Title
Around the canonical base property
Abstract
Internality is a fundamental notion in geometric model theory in order to understand a theory in terms of its building blocks, its minimal types of rank 1. An archetypal example of internality is the case of a finite dimensional vector space V over an algebraically closed field K, where every realization is definable over K after naming a basis of V. The Canonical Base Property (CBP) isolates a key notion behind various descent arguments for differential and difference varieties and states the following: Over a realization of a stationary type, its canonical base is almost internal to the family of all non-locally modular minimal types.
Though many relevant examples of theories satisfy the CBP, Hrushovski, Palacín and Pillay produced an example of an uncountably categorical theory without the CBP. In this talk, I will present an alternative description of their counterexample in terms of additive covers of an algebraically closed field. Covers are already present in early work of Ahlbrandt-Ziegler as well as of Hodges-Pillay. The approach in terms of covers will enable us to present a new proof of the failure of the CBP, exploiting a notion which already appeared in Chatzidakis’ work on the UCBP.
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