On the spectrum of limit models
Jeremy Beard, Marcos Mazari-Armida
TL;DR
This work develops a precise spectrum for limit models in nice AECs under a well-behaved independence notion. By introducing towers and their reduced/full variants, it proves that long limit models are isomorphic over a base when the limit's cofinality is large enough, while short limit models are non-isomorphic when cofinalities diverge, under tameness and independence assumptions. The main results yield a local-to-global picture: the isomorphism type of limit models is governed by the local invariants $ extchi( extsmile, extbf{K}, extle_K^u)$ and $ heta( extsmile, extbf{K})$, with corollaries for tame AECs and explicit applications to modules, torsion groups, and first-order theories. This framework provides robust tools for understanding the spectrum of limit models in natural AECs and offers a vehicle for deriving concrete classification-type conclusions in algebraic contexts.
Abstract
We study the spectrum of limit models assuming the existence of a nicely behaved independence notion. Under reasonable assumptions, we show that all `long' limit models are isomorphic, and all `short' limit models are non-isomorphic. $\textbf{Theorem.}$ Let $\mathbf{K}$ be a $\aleph_0$-tame abstract elementary class stable in $λ\geq \operatorname{LS}(\mathbf{K})$ with amalgamation, joint embedding and no maximal models. Suppose there is an independence relation on the models of size $λ$ that satisfies uniqueness, extension, non-forking amalgamation, universal continuity, and $(\geq κ)$-local character in a minimal regular $κ< λ^+$. Suppose $δ_1, δ_2 < λ^+$ with $\operatorname{cf}(δ_1) < \operatorname{cf}(δ_2)$. Then for any $N_1, N_2, M \in \mathbf{K}_λ$ where $N_l$ is a $(λ, δ_l)$-limit model over $M$ for $l = 1, 2$, \[N_1 \text{ is isomorphic to } N_2 \text{ over } M \iff \operatorname{cf}(δ_1) \geq κ\] Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the $\aleph_0$-tameness assumption and assuming the independence relation is defined only on high cofinality limit models. Low cofinality limits are non-isomorphic without assuming non-forking amalgamation. We show how our results can be used to study limit models in both abstract settings and in natural examples of abstract elementary classes.