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Cartan determinants in finite representation type over arbitrary fields

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

Let AA be a finite-dimensional representation-finite algebra over an arbitrary field, and let MM be a nonzero finite-dimensional right AA-module. We prove that End⁡A(M)\operatorname{End}_A(M) has Cartan determinant one whenever it has finite global dimension. In particular, the Cartan determinant conjecture holds for representation-finite algebras over arbitrary fields.

Provenance statement

This paper was roughly produced by running AI for about three hours and incorporating previous papers (recent papers have already been strictly reviewed by the AI). At the same time, the text has undergone independent, strongly adversarial reviews by GPT-6-Astra-Pro, as well as correctness verification by Rethlas-OSV (2 parallel runs, with reasoning intensity set to GPT-6-Astra-Xhigh). Rethlas-OSV: https://github.com/OSVerify/Rethlas-OSV

References

  1. L. Chen, The Cartan determinant conjecture for representation-finite algebras over perfect fields , preprint (2026), arXiv:2610.14638v1 (https://arxiv.org/abs/2610.14638v1).arXiv
  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. B. Doherty, E. Faber, and C. Ingalls, Computing global dimension of endomorphism rings via ladders , J. Algebra 458 (2016), 307--350. doi:10.1016/j.jalgebra.2016.03.020 (https://doi.org/10.1016/j.jalgebra.2016.03.020).DOI
  4. H. Enomoto, The Cartan determinant conjecture for representation-finite algebras , preprint (2026), arXiv:2608.23132v1 (https://arxiv.org/abs/2608.23132v1).arXiv
  5. O. Iyama, The relationship between homological properties and representation theoretic realization of artin algebras , Trans. Amer. Math. Soc. 357 (2005), 709--734. doi:10.1090/S0002-9947-04-03482-8 (https://doi.org/10.1090/S0002-9947-04-03482-8). Section references follow arXiv:math/0307392v1 (https://arxiv.org/abs/math/0307392v1) (2003).arXivDOI
  6. O. Iyama and M. Wemyss, The classification of special Cohen--Macaulay modules , Math. Z. 265 (2010), 41--83; arXiv:0809.1958v3 (https://arxiv.org/abs/0809.1958v3).arXiv
  7. G. V. Wilson, The Cartan map on categories of graded modules , J. Algebra 85 (1983), 390--398.

Version history

  1. v1Submitted by Qiyue TangInitial depositSupersededOct 19, 2026
  2. v2Submitted by Qiyue TangSubmitted as a new versionCurrentOct 20, 2026