HEXAGONbeta

A note of the Cartan determinant conjecture

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

Contributed by Qiyue Tang ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

We prove the Cartan determinant conjecture for every corner of finite global dimension in an Artin algebra of global dimension at most two. Consequently, the conjecture holds for torsionless-finite Artin algebras.

Provenance statement

The above results are partial results extracted from several lengthy, multi-turn conversations with AI. They have first been compiled into a Note for reference. The text has undergone some polishing and revision, and the content has been subjected to independent adversarial review by AI.

References

  1. W. D. Burgess and K. R. Fuller, The Cartan determinant and generalizations of quasihereditary rings , Proc. Edinburgh Math. Soc. 41 (1998), 23--32. doi:10.1017/S0013091500019404 (https://doi.org/10.1017/S0013091500019404).DOI
  2. L. Chen and Q. Guo, The Cartan determinant conjecture for representation-finite Artin algebras , preprint (2026), arXiv:2610.14638v2 (https://arxiv.org/abs/2610.14638v2).arXiv
  3. S. Eilenberg, Algebras of cohomologically finite dimension , Comment. Math. Helv. 28 (1954), 310--319.
  4. C. Ingalls and C. Paquette, Homological behavior of idempotent subalgebras and Ext algebras , arXiv:1703.08725v2 (https://arxiv.org/abs/1703.08725v2) (2017).arXiv
  5. C. M. Ringel, On the representation dimension of artin algebras , arXiv:1107.1861 (https://arxiv.org/abs/1107.1861) (2011); author's version (https://www.math.uni-bielefeld.de/~ringel/opus/repdim3.pdf).arXiv
  6. D. Zacharia, On the Cartan matrix of an Artin algebra of global dimension two , J. Algebra 82 (1983), 353--357. doi:10.1016/0021-8693(83)90156-4 (https://doi.org/10.1016/0021-8693(83)90156-4).DOI

Version history

  1. v1Submitted by Qiyue TangInitial depositCurrentOct 25, 2026