Shelah's eventual categoricity conjecture in tame abstract elementary classes with primes

A new case of Shelah's eventual categoricity conjecture is established: Let K be an abstract elementary class with amalgamation. Write mu := beth((LS(K))(2))+ and H-2 := beth((2 mu))+. Assume that K is H-2-tame and K->= H2 has primes over sets of the form M boolean OR {a}. If K is categorical in some lambda > H-2, then K is categorical in all lambda' >= H-2. The result had previously been established when the stronger locality assumptions of full tameness and shortness are also required. An application of the method of proof of the mentioned result is that Shelah's categoricity conjecture holds in the context of homogeneous model theory (this was known, but our proof gives new cases): If D be a homogeneous diagram in a first-order theory T and D is categorical in a lambda > |T|, then D is categorical in all lambda' >= min(lambda, beth(|T|)((2)())+). (C)2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.