{
  "abstract": "We construct an affine geometry for commutative $\\mathbb F_1$-algebras in the framework of Connes and Consani. Using their localization construction, we equip the Deitmar spectrum of the underlying multiplicative monoid $A(1_+)$ with a sheaf $\\mathcal O_A$ of $\\mathbb F_1$-algebras. We prove that $\\mathcal O_A(D(f))\\cong A_f$ and, in particular, that global sections recover $A$. The resulting spectrum construction yields an anti-equivalence between commutative $\\mathbb F_1$-algebras and absolute affine schemes. We compare our choice of covering families with the Connes--Consani topology and explain its role in affine reconstruction. Finally, we describe base change to $\\mathbb Z$, under which the absolute affine scheme associated to the Eilenberg--MacLane algebra $HR$ recovers the classical affine scheme $\\operatorname{Spec}R$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.04843",
  "author_total": 1,
  "authors": [
    {
      "name": "Luqiao Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a85fe23vs7kzy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.04843",
  "fulltext_url": null,
  "id": "a85fe23vs7kzy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.04843",
  "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.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7167581881",
  "pdf_url": "https://arxiv.org/pdf/2607.04843",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a85fe23vs7kzy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2607.04843",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.04843"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Localization and Affine Schemes over $\\mathbb{F}_1$",
  "title_text": "Localization and Affine Schemes over 𝔽_1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/a85fe23vs7kzy",
  "venue": "arXiv",
  "versions": []
}