Growing Spines Ad Infinitum

TL;DR

This work develops spine-based methods to study the model theory of ordered abelian groups, proving that every non-trivial group is augmentable by infinites and establishing a robust framework via coloured multi-orders. The main result leverages spines to show augmentability, yielding corollaries about $p$-divisibility and the structure of initial points in spines. An important application shows that, in characteristic $0$, not being $t$-henselian is equivalent to uniform definability of henselian valuations with a given residue field in the language of rings, connecting value-group theory to definability questions in henselian fields. The results advance understanding of definability of valuations and illuminate how spine and multi-order techniques can control elementary embeddings and definability in valued-field settings.

Abstract

We show that every non-trivial ordered abelian group $G$ is augmentable by infinite elements, i.e., we have $G\preccurlyeq H\oplus G$ for some non-trivial ordered abelian group $H$. As an application, we show that when $k$ is a field of characteristic 0, then $k$ is not $t$-henselian if and only if all henselian valuations with residue field $k$ are ($\emptyset$-)definable.

Theorems & Definitions (50)

• Theorem 1.1: Cf. Theorem \ref{['thm:strongleftaugmentabilityOAG']}
• Theorem 1.2: Cf. Theorem \ref{['def_fieldwise']}
• Definition 2.1
• Definition 2.2
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Corollary 2.7
• Example 2.8