Downward categoricity from a successor inside a good frame

We deduce improvements on the threshold of several categoricity transfers that do not mention frames. For example, the threshold in Shelah's transfer can be improved from ℶℶ(2LS(K))+ to ℶ(2LS(K))+ assuming that the AEC is LS(K)-tame. The successor hypothesis can also be removed from Shelah's result by assuming in addition either that the AEC has primes over sets of the form M∪{a} or (using an unpublished claim of Shelah) that the weak generalized continuum hypothesis holds.