On the existence of precovers

Abstract

It is proved consistent with ZFC + GCH that for every Whitehead group $A$ of infinite rank, there is a Whitehead group $H_{A}$ such that $\operatorname{Ext} (H_{A},A)\neq 0$. This is a strong generalization of the consistency of the existence of non-free Whitehead groups. A consequence is that it is undecidable in ZFC + GCH whether every $\mathbb{Z}$-module has a $^{\perp }\{ \mathbb{Z\}}$-precover. Moreover, for a large class of $\mathbb{Z}$-modules $ N$, it is proved consistent that a known sufficient condition for the existence of $^{\perp }\{N\}$-precovers is not satisfied.