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Pretorsion theories and image-closed subcategories

Primarily AI-generated textHuman understanding: some partsmath.RT — Representation Theorymath.CO — Combinatorics

Contributed by Haruhisa Enomoto ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

A pretorsion theory generalizes a torsion pair by allowing its two components to have a nonzero intersection, with morphisms factoring through the intersection playing the role of zero morphisms. In the module category of a finite-dimensional algebra, the first and second components of pretorsion theories are known to be the quotient-closed and submodule-closed subcategories, respectively. We show that the intersections of the two components are precisely the image-closed subcategories, namely those containing the image of every morphism between their objects, and that the pretorsion theories form a complete lattice. For a linearly oriented quiver of type A, we prove that the numbers of pretorsion theories and of image-closed subcategories are the Genocchi and median Genocchi numbers, respectively. Counting pretorsion theories by the number of indecomposable modules in their intersection, and image-closed subcategories by their number of indecomposable modules, yields the Gandhi polynomial refinements of Han and Zeng.

Author comments: Currently we are working Lean formalization, which will be released soon.

Provenance statement

- I directed Sol and later Astra to develop theory of the pretorsion theory and its relation to image-closed subcategories, and enumeration problem for type A - Sol proved almost all materials - Astra, Opus 5.5, Argon 4 wrote and revised manuscript heavily

Tools used

OpenAI
CodexVersion GPT-5.6-Sol
OpenAI
CodexVersion GPT-6-Astra
Anthropic
Claude OpusVersion Opus 5.5
Google
GeminiVersion Argon 4

References

  1. M. Auslander and S. O. Smalø, Preprojective modules over Artin algebras , J. Algebra 66 (1980), no. 1, 61--122.
  2. D. Barsky, Congruences pour les nombres de Genocchi de 2e espèce , Groupe d'étude d'analyse ultramétrique, 8e année (1980/81), Exposé no. 34, 13 pp.
  3. F. Campanini and F. Fedele, Building pretorsion theories from torsion theories , Algebr. Represent. Theory 28 (2025), 737--753.
  4. F. Campanini, F. Fedele, and E. Y ld r m, Lattices of pretorsion classes , arXiv:2511.19223v1 (2025).arXiv
  5. L. Carlitz, A conjecture concerning Genocchi numbers , Kongelige Norske Videnskabers Selskab Skrifter (Trondheim) (1971), no. 9, 1--4.
  6. G. Cerulli Irelli, E. Feigin, and M. Reineke, Quiver Grassmannians and degenerate flag varieties , Algebra Number Theory 6 (2012), no. 1, 165--194.
  7. P. H. Edelman and R. E. Jamison, The theory of convex geometries , Geom. Dedicata 19 (1985), no. 3, 247--270.
  8. H. Enomoto, An equidistribution conjecture for quotient-closed and submodule-closed subcategories , arXiv:2608.18024v1 (2026).arXiv
  9. H. Enomoto and A. Sakai, Image-extension-closed subcategories of module categories of hereditary algebras , J. Pure Appl. Algebra 227 (2023), no. 9, Paper No. 107372.
  10. A. Facchini and C. A. Finocchiaro, Pretorsion theories, stable category and preordered sets , Ann. Mat. Pura Appl. (4) 199 (2020), no. 3, 1073--1089.
Show all 21 referencesShow fewer references
  1. A. Facchini, C. A. Finocchiaro, and M. Gran, Pretorsion theories in general categories , J. Pure Appl. Algebra 225 (2021), no. 2, Paper No. 106503.
  2. E. Feigin, The median Genocchi numbers, qq-analogues and continued fractions , European J. Combin. 33 (2012), no. 8, 1913--1918.
  3. J. M. Gandhi, A conjectured representation of Genocchi numbers , Amer. Math. Monthly 77 (1970), 505--506.
  4. G.-N. Han and J. Zeng, qq-Polynômes de Gandhi et statistique de Denert , Discrete Math. 205 (1999), 119--143.
  5. G.-N. Han and J. Zeng, On a qq-sequence that generalizes the median Genocchi numbers , Ann. Sci. Math. Québec 23 (1999), no. 1, 63--72.
  6. C. Ingalls and H. Thomas, Noncrossing partitions and representations of quivers , Compos. Math. 145 (2009), no. 6, 1533--1562.
  7. OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences , entry A110501, https://oeis.org/A110501.link
  8. OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences , entry A005439, https://oeis.org/A005439.link
  9. V. Pilaud, Plumbing bijections , arXiv:2512.05001v2 (2025).arXiv
  10. J. Riordan and P. R. Stein, Proof of a conjecture on Genocchi numbers , Discrete Math. 5 (1973), 381--388.
  11. H. Thomas, The Tamari lattice as it arises in quiver representations , in Associahedra, Tamari Lattices and Related Structures, Prog. Math. 299 , Birkhäuser, Basel, 2012, 281--291.

Version history

  1. v1 · 35 pagesSubmitted by Haruhisa EnomotoCurrently we are working Lean formalization, which will be released soon.CurrentNov 08, 2026