{
  "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 \\(\\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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24007",
  "author_total": 1,
  "authors": [
    {
      "name": "Weiheng Xia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2jpt69hq13ger.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24007",
  "fulltext_url": null,
  "id": "2jpt69hq13ger",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24007",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7214056758",
  "pdf_url": "https://arxiv.org/pdf/2609.24007",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2jpt69hq13ger.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24007",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24007"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Auslander--Reiten conjecture for quantum complete intersections",
  "title_text": "The Auslander--Reiten conjecture for quantum complete intersections",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/2jpt69hq13ger",
  "venue": "arXiv",
  "versions": []
}