Stability results assuming tameness, monster model and continuity of nonsplitting

Abstract

Assuming the existence of a monster model, tameness and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let $μ>LS({\bf K})$ be a regular stability cardinal and let $χ$ be the local character of $μ$-nonsplitting. The following holds: 1. When $μ$-nonforking is restricted to $(μ,\geqχ)$-limit models ordered by universal extensions, it enjoys invariance, monotonicity, uniqueness, existence, extension and continuity. It also has local character $χ$. This generalizes Vasey's result which assumed $μ$-superstability to obtain same properties but with local character $\aleph_0$. 2. There is $λ\in[μ,h(μ))$ such that if ${\bf K}$ is stable in every cardinal between $μ$ and $λ$, then ${\bf K}$ has $μ$-symmetry while $μ$-nonforking in (1) has symmetry. In this case (a) ${\bf K}$ has the uniqueness of $(μ,\geqχ)$-limit models: if $M_1,M_2$ are both $(μ,\geqχ)$-limit over some $M_0\in K_μ$, then $M_1\cong_{M_0}M_2$; (b) any increasing chain of $μ^+$-saturated models of length $\geqχ$ has a $μ^+$-saturated union. These generalize VanDieren-Vasey's result and remove the symmetry assumption in Boney-VanDieren and Vasey's result. Under $(<μ)$-tameness, the conclusions of (1), (2)(a)(b) are equivalent to ${\bf K}$ having the $χ$-local character of $μ$-nonsplitting. Grossberg and Vasey gave eventual superstability criteria for tame AECs with a monster model. We remove the high cardinal threshold and reduce the cardinal jump between equivalent superstability criteria. We also add two new superstability criteria to the list: a weaker version of solvability and the boundedness of the $U$-rank.