$\aleph_k$-free cogenerators
Manfred Dugas, Daniel Herden, Saharon Shelah

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 is cotorsion if and only if for every -free group , and discuss some consequences and related results. This short note includes a condensed overview of the -Black Box for -free constructions in ZFC.
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