Finding a field in a Zariski-like structure

The starting point for this dissertation is whether the concept of Zariski geometry, introduced by Hrushovski and Zilber, could be generalized to the context of nonelementary classes. This leads to the axiomatization of Zariski-like structures. As our main result, we prove that if the canonical pregeometry of a Zariski-like structure is non locally modular, then the structure interprets either an algebraically closed field or a non-classical group. This is a counterpart to the result by Hrushovski and Zilber which states that an algebraically closed field can be found in a non locally modular Zariski geometry. It demonstrates that the concept of a Zariski-like structure captures one of the most essential features of a Zariski geometry. Finally, we give a non-trivial example by showing that the cover of the multiplicative group of an algebraically closed field of characteristic zero is Zariski-like. Since Zariski geometries are strongly minimal and the non-elementary analogue of strong minimality is quasiminimality, it seems natural to look into quasiminimal classes for the generalization. Thus, we define a Zariski-like structure as a quasiminimal pregeometry structure that has certain properties. Unlike in the context of Zariski geometries, we do not start from a single structure, but formulate our axioms to generalize the setting obtained after moving into an elementary extension. Instead of assuming underlying topologies, we formulate the axioms for a countable collection C of Galois definable sets that have some of the properties of irreducible closed sets from the Zariski geometry context. Quasiminimal classes are abstract elementary classes (AECs) that arise from a quasiminimal pregeometry structure. They are uncountably categorical, and have both the amalgamation property (AP) and the joint embedding property (JEP), and thus also a model homogeneous universal monster model, which we denote by M. These classes are excellent in the sense of Zilber (this is different from the original notion of excellence due to Shelah). To adapt Hrushovski’s and Zilber’s proof to our setting, we first generalize Hrushovski’s Group Configuration Theorem to the context of quasiminimal classes. For this, we develop an independence calculus that has all the usual properties of non-forking and works in our context. Here, we employ some ideas that Hyttinen and Lessman used when studying independence in context of classes that are excellent in the sense of Shelah. However, we cannot directly apply their results since the classes we are working in are not excellent in this stronger sense. We then prove the group configuration theorem and apply it to find a 1-dimensional group, assuming that the canonical pregeometry obtained from the bounded closure operator is non-trivial. A field can be found under the further assumptions that M does not interpret a non-classical group and the canonical pregeometry is non locally modular. Then, the group obtained previously is Abelian, and we can use its elements to find another group configuration that yields a 2-dimensional group. This allows us to interpret an algebraically closed field. To do so, we apply methods that originate from Hrushovski’s Ph.D. thesis and have been adopted to the the non-elementary context by Hyttinen, Lessman, and Shelah. Finally, we show that the cover of the multiplicative group of an algebraically closed field, studied by e.g. Boris Zilber and Lucy Burton, provides a non-trivial example of a Zariski-like structure. Burton obtained a topology on the cover by taking sets definable by positive, quantifier-free first order formulae as the basic closed sets. This is called the PQF-topology, and the sets that are closed with respect to it are called the PQF-closed sets. We show that the cover becomes Zariski-like after adding names for a countable number of elements to the language. The axioms for a Zariski-like structure are then satisfied if the collection C is taken to consist of the PQF-closed sets that are definable over the empty set.