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$δ$-lifting and $1$-dimensional analytic fields

Jiahong Yu

TL;DR

The work determines when the valuation ring of a one-dimensional analytic field over a complete non-Archimedean base admits a flat $\delta$-lifting over $A_{\inf}$, tying liftability to the Abhyankar (not type $4$) condition. It establishes that, in characteristic $p$, this liftability is equivalent to $F$-splitting and that type $2$ points correspond to $F$-finite $\mathcal{K}^+/p$ (and finite Frobenius). The paper develops a rubric linking prismatization, de Rham–Frobenius theory, and Berkovich geometry, using Temkin's uniformization and deformation theory to extend the liftability criteria to mixed characteristic. A key correction to prior results on $F$-finiteness is provided via explicit counterexamples, and the results have implications for prismatization of geometric valuation rings in $p$-adic Hodge theory.

Abstract

Let $k$ be an algebraically closed complete non-Archimedean field, and let $K$ be a finitely generated field extension over $k$ with transcendence degree $1$. Equip $K$ a non-Archimedean norm extending the one on $k$, and let $\mathcal{K}$ denote the completion of $K$. We will prove that the valuation ring $\mathcal{K}^+$ admits a flat $δ$-lifting over $\mathbb{A}_{\mathrm{inf}}(k^+)$ if and only if $\mathcal{K}$ is not of type 4.

$δ$-lifting and $1$-dimensional analytic fields

TL;DR

The work determines when the valuation ring of a one-dimensional analytic field over a complete non-Archimedean base admits a flat -lifting over , tying liftability to the Abhyankar (not type ) condition. It establishes that, in characteristic , this liftability is equivalent to -splitting and that type points correspond to -finite (and finite Frobenius). The paper develops a rubric linking prismatization, de Rham–Frobenius theory, and Berkovich geometry, using Temkin's uniformization and deformation theory to extend the liftability criteria to mixed characteristic. A key correction to prior results on -finiteness is provided via explicit counterexamples, and the results have implications for prismatization of geometric valuation rings in -adic Hodge theory.

Abstract

Let be an algebraically closed complete non-Archimedean field, and let be a finitely generated field extension over with transcendence degree . Equip a non-Archimedean norm extending the one on , and let denote the completion of . We will prove that the valuation ring admits a flat -lifting over if and only if is not of type 4.

Paper Structure

This paper contains 15 sections, 32 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Motivation from $p$-adic Hodge Theory: Prismatization of Geometric Valuation Rings
  3. Main results: Answering Question \ref{['ques: delta lift of val ring']} for one dimensional analytic field
  4. Acknowledgement
  5. Preliminaries
  6. General theory of valuation rings
  7. Commutative algebra in positive characteristic
  8. $\delta$-liftings
  9. Berkovich $\mathbb{A}^1$
  10. Proof of the main theorems
  11. From $F$-lifting to $F$-splitting
  12. A key theorem
  13. Proof of Theorem \ref{['main theo in intro']}
  14. Proof of Theorem \ref{['theo: main 2 f finite']}
  15. Appendix: Description of the ring of integers of Gauss norms

Key Result

Theorem 1.5

With the notation as above, the following are equivalent: If in addition $\mathrm{char}(k)=p$, the above conditions are further equivalent to $\mathcal{K}^+$ being $F$-split.

Theorems & Definitions (79)

  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5: Subsection \ref{['pr to main 1']}
  • Theorem 1.6: Subsection \ref{['pf to main 2']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • ...and 69 more