Stability results assuming tameness, monster model and continuity of nonsplitting

TL;DR

This paper extends superstability results in tame abstract elementary classes assuming a monster model and continuity of nonsplitting, generalizing previous results and introducing new criteria for superstability.

Contribution

It generalizes known superstability results by removing high cardinal thresholds, reducing cardinal jumps, and adding new superstability criteria under tameness and continuity assumptions.

Findings

Invariance, monotonicity, and other properties of nonforking in limit models

Existence of a cardinal where stability implies symmetry and model uniqueness

Equivalence of local character and superstability criteria under tameness

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 $\mu>LS({\bf K})$ be a regular stability cardinal and let $\chi$ be the local character of $\mu$-nonsplitting. The following holds: 1. When $\mu$-nonforking is restricted to $(\mu,\geq\chi)$-limit models ordered by universal extensions, it enjoys invariance, monotonicity, uniqueness, existence, extension and continuity. It also has local character $\chi$. This generalizes Vasey's result which assumed $\mu$-superstability to obtain same properties but with local character $\aleph_0$. 2. There is $\lambda\in[\mu,h(\mu))$ such that if ${\bf K}$ is stable in every cardinal between $\mu$ and $\lambda$, then ${\bf K}$ has $\mu$-symmetry while $\mu$-nonforking in (1) has symmetry. In this case (a) ${\bf K}$ has the uniqueness of $(\mu,\geq\chi)$-limit models: if $M_1,M_2$ are both $(\mu,\geq\chi)$-limit over some $M_0\in K_\mu$, then $M_1\cong_{M_0}M_2$; (b) any increasing chain of $\mu^+$-saturated models of length $\geq\chi$ has a $\mu^+$-saturated union. These generalize VanDieren-Vasey's result and remove the symmetry assumption in Boney-VanDieren and Vasey's result. Under $(<\mu)$-tameness, the conclusions of (1), (2)(a)(b) are equivalent to ${\bf K}$ having the $\chi$-local character of $\mu$-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.