Submission date: 23 September 2015.

Abstract:
We show:

$\mathbf{Theorem}$
Let $K$ be a fully $\text{LS} (K)$-tame and short AEC with amalgamation and no maximal models. Write $H_2 := \beth_{\left(2^{\beth_{\left(2^{\text{LS} (K)}\right)^+}}\right)^+}$. The following are equivalent:
1) $K_{\ge H_2}$ has primes over sets of the form $M \cup \{a\}$ and $K$ is categorical in some $\lambda > H_2$.
2) $K$ is categorical in all $\lambda \ge H_2$.
Note that (1) implies (2) appears in an earlier paper. Here we prove (2) implies (1), generalizing an argument of Shelah who proved the existence of primes at successor cardinals. Assuming a large cardinal axiom, we deduce an equivalence between Shelah's eventual categoricity conjecture and the statement that every abstract elementary class (AEC) categorical in a proper class of cardinals eventually has prime models over sets of the form $M \cup \{a\}$.

$\mathbf{Corollary}$
Assume there exists a proper class of almost strongly compact cardinals. Let $K$ be an AEC categorical in a proper class of cardinals. The following are equivalent:
1) There exists $\lambda_0$ such that $K_{\ge \lambda_0}$ has primes over sets of the form $M \cup \{a\}$.
2) There exists $\lambda_1$ such that $K$ is categorical in all $\lambda \ge \lambda_1$.

Mathematics Subject Classification: 03C48 (Primary), 03C45, 03C52, 03C55, 03C75, 03E55 (Secondary)

Keywords and phrases:

Full text arXiv 1509.07024: pdf, ps.