Shelah's eventual categoricity conjecture in tame AECs with primes

Sebastien Vasey

Published 2015-09-14Version 1

Two new cases of Shelah's eventual categoricity conjecture are established: $\mathbf{Theorem}$ Let $K$ be an AEC which is tame and has primes over sets of the form $M \cup \{a\}$. If $K$ is categorical in a high-enough cardinal, then $K$ is categorical on a tail of cardinals. We do not assume amalgamation (however the hypotheses imply that there exists a cardinal $\lambda$ so that $K_{\ge \lambda}$ has amalgamation). The result had previously been established when the stronger locality assumptions of full tameness and shortness are also required. An application of the theorem is that Shelah's categoricity conjecture holds in the context of homogeneous model theory: $\mathbf{Theorem}$ Let $D$ be a homogeneous diagram in a first-order theory $T$. If $D$ is categorical in a $\lambda > |T|$, then $D$ is categorical in all $\lambda' \ge \min (\lambda, \beth_{(2^{|T|})^+})$.