${\aleph }_k$-free cogenerators

Abstract

We prove in ZFC that an abelian group $C$ is cotorsion if and only if $\mathrm{Ext}(F,C)=0$ 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.