A contribution to the model theory of fields with free operators

Abstract

This thesis makes a contribution to the model theory of fields with free operators, as introduced by Moosa and Scanlon. The classical Weil restriction, a result of algebraic geometry, establishes the existence of a left adjoint to base extension of algebras. Generalising the corresponding differential result of Leon Sanchez and Tressl, we extend this to the case of algebras equipped with free operators - given an extension of rings with free operators whose underlying extension of rings is free and of finite rank, and subject to a mild algebraic condition on the endomorphisms definable in the free operator structure, we show that there is a unique sequence of free operators on the classical Weil restriction that ensures the unit and counit of the classical adjunction preserve the free operator structure. Thus base change in the category of algebras with free operators has a left adjoint, which we call the D-Weil restriction. Properties of the free operator structure preserved under the D-Weil restriction are investigated, including triviality of the associated endomorphisms and commutativity of the operators, and a partial converse to the main adjunction result is shown: the existence of a left adjoint to base change over a field implies the associated endomorphisms must have the aforementioned algebraic condition. The theory UCD in the language of rings with free operators is introduced as a suitable weakening of the geometric axiom of Moosa and Scanlon's theory of D-closed fields D-CF0, the model companion of the theory of fields of characteristic zero with free operators. We show that whenever T is a model complete theory of difference large fields of characteristic zero - a notion of Cousins - T + UCD is the model companion of the theory T + "free operators", establishing the existence of the uniform companion for theories of difference large fields of characteristic zero with free operators, following Tressl's result in the differential context. We show that quantifier elimination transfers from T to T + UCD - from which it immediately follows that stability and NIP do as well - and we use the D-Weil restriction to show that the algebraic closure of a model of UCD is a model of D-CF0. We provide an axiomatic framework for proving the transfer of various neostability properties from theories of fields to theories of fields with operators, show that this unifies many proofs of stability and simplicity of theories of fields with operators existing already in the literature, and use it to characterise forking in the theory of separably differentially closed fields of infinite differential degree of imperfection, as defined by Ino and Leon Sanchez. Finally, we introduce the class of bounded pseudo D-closed fields in analogy to the class of bounded pseudo-differentially closed fields as a case study for some of the general results just described.

Date of Award	31 Dec 2024
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Omar Leon Sanchez (Supervisor) & Gareth Jones (Supervisor)