Stationarity and elimination of imaginaries in stable and simple theories

TL;DR

This paper investigates stationarity and elimination of imaginaries in stable and simple theories, providing new general criteria and extending known results to broader classes of theories without relying on specific field properties.

Contribution

It introduces a general framework for stationarity over real algebraically closed sets and extends the failure results of elimination of imaginaries to new classes of theories.

Findings

Types over real algebraically closed sets are stationary in certain theories.

Separably closed fields of infinite degree of imperfection lack elimination of imaginaries.

A general criterion for the failure of geometric elimination of imaginaries is established.

Abstract

We show that types over real algebraically closed sets are stationary, both for the theory of separably closed fields of infinite degree of imperfection and for the theory of beautiful pairs of algebraically closed field. The proof is given in a general setup without using specific features of theories of fields. Moreover, we generalize results of Delon as well as of Messmer and Wood that separably closed fields of infinite degree of imperfection and differentially closed fields of positive characteristic do not have elimination of imaginaries. Using work of Wagner on subgroups of stable groups, we obtain a general criterion yielding the failure of geometric elimination of imaginaries. This criterion applies in particular to beautiful pairs of algebraically closed fields, giving an alternative proof of the corresponding result of Pillay and Vassiliev.