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PreprintCC BY-NC-ND 4.0arXiv

Frobenius splitting of valuation rings revisited

  • Rankeya Datta
  • Karen E. Smith
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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.

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Cite

BibTeXRIS
BibTeX
@misc{datta2026frobenius,
  title = {{Frobenius splitting of valuation rings revisited}},
  author = {Rankeya Datta and Karen E. Smith},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11032},
  eprint = {2610.11032},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11032},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Frobenius splitting of valuation rings revisited
AU  - Rankeya Datta
AU  - Karen E. Smith
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11032
UR  - https://arxiv.org/abs/2610.11032
AB  - 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.
ER  -