{
  "abstract": "A finite poset is the prime spectrum of a Noetherian ring exactly when it has no chain of three elements, that is, it is the poset of a finite bipartite graph. This answers Conjecture 5.3 of Alkass. The substance is a construction: minimal elements become curves y=g_i(x), maximal elements points, edges the incidences, arranged by interpolation. The coordinate ring of the union of the curves, localized at the chosen points, is essentially of finite type over Q and of dimension at most one. Dropping finiteness costs two conditions, both isolated by Colbert: a poset with no chain of three elements is a Noetherian spectrum exactly when it has finitely many minimal elements and any two distinct ones have finitely many common upper bounds. A variant keeping the shape of the ideals of Alkass but changing their exponents shows why that construction produces only nested neighborhoods. Constraining the ring destroys most of this freedom: normal rings leave finite disjoint unions of stars, domains a single star, local rings a point or a finite co-star. In height two the question is open in general, but the conditions conjectured by Wiegand and Wiegand are not sufficient",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33675",
  "author_total": 1,
  "authors": [
    {
      "name": "Xin Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2tq7akkfshm1j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33675",
  "fulltext_url": null,
  "id": "2tq7akkfshm1j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33675",
  "language": null,
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  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
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      "arxiv",
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    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
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  ],
  "openalex": "W7224000908",
  "pdf_url": "https://arxiv.org/pdf/2610.33675",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2tq7akkfshm1j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33675",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33675"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Which posets are spectra of Noetherian rings? The case of height at most one",
  "title_text": "Which posets are spectra of Noetherian rings? The case of height at most one",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/2tq7akkfshm1j",
  "venue": "arXiv",
  "versions": []
}