Coordinatization in superstable theories. I. Stationary types

Abstract

Suppose T T is superstable and P P is a complete type over some finite set with U ( p ) = α + 1 U(p) = \alpha + 1 for some α \alpha . We show how to associate with p p an incidence geometry which measures the complexity of the family of extensions of p p of rank α \alpha . When p p is stationary we give a characterization of the possible incidence geometries. As an application we prove Theorem. Suppose M M is superstable and has only one 1 1 -type p ∈ S ( ∅ ) p \in S(\emptyset ) . Further suppose p p is stationary with U ( p ) = α + 1 U(p) = \alpha + 1 for some α \alpha . Then one of the following holds: (i) There is an equivalence relation E ⊂ M 2 E \subset {M^2} with infinitely many infinite classes definable over ∅ \emptyset . (ii) M M is the algebraic closure of a set of Morley rank 1 1 . In particular, M M is ℵ 0 {\aleph _0} -stable of finite rank.