Local stability in generic models (Q2753068)

In this dissertation, Ingo Kraus develops local stability theory in non-elementary classes. In chapter 1 he defines \textit{\(\kappa\)-\textit{generic models}} as existentially closed structures which are \(\kappa\)-saturated for existential types, and analyses their basic properties. This slightly generalizes and unifies the work on Robinson theories by \textit{Ehud Hrushovski} [Simplicity and the Lascar group, preprint 1998] and on e-universal domains by \textit{A. Pillay} [Forking in the category of existentially closed structures, preprint 1999].NEWLINENEWLINENEWLINEIn chapter 2 Kraus introduces local formulas and local types (where parameters may be restricted to come from a type-definable set), free extensions of local types and their definability, and obtains the basic equivalences between the stability of a set \(\Delta\) of local formulas, definability of \(\Delta\)-types, and existence of few \(\Delta\)-types. Assuming stability, he then shows the basic properties of non-forking, except for symmetry of course. All this is done in a non-elementary context, in what he calls a pseudomodel, namely a suitable class of generic models.NEWLINENEWLINENEWLINEIn chapter 3 Kraus proves some further results about his principal example of a generic structure, namely a structure with a generic automorphism. The dissertation closes with three appendices on existentially closed structures and universal theories, local stable group theory, and general Galois theory.