{
  "abstract": "Let $K$ be a function field over an $F$-finite field $k$ of characteristic $p > 0$ and let $ν$ be a valuation of $K/k$. We show $ν$ is an Abhyankar valuation of $K/k$ if and only if the corresponding valuation ring is Frobenius split. The proof proceeds by analyzing when ideals of a valuation ring are uniformly $F$-compatible and establishing a general ramification-theoretic characterization of Frobenius split valuation rings of $F$-finite fields. In addition, when the ground field $k$ is not $F$-finite, we show that the equivalence between Frobenius splitting and the Abhyankar property can fail by constructing an example of an excellent Frobenius split DVR of a function field whose corresponding valuation is not divisorial. Our methods do not rely on local uniformization results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11032",
  "author_total": 2,
  "authors": [
    {
      "name": "Rankeya Datta"
    },
    {
      "name": "Karen E. Smith"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/emk2mf6j188aa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11032",
  "fulltext_url": null,
  "id": "emk2mf6j188aa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11032",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221117628",
  "pdf_url": "https://arxiv.org/pdf/2610.11032",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/emk2mf6j188aa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11032",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11032"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Frobenius splitting of valuation rings revisited",
  "title_text": "Frobenius splitting of valuation rings revisited",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/emk2mf6j188aa",
  "venue": "arXiv",
  "versions": []
}