HEXAGONbeta

Reduction of the Gorenstein symmetry conjecture to finite rings

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

Contributed by Qiyue Tang

Version 2 / Oct 20, 2026 / CC BY 4.0

Abstract

We prove that the Gorenstein symmetry conjecture for Artin algebras is equivalent to its restriction to finite rings of prime-power cardinality, preserving the finite self-injective dimension of a counterexample. For finite-dimensional algebras over fields, the conjecture reduces to finite prime fields. In characteristic zero, a counterexample yields counterexamples of one fixed dimension over every prime field outside a finite set.

Provenance statement

Some of the results in this paper were obtained by AI. The result has passed Rethlas-OSV’s strong adversarial review (2-way parallel, reasoning intensity: astra-Xhigh).

References

  1. M. Auslander, I. Reiten, and S. O. Smalø, Representation Theory of Artin Algebras , Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, Cambridge, 1995.
  2. M. Hoshino, On self-injective dimensions of artinian rings , Tsukuba J. Math. 18 (1994), no. 1, 1--8. author's article (https://tsukuba.repo.nii.ac.jp/records/16259).link
  3. Y. Han, Y. Qin, and X. Wang, Singular equivalences of nn-adjoint type and standard eventually homological isomorphisms , arXiv:2609.03365v2 (https://arxiv.org/abs/2609.03365v2), 17 October 2026.arXiv
  4. K. Igusa, Comparison of views on finitistic projective dimension , lecture notes, corrected Montevideo version (2010), author's manuscript (https://people.brandeis.edu/~igusa/Papers/Uruguay2010b.pdf).link
  5. Y. Y. Qin and Y. Han, Reducing homological conjectures by nn-recollements , Algebr. Represent. Theory 19 (2016), 377--395. doi:10.1007/s10468-015-9578-z (https://doi.org/10.1007/s10468-015-9578-z). Preprint: arXiv:1410.3223v2 (https://arxiv.org/abs/1410.3223v2).arXivDOI
  6. The Stacks Project Authors, The Stacks Project , Section 10.160 (https://stacks.math.columbia.edu/tag/0323) and Section 16.5 (https://stacks.math.columbia.edu/tag/07CJ), accessed 27 October 2026.link
  7. O. Zariski, A new proof of Hilbert's Nullstellensatz , Bull. Amer. Math. Soc. 53 (1947), 362--368. doi:10.1090/S0002-9904-1947-08801-7 (https://doi.org/10.1090/S0002-9904-1947-08801-7).DOI
  8. Y. Zhang, Excellent extensions and homological conjectures , Algebra Colloq. 25 (2018), no. 4, 619--626. doi:10.1142/S1005386718000433 (https://doi.org/10.1142/S1005386718000433). Preprint: arXiv:1702.05902 (https://arxiv.org/abs/1702.05902).arXivDOI

Version history

  1. v1Submitted by Qiyue TangInitial depositSupersededOct 19, 2026
  2. v2Submitted by Qiyue TangSubmitted as a new versionCurrentOct 27, 2026