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Counterexamples to the weakly Gorenstein symmetry conjecture

A mix of human-written and AI-generated textHuman understanding: some partsmath.RT — Representation Theory

Contributed by Qiyue Tang ↗

Version 1 / Oct 21, 2026 / CC BY 4.0

Abstract

We construct a 29-dimensional algebra which is left weakly Gorenstein and is not right weakly Gorenstein. We also construct a 145-dimensional gendo-symmetric algebra with the same property. Both constructions work over every field containing an element of infinite multiplicative order. These examples disprove the weakly Gorenstein symmetry conjecture.

Provenance statement

This result was obtained through approximately three hours of dialogue with GPT-6-Astra-Pro, followed by a Codex review and repeated polishing using our built-in writing skill. This article has been reviewed and approved by Rethlas-OSV (2 concurrent, Astra-Xhigh thinking intensity). Rethlas-OSV: https://github.com/OSVerify/Rethlas-OSV 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

References

  1. Z. Huang, Auslander-type conditions and weakly Gorenstein algebras , arXiv:2408.05468v1 (https://arxiv.org/abs/2408.05468v1), 2024.arXiv
  2. D. A. Jorgensen and L. M. Şega, Asymmetric complete resolutions and vanishing of Ext over Gorenstein rings , Int. Math. Res. Not. 2005 (2005), no. 56, 3459--3477. doi:10.1155/IMRN.2005.3459 (https://doi.org/10.1155/IMRN.2005.3459).DOI
  3. D. A. Jorgensen and L. M. Şega, Independence of the total reflexivity conditions for modules , arXiv:math/0410257v2 (https://arxiv.org/abs/math/0410257v2), 2004.arXiv
  4. R. Marczinzik, Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra , arXiv:1608.04212v2 (https://arxiv.org/abs/1608.04212v2), 2021.arXiv
  5. C. M. Ringel and P. Zhang, Gorenstein-projective and semi-Gorenstein-projective modules , Algebra Number Theory 14 (2020), no. 1, 1--36. Author's manuscript (https://www.math.uni-bielefeld.de/~ringel/opus/jul22.pdf).link
  6. P. Zhang, Monic modules and semi-Gorenstein-projective modules , J. Pure Appl. Algebra 227 (2023), no. 2, 107181. arXiv:2208.05690v1 (https://arxiv.org/abs/2208.05690v1).arXiv

Version history

  1. v1Submitted by Qiyue TangInitial depositCurrentOct 21, 2026