Generic Automorphisms and Green Fields

TL;DR

This paper demonstrates the axiomatisability of the generic automorphism in green fields and related structures, providing geometric axioms and a general framework applicable to various Hrushovski amalgamation theories.

Contribution

It introduces a general framework for axiomatising generic automorphisms in green fields and similar structures, extending previous results and closing gaps in the construction of bad fields.

Findings

Axiomatisability of the generic automorphism in green fields and bad fields.

Development of geometric axioms for these theories.

Application of the framework to other Hrushovski amalgamation theories.

Abstract

We show that the generic automorphism is axiomatisable in the green field of Poizat (once Morleyised) as well as in the bad fields which are obtained by collapsing this green field to finite Morley rank. As a corollary, we obtain "bad pseudofinite fields" in characteristic 0. In both cases, we give geometric axioms. In fact, a general framework is presented allowing this kind of axiomatisation. We deduce from various constructibility results for algebraic varieties in characteristic 0 that the green and bad fields fall into this framework. Finally, we give similar results for other theories obtained by Hrushovski amalgamation, e.g. the free fusion of two strongly minimal theories having the definable multiplicity property. We also close a gap in the construction of the bad field, showing that the codes may be chosen to be families of strongly minimal sets.