HEXAGONbeta

Cartan determinants for almost laura, quasi-laura, semilinear clannish and piecewise Bäckström algebras

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

Contributed by Qiyue Tang ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

We prove the Cartan determinant conjecture for corners of finite global dimension in almost laura, quasi-laura and (m)-quasi-laura Artin algebras, finite-dimensional piecewise Bäckström algebras, and semilinear clannish Artin algebras of finite dimension over their defining division ring.

Provenance statement

AI systems were used to generate and review the research arguments. Codex was used for language polishing, manuscript preparation and adversarial review of the proofs and their dependencies. The manuscript passed an automated review using Rethlas-OSV, which incorporates OSVerify (Nie et al., arXiv:2610.18089v2; https://github.com/OSVerify/Rethlas-OSV).

References

  1. E. R. Alvares, I. Assem, F. U. Coelho, M. I. Peña and S. Trepode, From trisections in module categories to quasi-directed components , J. Algebra Appl. 10 (2011), 409--433. doi:10.1142/S0219498811004653 (https://doi.org/10.1142/S0219498811004653).DOI
  2. I. Assem and F. U. Coelho, Two-sided gluings of tilted algebras , J. Algebra 269 (2003), 456--479.
  3. I. Assem and F. U. Coelho, Endomorphism algebras of projective modules over laura algebras , J. Algebra Appl. 3 (2004), 49--60. doi:10.1142/S0219498804000691 (https://doi.org/10.1142/S0219498804000691).DOI
  4. R. Bennett-Tennenhaus and W. Crawley-Boevey, Semilinear clannish algebras , Proc. Lond. Math. Soc. (2024), doi:10.1112/plms.12637 (https://doi.org/10.1112/plms.12637); version used: arXiv:2204.12138v3 (https://arxiv.org/abs/2204.12138v3).arXivDOI
  5. C. Bessenrodt and T. Holm, qq-Cartan matrices and combinatorial invariants of derived categories for skewed-gentle algebras , Pacific J. Math. 229 (2007), 25--47.
  6. W. D. Burgess and K. R. Fuller, The Cartan determinant and generalizations of quasihereditary rings , Proc. Edinburgh Math. Soc. 41 (1998), 23--32.
  7. 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
  8. X.-W. Chen, A note on standard equivalences , Bull. Lond. Math. Soc. 48 (2016), 797--801. doi:10.1112/blms/bdw038 (https://doi.org/10.1112/blms/bdw038).DOI
  9. Y. Drozd, Backström algebras , arXiv:2206.12875v2 (https://arxiv.org/abs/2206.12875v2) (2023).arXiv
  10. S. Eilenberg, Algebras of cohomologically finite dimension , Comment. Math. Helv. 28 (1954), 310--319.
Show all 22 referencesShow fewer references
  1. D. Happel, I. Reiten and S. O. Smalø, Piecewise hereditary algebras , Arch. Math. (Basel) 66 (1996), 182--186. doi:10.1007/BF01195702 (https://doi.org/10.1007/BF01195702).DOI
  2. T. Holm, Cartan determinants for gentle algebras , Arch. Math. (Basel) 85 (2005), 233--239.
  3. C. Ingalls and C. Paquette, Homological behavior of idempotent subalgebras and Ext algebras , arXiv:1703.08725v2 (https://arxiv.org/abs/1703.08725v2) (2017).arXiv
  4. F. Nie, H. Ye, S. Liu, Z. Wang, S. Ermon and J. Zou, One-Sided Verification Under Limited Expert Review , preprint (2026), arXiv:2610.18089v2 (https://arxiv.org/abs/2610.18089v2).arXiv
  5. OSVerify, Rethlas-OSV , software repository, https://github.com/OSVerify/Rethlas-OSV, accessed 3 November 2026.link
  6. J. A. de la Peña and I. Reiten, Trisections of module categories , Colloq. Math. 107 (2007), 191--219.
  7. Y. Qin and Y. Han, Reducing homological conjectures by nn-recollements , Algebr. Represent. Theory 19 (2016), 377--395; arXiv:1410.3223v2 (https://arxiv.org/abs/1410.3223v2).arXiv
  8. I. Reiten and A. Skowroński, Generalized double tilted algebras , J. Math. Soc. Japan 56 (2004), 269--288.
  9. D. Smith, Almost laura algebras , J. Algebra 319 (2008), 432--456; arXiv:math/0702562v3 (https://arxiv.org/abs/math/0702562v3).arXiv
  10. Q. Tang, A note of the Cartan determinant conjecture , hexagon:2610.00537v1 (https://hexagonmath.org/2610.00537v1) (2026).hexagon
  11. G. V. Wilson, The Cartan map on categories of graded modules , J. Algebra 85 (1983), 390--398.
  12. 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 depositCurrentNov 03, 2026