{
  "abstract": "In this paper, we study perfectoid ideals using the reduction modulo $p$ of perfectoid rings. We show that perfectoid ideals correspond to ideals in the reduction modulo $p$ on which Frobenius is surjective. We establish the existence of universal $p$-adic power series whose evaluations, together with their compatible $p$-power roots, generate perfectoidizations of ideals. In the special case of the perfectoidization of $\\mathbb{Z}_p^{\\mathrm{cycl}}\\langle t^{1/p^{\\infty}}\\rangle/(t-1)$, we describe a linear algebra algorithm for computing finite approximations of such generators. We illustrate this by an explicit approximation modulo $8$ of an element $σ\\in (t-1)_{\\mathrm{perfd}}\\subseteq \\mathbb{Z}_2^{\\mathrm{cycl}}\\langle t^{1/2^{\\infty}}\\rangle$ which admits compatible sequences of $2$-power roots and satisfies $σ\\equiv t-1\\pmod{2}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31964",
  "author_total": 1,
  "authors": [
    {
      "name": "Léo Navarro Chafloque"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c26bt8hp76hk1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31964",
  "fulltext_url": null,
  "id": "c26bt8hp76hk1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31964",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "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"
    ],
    [
      "arxiv",
      "math.AG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.31964",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c26bt8hp76hk1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.31964",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31964"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On perfectoid ideals",
  "title_text": "On perfectoid ideals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/c26bt8hp76hk1",
  "venue": "arXiv",
  "versions": []
}