The model theory of Cohen rings

TL;DR

This paper provides a comprehensive account of Cohen rings, generalizing Witt rings to imperfect residue fields, and establishes model-theoretic results that extend Ax-Kochen/Ershov theorems to these structures.

Contribution

It introduces a self-contained algebraic and model-theoretic framework for Cohen rings, including relative completeness and model completeness results.

Findings

Proves relative completeness for Cohen rings

Establishes relative model completeness for Cohen rings

Extends Ax-Kochen/Ershov theorems to imperfect residue fields

Abstract

The aim of this article is to give a self-contained account of the algebra and model theory of Cohen rings, a natural generalization of Witt rings. Witt rings are only valuation rings in case the residue field is perfect, and Cohen rings arise as the Witt ring analogon over imperfect residue fields. Just as one studies truncated Witt rings to understand Witt rings, we study Cohen rings of positive characteristic as well as of characteristic zero. Our main results are a relative completeness and a relative model completeness result for Cohen rings, which imply the corresponding Ax-Kochen/Ershov type results for unramified henselian valued fields also in case the residue field is imperfect.