Deconstructible abstract elementary classes of modules and categoricity

Abstract

We prove a version of Shelah's Categoricity Conjecture for arbitrary deconstructible classes of modules. Moreover, we show that if $\mathcal{A}$ is a deconstructible class of modules that fits in an abstract elementary class $(\mathcal{A},\preceq)$ such that (1) $\mathcal{A}$ is closed under direct summands and (2) $\preceq$ refines direct summands, then $\mathcal{A}$ is closed under arbitrary direct limits. In an Appendix, we prove that the assumption (2) is not needed in some models of ZFC.