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A note of refinements to an algebra of infinite little finitistic dimension

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

Contributed by 琪越 唐 ↗claimed

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Version 5 / Oct 07, 2026 / CC BY 4.0

Abstract

Building on OpenAI’s An algebra of infinite little finitistic dimension, we simplify the exposition and extend the construction to every field. 2026/10/10：We have added formal verification：https://github.com/tqyqiqi1509/Note198-formalization

Provenance statement

Based on Result No. 198 released by OpenAI, we first used GPT-6-Astra-Pro to conduct a full adversarial review (see the prompts below), and also asked it to make some strengthenings. After obtaining the review comments and strengthenings, we independently used GPT-6-Astra to conduct an adversarial review. Finally, using the built-in writing Skill, we refined the original proof and added strengthenings. Prompt: “Please read our paper and the related supporting materials. I am inclined to think 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. Method: Please conduct a strong adversarial check, attacking each Lemma, Prop, Thm, etc. in order, without letting any detail slip, and do not trust any memory in the historical records. I hope you will re-derive the entire paper completely and meticulously. When encountering citations, please carefully verify whether the citation scope is correct, whether there are mis-citations, indiscriminate citations, etc. 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 want to generalize). If so, please include them in the report as well. What do you think of the writing in this paper? Is its readability very poor? If you can improve it further (by emulating the best human mathematical papers), please write revision suggestions. Finally, just generate an overall report in md.” Uploading this paper to this platform does not mean that I claim ownership or copyright over it. My sole intention is to share this AI-generated solution and make it available for independent review. Once its correctness has been independently verified, the paper may be cited directly. I welcome anyone who would like to organize this work; if you do so, the credit will be yours. I also welcome further research building on this work, including extensions, new applications, and related questions. For additional references or detailed research materials, please contact me by email. Email: tangqy24@mails.tsinghua.edu.cn

Tools used

OpenAI
ChatGPTVersion GPT-6-Astra
Lean
LeanVersion 4.34.1

References

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  7. OpenAI . An algebra of infinite little finitistic dimension. OpenAI Math preprint (https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026), 2026. September 23, 2026. Source version: repository commit adc7f1241b42e322a6451854ab7e4b4c146bf78a .link
  8. Jeremy Rickard. Unbounded derived categories and the finitistic dimension conjecture. Advances in Mathematics , 354:106735, 2019. Publisher DOI (https://doi.org/10.1016/j.aim.2019.106735); arXiv:1804.09801v1 (https://arxiv.org/abs/1804.09801v1).arXivDOI
  9. Yuri Santos Rego. On the finiteness length of some soluble linear groups. Canadian Journal of Mathematics , 74(5):1209--1243, 2022. Publisher DOI (https://doi.org/10.4153/S0008414X21000213); arXiv:1901.06704v3 (https://arxiv.org/abs/1901.06704v3).arXivDOI
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  1. Birge Zimmermann Huisgen. The finitistic dimension conjectures---a tale of 3.5 decades. In Alberto Facchini and Claudia Menini, editors, Abelian Groups and Modules , volume 343 of Mathematics and Its Applications , pages 501--517. Kluwer Academic Publishers, Dordrecht, 1995. Publisher DOI (https://doi.org/10.1007/978-94-011-0443-2_41); arXiv:1407.2383v1 (https://arxiv.org/abs/1407.2383v1).arXivDOI

Version history

  1. v1Submitted by 琪越 唐Initial depositSupersededOct 07, 2026
  2. v2Submitted by 琪越 唐Submitted as a new versionSupersededOct 07, 2026
  3. v3Submitted by 琪越 唐Submitted as a new versionSupersededOct 08, 2026
  4. v4Submitted by 琪越 唐Submitted as a new versionSupersededOct 10, 2026
  5. v5Submitted by 琪越 唐Submitted as a new versionCurrentOct 13, 2026