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The Benson-Etingof-Ostrik conjecture

Primarily AI-generated textHuman understanding: some partsmath.RT — Representation Theorymath.CT — Category Theory

Contributed by Kevin Coulembier ↗, Pavel Etingof ↗, Johannes Flake ↗, Joseph Newton ↗, Victor Ostrik

Version 1 / Nov 11, 2026 / CC BY-SA 4.0

Abstract

We prove the conjecture of Benson--Etingof--Ostrik (2023): every symmetric tensor category of moderate growth over an algebraically closed field of positive characteristic admits a symmetric tensor functor to a higher Verlinde category, possibly of infinite height. Using maximal nilpotence and geometric reductivity, we reduce the problem to proving that finitely generated maximally nilpotent tensor categories are finite. The main ingredients are the construction of generic points using alternating norms and a Frobenius reduction of the fundamental coend to an ordinary Hopf subalgebra.

Author comments: The authors plan to turn this AI generated proof into a human paper.

Provenance statement

The paper was generated by GPT-6 Astra. The contributors asked questions, suggested reduction arguments, and shared proof ideas based on their experience from previous projects and on discussions among the contributors.

Tools used

OpenAI
CodexVersion 6 Astra
Anthropic
Claude FableVersion 5.2
Julia
julia
Python
python

References

  1. D. Benson and P. Etingof, Symmetric tensor categories in characteristic 2 , Adv. Math. 351 (2019), 967--999. arXiv:1807.05549 (https://arxiv.org/abs/1807.05549).arXiv
  2. D. Benson, P. Etingof, and V. Ostrik, New incompressible symmetric tensor categories in positive characteristic , Duke Math. J. 172 (2023), no. 1, 105--200. arXiv:2003.10499 (https://arxiv.org/abs/2003.10499).arXiv
  3. K. Coulembier, P. Etingof, and V. Ostrik, On Frobenius exact symmetric tensor categories , with Appendix A by A. Kleshchev, Ann. of Math. (2) 197 (2023), no. 3, 1235--1279. arXiv:2107.02372 (https://arxiv.org/abs/2107.02372).arXiv
  4. K. Coulembier, P. Etingof, and V. Ostrik, Incompressible tensor categories , Adv. Math. 457 (2024), 109935. arXiv:2306.09745 (https://arxiv.org/abs/2306.09745).arXiv
  5. K. Coulembier and J. Flake, The Benson--Etingof--Ostrik conjecture for finite tensor categories , preprint (2026). arXiv:2610.04096 (https://arxiv.org/abs/2610.04096).arXiv
  6. K. Coulembier, J. Flake, J. Newton, and V. Ostrik, Maximal nilpotence for higher Verlinde categories , paper in preparation.
  7. K. Coulembier and A. Sherman, Homogeneous spaces in tensor categories , arXiv:2505.04848 (https://arxiv.org/abs/2505.04848) (2025; revised 2026).arXiv
  8. K. Coulembier and A. Sherman, Absolutely flat algebras in tensor categories , arXiv:2608.13137 (https://arxiv.org/abs/2608.13137) (2026).arXiv
  9. P. Etingof and S. Gelaki, Finite symmetric integral tensor categories with the Chevalley property , with an appendix by K. Coulembier and P. Etingof, Int. Math. Res. Not. IMRN 2021, no. 12, 9083--9121. arXiv:1901.00528 (https://arxiv.org/abs/1901.00528).arXiv
  10. P. Etingof and S. Gelaki, Finite symmetric tensor categories with the Chevalley property in characteristic 2 , J. Algebra Appl. 20 (2021), 2140010. arXiv:1904.07576 (https://arxiv.org/abs/1904.07576).arXiv
Show all 14 referencesShow fewer references
  1. P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor categories , Mathematical Surveys and Monographs, vol. 205, American Mathematical Society, Providence, RI, 2015. Authors' final version (https://math.mit.edu/~etingof/egnobookfinal.pdf).link
  2. P. Etingof and V. Ostrik, On the Frobenius functor for symmetric tensor categories in positive characteristic , J. Reine Angew. Math. 773 (2021), 165--198. arXiv:1912.12947 (https://arxiv.org/abs/1912.12947).arXiv
  3. G. Garkusha, Grothendieck categories , preprint (1999). arXiv:math/9909030 (https://arxiv.org/abs/math/9909030).arXiv
  4. M. Takeuchi, Finite Hopf algebras in braided tensor categories , J. Pure Appl. Algebra 138 (1999), no. 1, 59--82. doi:10.1016/S0022-4049(97)00207-7 (https://doi.org/10.1016/S0022-4049(97)00207-7).DOI

Version history

  1. v1 · 19 pagesSubmitted by Johannes FlakeThe authors plan to turn this AI generated proof into a human paper.CurrentNov 10, 2026