Forking in Short and Tame Abstract Elementary Classes

We develop a notion of forking for Galois-types in the context of Abstract Elementary Classes (AECs). Under the hypotheses that an AEC K is tame, type-short, and failure of an order-property, we consider

Definition 1. Let M0 ≺ N be models from K and A be a set. We say that the Galois-type of A over N does not fork over M0, written A^M0 N, iff for all small a ∈ A and all small N − ≺ N, we have that Galois-type of a over N − is realized in M0.

Assuming property (E) (Existence and Extension, see Definition 3.3) we show that this non-forking is a well behaved notion of independence, in particular satisfies symmetry and uniqueness and has a corresponding U-rank. We find conditions for a universal local character, in particular derive superstability-like property from little more than categoricity in a “big cardinal”. Finally, we show that under large cardinal axioms the proofs are simpler and the non-forking is more powerful.

In [BGKV], it is established that, if this notion is an independence notion, then it is the only one.