{
  "schema_version": 1,
  "problem_number": "OWR-12697690-008",
  "title": "A Kupisch Criterion for Higher Auslander Koszul Duals",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a connected nonsemisimple split basic Nakayama algebra B, let c_i and d_i be the lengths of the projective cover and injective envelope of the same simple module. We prove that the Koszul dual E of its Auslander algebra has global dimension max_i(c_i+d_i-2) and dominant dimension min_i(c_i+d_i-2). Thus the higher Auslander condition is exactly constancy of c_i+d_i, with the dimension-one convention stated separately. The proof combines an explicit graded morphism calculation with the credited Auslander-Koszul transfer theorem of Chan, Iyama and Marczinzik, version 2. We deduce the hereditary characterization for linear Nakayama algebras and a complete two-simple cyclic classification. Nonselfinjective positive examples include (2r+1,2r+2). The (3,4) series has both dimensions equal to 5, contrary to a negative example in the earlier preprint; independent exact quadratic-dual resolutions are supplied. This is a complete classification of the Nakayama subclass, not a solution of the general representation-finite question in UnsolvedMath record OWR-12697690-008. The English five-page AI-assisted preprint includes739 exact radical checks and97 independent minimal-resolution checks. It is unrefereed; no absolute-priority or formal-verification claim is made.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "keywords": [
    "OWR-12697690-008",
    "Nakayama algebra",
    "Kupisch series",
    "coKupisch series",
    "Auslander algebra",
    "Koszul dual",
    "dominant dimension"
  ],
  "manuscript_version_date": "2026-10-26",
  "publication_date": "2026-10-26",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-26",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-12697690-008/",
  "pdf_url": "https://eulersolve.org/papers/owr-12697690-008/paper.pdf?v=d50a7588ff14",
  "doi": "10.5281/zenodo.23671148",
  "zenodo_record_url": "https://zenodo.org/records/23671148",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete classification for connected nonsemisimple split basic Nakayama algebras B: if c_i and d_i are projective and injective lengths at the same simple, the Koszul dual E of the Auslander algebra has global dimension max(c_i+d_i-2) and dominant dimension min(c_i+d_i-2). Hence the higher Auslander condition is constancy of c_i+d_i, with the dimension-one convention stated. Includes the linear hereditary characterization and the two-simple cyclic classification. Complete classification of the connected nonsemisimple split basic Nakayama subclass. The general representation-finite classification in OWR-12697690-008 remains unresolved; no whole source-problem closure is claimed. The Auslander-Koszul transfer theorem is explicitly credited to Chan, Iyama and Marczinzik, arXiv2210.06180v2; classical radical methods are credited. No absolute-priority or formal-verification claim is made. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "d50a7588ff141c0d773f3b5d7951ebfcb992173a9b5248d8bede6abb24d372a2"
    },
    "source.zip": {
      "sha256": "1b07c53cf48e8a73595ad444abbf607ff84d183b4e1538028a1ba211bd5ba024"
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    "verification_report.md": {
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}