Forking in Short and Tame Abstract Elementary Classes

TL;DR

This paper develops a forking notion for Galois-types in tame, type-short AECs, establishing its properties and implications for stability and categoricity, with simplified proofs under large cardinal axioms.

Contribution

It introduces a well-behaved non-forking concept for Galois-types in AECs and explores its properties, including symmetry, uniqueness, and local character, extending stability theory.

Findings

Non-forking satisfies symmetry and uniqueness.

Derives superstability-like properties from categoricity.

Simplifies proofs under large cardinal assumptions.

Abstract

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 {\bf Definition.} Let $M_0 \prec N$ be models from $K$ and $A$ be a set. We say that the Galois-type of $A$ over $M$ \emph{does not fork over $M_0$} iff for all small $a \in A$ and all small $N^- \prec N$, we have that Galois-type of $a$ over $N^-$ is realized in $M_0$. Assuming property (E) (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 this notion of non-forking is the only independence relation possible.