The model theory of Cohen rings

Sylvy Anscombe, Franziska Jahnke

Published 2019-04-17Version 1

The aim of this article is to give an 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. The key to these results is a proof of relative quantifier elimination down to the residue field in an appropriate language which holds in any unramified henselian valued field.