Coordinatization in superstable theories. I. Stationary types

Abstract:

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