An AEC framework for fields with commuting automorphisms

Abstract

In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms generalise difference fields. Whereas in a difference field, there is one distinguished automorphism, a field with commuting automorphisms can have several of them, and they are required to commute. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA, the model companion of difference fields. Hrushovski has proved that in the case of fields with two or more commuting automorphisms, the existentially closed models do not necessarily form a first order model class. In the present paper, we introduce FCA-classes, an AEC framework for studying the existentially closed models of the theory of fields with commuting automorphisms. We prove that an FCA-class has AP and JEP and thus a monster model, that Galois types coincide with existential types in existentially closed models, that the class is homogeneous, and that there is a version of type amalgamation theorem that allows to combine three types under certain conditions. Finally, we use these results to show that our monster model is a simple homogeneous structure in the sense of S. Buechler and O. Lessman (this is a non-elementary analogue for the classification theoretic notion of a simple first order theory). Contents 1. Introduction 1 2. A remark on existential types in multiuniversal classes 5 3. The AEC framework 13 4. Independence 18 5. Simplicity 24 References 28