Categoricity for transfinite extensions of modules

TL;DR

This paper proves that for certain classes of modules, categoricity in large cardinals implies categoricity in all sufficiently large cardinals, and confirms Shelah's Categoricity Conjecture for specific classes of modules.

Contribution

It establishes the equivalence of categoricity in a big cardinal and a tail of cardinals for deconstructible classes of modules and proves Shelah's Conjecture for classes of roots of Ext.

Findings

Categoricity in a big cardinal implies categoricity in a tail of cardinals.

Confirmed Shelah's Categoricity Conjecture for classes of roots of Ext.

Established equivalence of categoricity conditions for deconstructible classes.

Abstract

For each deconstructible class of modules $\mathcal D$, we prove that the categoricity of $\mathcal D$ in a big cardinal is equivalent to its categoricity in a tail of cardinals. We also prove Shelah's Categoricity Conjecture for $(\mathcal D, \prec)$, where $(\mathcal D, \prec)$ is any abstract elementary class of roots of Ext.