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A note on counterexamples to homological conjectures

A mix of human-written and AI-generated textHuman understanding: no partsmath.RT — Representation Theorymath.RA — Rings and Algebras

Contributed by 琪越 唐

Version 1 / Oct 07, 2026 / CC BY 4.0

Abstract

Building on the constructions of OpenAI, we extend these constructions to any base field containing an element of infinite multiplicative order. Over each such field, we obtain finite-dimensional algebras giving counterexamples to both Tachikawa conjectures, the Auslander–Reiten conjecture, the Gorenstein-projective conjecture, the Nakayama, generalized Nakayama, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, the little and big finitistic dimension conjectures, and the Wakamatsu tilting conjecture.

Provenance statement

Based on Open AI’s Result No. 199 released today, we conducted a complete adversarial review using GPT-6-Astra-Pro (the prompts are given below) and attempted to have it generalize the counterexample to a more general domain. After obtaining the results, we independently conducted a strong adversarial review using GPT-6-Astra and used the built-in Writing Skill to polish the writing. Prompt: "Please read our paper and the related supporting materials. I am inclined to believe that this paper has errors or gaps. Please help me identify them. If there are errors, you do not need to correct them. If there are gaps that can be fixed, please fix them and write detailed revision suggestions. 1. Method: Please conduct strong adversarial checking, attacking each Lemma, Prop, Thm, etc. in order. Do not overlook any detail. Do not trust any memory in the historical record. I hope you will re-derive this entire paper completely and meticulously from scratch. When encountering citations, please carefully verify whether the citation scope is correct, and whether there are mis-citations, indiscriminate citations, etc. 2. Check whether the proofs are repetitive and verbose, whether they can be optimized, and whether more generalizations can be made based on this counterexample. (We make generalizations, for example, generalizing 199A to arbitrary fields; for 199B, see whether it can be used to refute other major conjectures, such as SNC, GNC, ARC, NC, etc.) If possible, please include them in the report as well. 3. What do you think of the writing in this paper? Is its readability very poor? If you can improve it further (by imitating the best human mathematical papers), please write out revision suggestions. Finally, just generate one overall report in md." This paper is not intended for journal submission.

References

  1. OpenAI, An explicit counterexample to the Auslander--Reiten conjecture , preprint, September 23, 2026. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/paper.pdf Fixed repository version .link
  2. OpenAI, A counterexample to Tachikawa's second conjecture , preprint, September 23, 2026. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf Fixed repository version . Both OpenAI references use repository commit adc7f1241b42 .link
  3. M. Linckelmann, Tate duality and transfer in Hochschild cohomology , J. Pure Appl. Algebra 217 (2013), 2387--2399. https://arxiv.org/abs/1211.5999v1 arXiv:1211.5999v1 .arXiv
  4. N. Spaltenstein, Resolutions of unbounded complexes , Compos. Math. 65 (1988), 121--154. https://www.numdam.org/item/CM_1988__65_2_121_0/ Original article .link
  5. H. Enomoto and R. Marczinzik, On the homological conjectures for Artin algebras , preprint, September 24, 2026. https://arxiv.org/abs/2609.19172v2 arXiv:2609.19172v2 .arXiv
  6. X. Zhang and P. Zhou, Igusa--Todorov algebras and the Auslander--Reiten conjecture , preprint, September 15, 2026. https://arxiv.org/abs/2609.14453v2 arXiv:2609.14453v2 .arXiv
  7. W. Xia, The Auslander--Reiten conjecture for quantum complete intersections , preprint, September 21, 2026. https://arxiv.org/abs/2609.24007v1 arXiv:2609.24007v1 .arXiv
  8. X. Zhang and P. Zhou, The Auslander--Reiten conjecture for algebras with radical cube zero , preprint, 2026. https://arxiv.org/abs/2609.08679v1 arXiv:2609.08679v1 .arXiv
  9. L. W. Christensen and H. Holm, Algebras that satisfy Auslander's condition on vanishing of cohomology , Math. Z. 265 (2010), 21--40. https://arxiv.org/abs/0801.0401v2 arXiv:0801.0401v2 .arXiv

Version history

  1. v1Submitted by 琪越 唐Initial depositSupersededOct 07, 2026
  2. v2Submitted by 琪越 唐Submitted as a new versionCurrentOct 08, 2026