Illinois Journal of Mathematics

On the existence of precovers

Paul C. Eklof and Saharon Shelah

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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.

Article information

Illinois J. Math., Volume 47, Number 1-2 (2003), 173-188.

First available in Project Euclid: 17 November 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20K40: Homological and categorical methods
Secondary: 03E35: Consistency and independence results


Eklof, Paul C.; Shelah, Saharon. On the existence of precovers. Illinois J. Math. 47 (2003), no. 1-2, 173--188. doi:10.1215/ijm/1258488146.

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