Abstract Decomposition Theorem and Applications

DECOMPOSITION THEOREM AND APPLICATIONS RAMI GROSSBERG AND OLIVIER LESSMANN Abstract. In this paper, we prove a decomposition theorem for abstract elementary classes K with the amalgamation property, under the assumption that certain axioms regarding independence, existence of some prime models, and regular types are satisfied. This context encompasses the following: (1) K is the class of models of an א0-stable first order theory. (2) K is the class of Faא0 -saturated models of a superstable first order theory. (3) K is the class of models of an excellent Scott sentence ψ ∈ Lω1,ω. (4) K the class of locally saturated models of a superstable good diagram D. (5) K is the class of (D,א0)-homogeneous models of a totally transcendental good diagram D. We also prove the nonstructure part necessary to obtain a Main Gap theorem for (5), which appears in the second author’s Ph.D. thesis [Le]. The main gap in the contexts (1) and (2) are theorems of Shelah [Sh a]; (3) is by Grossberg and Hart [GrHa]; (4) by Hyttinen and Shelah [HySh2]. In this paper, we prove a decomposition theorem for abstract elementary classes K with the amalgamation property, under the assumption that certain axioms regarding independence, existence of some prime models, and regular types are satisfied. This context encompasses the following: (1) K is the class of models of an א0-stable first order theory. (2) K is the class of Faא0 -saturated models of a superstable first order theory. (3) K is the class of models of an excellent Scott sentence ψ ∈ Lω1,ω. (4) K the class of locally saturated models of a superstable good diagram D. (5) K is the class of (D,א0)-homogeneous models of a totally transcendental good diagram D. We also prove the nonstructure part necessary to obtain a Main Gap theorem for (5), which appears in the second author’s Ph.D. thesis [Le]. The main gap in the contexts (1) and (2) are theorems of Shelah [Sh a]; (3) is by Grossberg and Hart [GrHa]; (4) by Hyttinen and Shelah [HySh2].