{
  "abstract": "We study general properties of the perfectoidization of finite algebras over a perfectoid ring, which helps to understand some precise and explicit descriptions. For example, we prove that if $A=R[t]/(m(t))$ where $m(t)$ is monic, $R$ is perfectoid and the discriminant $d$ of $m(t)$ is a non-zero divisor of $R$ satisfying a bounded torsion condition, then $dA_{\\mathrm{pfd}}\\subset A$. We also prove a density criterion reducing the construction of the perfectoidization to adjoining suitable $p$-power roots modulo $p$. In the second part of the paper, we compute perfectoidizations in several families of examples, including Kummer-type extensions and split finite algebras.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.12229",
  "author_total": 2,
  "authors": [
    {
      "name": "Ryo Ishizuka"
    },
    {
      "name": "Léo Navarro Chafloque"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c4vf4fqybp17s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.12229",
  "fulltext_url": null,
  "id": "c4vf4fqybp17s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.12229",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7164345459",
  "pdf_url": "https://arxiv.org/pdf/2606.12229",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c4vf4fqybp17s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2606.12229",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.12229"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Perfectoidization of Finite Algebras over a Perfectoid Ring",
  "title_text": "On Perfectoidization of Finite Algebras over a Perfectoid Ring",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/c4vf4fqybp17s",
  "venue": "arXiv",
  "versions": []
}