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PreprintCC BY 4.0arXiv

The Auslander--Reiten conjecture for quantum complete intersections

  • Weiheng Xia
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Abstract

Let A be a finite-dimensional quantum complete intersection over an arbitrary field. We prove that a finite-dimensional left A-module M is projective whenever ExtA1(M,M)=ExtA2(M,M)=0, with no restriction on the multiplicative orders of the commutation parameters. Consequently, A satisfies the Auslander--Reiten conjecture and Tachikawa's second conjecture. The proof combines the gradability of rigid modules with successive graded twists and the self-extension criterion of Avramov and Buchweitz for commutative complete intersections. We derive a formula comparing ordinary self-extension groups under graded twisting, apply the resulting criterion to the Liu--Schulz algebras, and describe the successive twists explicitly in the four-generator case.

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Cite

BibTeXRIS
BibTeX
@misc{xia2026auslanderreiten,
  title = {{The Auslander--Reiten conjecture for quantum complete intersections}},
  author = {Weiheng Xia},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24007},
  eprint = {2609.24007},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24007},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Auslander--Reiten conjecture for quantum complete intersections
AU  - Weiheng Xia
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24007
UR  - https://arxiv.org/abs/2609.24007
AB  - Let A be a finite-dimensional quantum complete intersection over an arbitrary field. We prove that a finite-dimensional left A-module M is projective whenever Ext_A^1(M,M)=Ext_A^2(M,M)=0, with no restriction on the multiplicative orders of the commutation parameters. Consequently, A satisfies the Auslander--Reiten conjecture and Tachikawa's second conjecture. The proof combines the gradability of rigid modules with successive graded twists and the self-extension criterion of Avramov and Buchweitz for commutative complete intersections. We derive a formula comparing ordinary self-extension groups under graded twisting, apply the resulting criterion to the Liu--Schulz algebras, and describe the successive twists explicitly in the four-generator case.
ER  -