Toward a stability theory of tame abstract elementary classes

We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. Assuming the singular cardinal hypothesis (SCH), we prove a full characterization of the (high-enough) stability cardinals, and connect the stability spectrum with the behavior of saturated models. We deduce (in ZFC) that if a class is stable on a tail of cardinals, then it has no long splitting chains (the converse is known). This indicates that there is a clear notion of superstability in this framework. We also present an application to homogeneous model theory: for $D$ a homogeneous diagram in a first-order theory $T$, if $D$ is both stable in $\mathrm{\mid }T\mathrm{\mid }$ and categorical in $\mathrm{\mid }T\mathrm{\mid }$ then $D$ is stable in all $\lambda \ge \mathrm{\mid }T\mathrm{\mid }$.