Isolated types of finite rank: an abstract Dixmier–Moeglin equivalence

Suppose T is a totally transcendental first-order theory and every minimal non-locally-modular type is nonorthogonal to a nonisolated minimal type over the empty set. It is shown that a finite rank type p = tp ( a / A ) is isolated if and only if for every b ∈ acl ( A a ) and q ∈ S ( A b ) nonisolated and minimal. This applies to the theory of differentially closed fields—where it is motivated by the differential Dixmier–Moeglin equivalence problem—and the theory of compact complex manifolds.