$\aleph_k$-free cogenerators

TL;DR

This paper characterizes cotorsion abelian groups via Ext functors with $\uplus ext{aleph}_k$-free groups in ZFC and discusses related constructions and consequences.

Contribution

It provides a ZFC proof that cotorsion groups are characterized by Ext vanishing with $\uplus ext{aleph}_k$-free groups, including a condensed overview of the $ar ext{lambda}$-Black Box.

Findings

Characterization of cotorsion groups via Ext and $\uplus ext{aleph}_k$-free groups

Development of $ar ext{lambda}$-Black Box in ZFC for $\uplus ext{aleph}_k$-free constructions

Implications for the structure theory of abelian groups

Abstract

We prove in ZFC that an abelian group $C$ is cotorsion if and only if $\operatorname{Ext}(F,C) = 0$ for every $\aleph_k$-free group $F$, and discuss some consequences and related results. This short note includes a condensed overview of the $\bar\lambda$-Black Box for $\aleph_k$-free constructions in ZFC.