$\delta$-lifting and $1$-dimensional analytic fields

TL;DR

This paper investigates conditions under which the valuation ring of a completed one-dimensional field extension admits a flat lifting over a certain algebraic structure, revealing a precise criterion related to the field's type.

Contribution

It establishes a necessary and sufficient condition for the existence of a flat $ ext{ extdelta}$-lifting of valuation rings in one-dimensional non-Archimedean fields.

Findings

Valuation ring admits flat $ extdelta$-lifting iff field is not of type 4.

Provides a characterization of $ extdelta$-liftings in one-dimensional analytic fields.

Connects field type classification with lifting properties.

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 $\delta$-lifting over $\mathbb{A}_{\mathrm{inf}}(k^+)$ if and only if $\mathcal{K}$ is not of type 4.