Categoricity and infinitary logics

We point out a gap in Shelah's proof of the following result: $\mathbf{Claim}$ Let $K$ be an abstract elementary class categorical in unboundedly many cardinals. Then there exists a cardinal $λ$ such that whenever $M, N \in K$ have size at least $λ$, $M \le N$ if and only if $M \preceq_{L_{\infty, \text{LS} (K)^+}} N$. The importance of the claim lies in the following theorem, implicit in Shelah's work: $\mathbf{Theorem}$ Assume the claim. Let $K$ be an abstract elementary class categorical in unboundedly many cardinals. Then the class of $λ$ such that: 1) $K$ is categorical in $λ$; 2) $K$ has amalgamation in $λ$; and 3) there is a good $λ$-frame with underlying class $K_λ$ is stationary. We give a proof and discuss some related questions.

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