An AEC framework for fields with commuting automorphisms

In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms are closely related to difference fields. Some authors define a difference ring (or field) as a ring (or field) together with several commuting endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA, the model companion of difference fields with one automorphism. Our fields with commuting automorphisms generalize this setting. We have several automorphisms and they are required to commute. 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).

Automated summary

In this paper, an AEC framework for studying fields with commuting automorphisms is introduced. Fields with commuting automorphisms are connected to difference fields. The paper extends the definition that some authors use for difference rings (or fields) to include several commuting endomorphisms and proves various properties of FCA-classes, including the existence of AP and JEP, the coincidence of Galois types and existential types in existentially closed models, the homogeneity of the class, and a version of type amalgamation theorem. Furthermore, the paper shows that the monster model in this framework is a simple homogeneous structure.