Title: Tameness from two successive good frames,
Abstract: We show, assuming a mild set-theoretic hypothesis, that if an abstract
elementary class (AEC) has a superstable-like forking notion for models of
cardinality $\lambda$ and a superstable-like forking notion for models of
cardinality $\lambda^+$, then orbital types over models of cardinality
$\lambda^+$ are determined by their restrictions to submodels of cardinality
$\lambda$. By a superstable-like forking notion, we mean here a good frame, a
central concept of Shelah's book on AECs.
It is known that locality of orbital types together with the existence of a
superstable-like notion for models of cardinality $\lambda$ implies the
existence of a superstable-like notion for models of cardinality $\lambda^+$,
but here we prove the converse. An immediate consequence is that forking in
$\lambda^+$ can be described in terms of forking in $\lambda$.