503 ms · 2026-05-10T04:01:07.535135+00:00 · methodology

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Works this paper leans on

56 extracted references · 56 canonical work pages

  1. [1]

    Adachi, O

    T. Adachi, O. Iyama, I. Reiten, -tilting theory, Compos. Math., 150.3 (2014), 415--452

    work page 2014

  2. [2]

    Adachi, H

    T. Adachi, H. Enomoto, M. Tsukamoto, Intervals of s -torsion pairs in extriangulated categories with negative first extensions, Math. Proc. Cambridge Philos. Soc., 174.3 (2023), 451--469

    work page 2023

  3. [3]

    Aihara, Tilting-connected symmetric algebras, Algebr

    T. Aihara, Tilting-connected symmetric algebras, Algebr. Represent. Theory, 16.3 (2013), 873--894

    work page 2013

  4. [4]

    Aihara, O

    T. Aihara, O. Iyama, Silting mutation in triangulated categories, J. Lond. Math. Soc. (2), 85.3 (2012), 633--668

    work page 2012

  5. [5]

    Al-Nofayee, Simple objects in the heart of a t -structure, J

    S. Al-Nofayee, Simple objects in the heart of a t -structure, J. Pure Appl. Algebra, 213.1 (2009), 54--59

    work page 2009

  6. [6]

    Angeleri-H\" u gel, R

    L. Angeleri-H\" u gel, R. Laking, F. Sentieri, Torsion pairs via the Ziegler spectrum, arXiv:2403.00475

    work page arXiv

  7. [7]

    T. Aoki, A. Higashitani, O. Iyama, R. Kase, Y. Mizuno, Fans and polytopes in tilting theory I: Foundations, arXiv:2203.15213v4, to appear in Compos. Math

    work page arXiv

  8. [8]

    Asai, Semibricks, Int

    S. Asai, Semibricks, Int. Math. Res. Not., 2020.16 (2020), 4993--5054

    work page 2020

  9. [9]

    Asai, The wall-chamber structures of the real Grothendieck groups, Adv

    S. Asai, The wall-chamber structures of the real Grothendieck groups, Adv. Math. 381 (2021), Paper No. 107615

    work page 2021

  10. [10]

    Asai, Mutation of simple-minded collections revisited, in preparation

    S. Asai, Mutation of simple-minded collections revisited, in preparation

    work page

  11. [11]

    S. Asai, O. Iyama, Semistable torsion classes and canonical decompositions in Grothendieck groups, Proc. Lond. Math. Soc. (3), 129.5 (2024), e12639

    work page 2024

  12. [12]

    S. Asai, O. Iyama, M -TF equivalence in the real Grothendieck group, arxiv:2404.13232v3

    work page arXiv

  13. [13]

    S. Asai, C. Pfeifer, Wide subcategories and lattices of torsion classes, Algebr. Represent. Theory, 25 (2022), 1611--1629

    work page 2022

  14. [14]

    Auslander, S

    M. Auslander, S. O. Smal , Almost split sequences in subcategories, J. Algebra, 69.2 (1981), 426--454

    work page 1981

  15. [15]

    Barnard, A

    E. Barnard, A. Carroll, S. Zhu, Minimal inclusions of torsion classes, Algebr. Comb., 2.5 (2019), 879--901

    work page 2019

  16. [16]

    Barnard, C

    E. Barnard, C. Defant, E. Hanson, Pop-Stack Operators for Torsion Classes and Cambrian Lattices, arXiv:2312.03959v1

    work page arXiv

  17. [17]

    Baumann, J

    P. Baumann, J. Kamnitzer, P. Tingley, Affine Mirkovi\' c -Vilonen polytopes , Publ. Math. Inst. Hautes \' E tudes Sci., 120 (2014), 113--205

    work page 2014

  18. [18]

    Bazier-Matte, N

    V. Bazier-Matte, N. Chapelier-Laget, G. Douville, K. Mousavand, H. Thomas, E. Y ld r m, ABHY Associahedra and Newton polytopes of -polynomials for cluster algebras of simply laced finite type, J. Lond. Math. Soc. (2), 109.1 (2024), e12817

    work page 2024

  19. [19]

    A. A. Be linson, J. Bernstein, P. Deligne, Faisceaux pervers, Analysis and Topology on Singular Spaces, I, Luminy, 1981, Ast\' e risque 100, Soci\' e t\' e math\' e matique de France, 1982, 5--171

    work page 1981

  20. [20]

    Bridgeland, Scattering diagrams, Hall algebras and stability conditions, Algebr

    T. Bridgeland, Scattering diagrams, Hall algebras and stability conditions, Algebr. Geom., 4.5 (2017), 523--561

    work page 2017

  21. [21]

    Broomhead, R

    N. Broomhead, R. Coelho Sim\ o es, D. Pauksztello, J. Woolf, Simple tilts of length hearts and simple-minded mutation, arXiv:2401.02947v2

    work page arXiv

  22. [22]

    Broomhead, D

    N. Broomhead, D. Pauksztello, D. Ploog, J. Woolf, The heart fan of an abelian category, arXiv:2310.02844v2

    work page arXiv

  23. [23]

    Br\" u stle, D

    T. Br\" u stle, D. Smith, H. Treffinger, Wall and chamber structure for finite-dimensional algebras, Adv. Math., 354 (2019), Paper No. 106746

    work page 2019

  24. [24]

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

    T. Br\"ustle, D. Yang, Ordered exchange graphs, Advances in representation theory of algebras, 135--193, EMS Ser. Congr. Rep., Eur. Math. Soc., Z\"urich, 2013

    work page 2013

  25. [25]

    P. Cao, Y. Wang, H. Zhang, Relative left Bongartz completions and their compatibility with mutations, Math. Z., 305 (2023), article no. 27

    work page 2023

  26. [26]

    D. Cox, J. Little, H. Schenck, Toric varieties, Graduate Studies in Mathematics, 124. American Mathematical Society, Providence, RI, 2011

    work page 2011

  27. [27]

    Demonet, O

    L. Demonet, O. Iyama, G. Jasso, -Tilting Finite Algebras, Bricks, and g-Vectors, Int. Math. Res. Not., 2019.3 (2019), 852--892

    work page 2019

  28. [28]

    R. Dehy, B. Keller, On the Combinatorics of Rigid Objects in 2--Calabi--Yau Categories, Int. Math. Res. Not., 2008 (2008), rnn029

    work page 2008

  29. [29]

    Derksen, J

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

    work page 2015

  30. [30]

    Demonet, O

    L. Demonet, O. Iyama, N. Reading, I. Reiten, H. Thomas, Lattice theory of torsion classes: Beyond -tilting theory, Trans. Amer. Math. Soc. Ser. B, 10 (2023), 542--612

    work page 2023

  31. [31]

    Enomoto, A

    H. Enomoto, A. Sakai, ICE-closed subcategories and wide -tilting modules, Math. Z., 300 (2022), 541--577

    work page 2022

  32. [32]

    Fomin, A

    S. Fomin, A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc., 15.2 (2002), 497--529

    work page 2002

  33. [33]

    Garcia, On thick subcategories of the category of projective presentations, Math

    M. Garcia, On thick subcategories of the category of projective presentations, Math. Z., 312 (2026), article no. 110

    work page 2026

  34. [34]

    Gnedin, Silting theory under change of rings, arXiv:2204.00608v1

    W. Gnedin, Silting theory under change of rings, arXiv:2204.00608v1

    work page arXiv

  35. [35]

    Gupta, Y

    E. Gupta, Y. Zhou, Semibricks and wide subcategories in extended module categories, arXiv:2511.08157v1

    work page arXiv

  36. [36]

    Hanihara, O

    N. Hanihara, O. Iyama, Silting correspondences and Calabi-Yau dg algebras, arXiv:2508.12836v2

    work page arXiv

  37. [37]

    Hanson, K

    E. Hanson, K. Igusa, M. Kim, G. Todorov, Infinitesimal semi-invariant pictures and co-amalgamation, J. Lond. Math. Soc. (2), 109.1 (2024), e12786

    work page 2024

  38. [38]

    Hassoun, S

    S. Hassoun, S. Roy, Admissible intersection and sum property, arXiv:1906.03246v3

    work page arXiv 1906

  39. [39]

    Ingalls, H

    C. Ingalls, H. Thomas, Noncrossing partitions and representations of quivers, Compos. Math., 145.6 (2009), 1533--1562

    work page 2009

  40. [40]

    Iyama, Y

    O. Iyama, Y. Kimura, Classifying subcategories of modules over Noetherian algebras, Adv. Math., 446 (2024), Paper No. 109631

    work page 2024

  41. [41]

    Iyama, D

    O. Iyama, D. Yang, Silting reduction and Calabi--Yau reduction of triangulated categories, Trans. Amer. Math. Soc., 370 (2018), 7861--7898

    work page 2018

  42. [42]

    Jasso, Reduction of -tilting modules and torsion pairs, Int

    G. Jasso, Reduction of -tilting modules and torsion pairs, Int. Math. Res. Not., 2015.16 (2015), 7190--7237

    work page 2015

  43. [43]

    Kaipel, H

    M. Kaipel, H. Treffinger, Wall-and-chamber structures for finite-dimensional algebras and -tilting theory, Representations of Algebras and Related Topics, Proceedings of the Workshop and the 20th International Conference on Representations of Algebras, 261--296

    work page

  44. [44]

    Keller, D

    B. Keller, D. Vossieck, Aisles in derived categories, Deuxi\` e me Contact Franco-Belge en Alg\` e bre (Faulx-les-Tombes, 1987), Bull. Soc. Math. Belg. S\' e r. A, 40.2 (1988), 239--253

    work page 1987

  45. [45]

    Keller, D

    B. Keller, D. Yang, Derived equivalences from mutations of quivers with potential, Adv. Math., 226.3 (2011), 2118--2168

    work page 2011

  46. [46]

    A. D. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford Ser. (2), 45.180 (1994), 515--530

    work page 1994

  47. [47]

    Koenig, D

    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

    work page 2014

  48. [48]

    Marks, J

    F. Marks, J. S t ov\' i c ek, Torsion classes, wide subcategories and localisations, Bull. Lond. Math. Soc., 49.3 (2017), 405--416

    work page 2017

  49. [49]

    Mizuno, Shard theory of g -fans, Int

    Y. Mizuno, Shard theory of g -fans, Int. Math. Res. Not., 2024.19 (2024), 13106--13126

    work page 2024

  50. [50]

    Plamondon, Generic bases for cluster algebras from the cluster category, Int

    P.-G. Plamondon, Generic bases for cluster algebras from the cluster category, Int. Math. Res. Not., 2013.10 (2013), 2368--2420

    work page 2013

  51. [51]

    Plamondon, T

    P.-G. Plamondon, T. Yurikusa, B. Keller, Tame Algebras Have Dense g -Vector Fans, Int. Math. Res. Not., 2023.4 (2023), 2701--2747

    work page 2023

  52. [52]

    Rickard, Morita theory for derived categories, J

    J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2), 39.3 (1989), 436--456

    work page 1989

  53. [53]

    Rudakov, Stability for an abelian category, J

    A. Rudakov, Stability for an abelian category, J. Algebra, 197.1 (1997), 231--245

    work page 1997

  54. [54]

    Schroll, A

    S. Schroll, A. Tattar, H. Treffinger, N. Williams, A geometric perspective on the -cluster morphism category, Math. Z., 312 (2026), article no. 77

    work page 2026

  55. [55]

    Tattar, Torsion Pairs and Quasi-abelian Categories, Algebr

    A. Tattar, Torsion Pairs and Quasi-abelian Categories, Algebr. Represent. Theory, 24 (2021), 1557--1581

    work page 2021

  56. [56]

    Yurikusa, Wide subcategories are semistable, Doc

    T. Yurikusa, Wide subcategories are semistable, Doc. Math., 23 (2018), 35--47

    work page 2018