## On the existence of precovers.(English)Zbl 1033.20065

This very nice paper deals with a version of the Whitehead problem under ZFC + GCH which also provides new results towards the existence of covers – a topic in the center of research for the last decade in module theory: Assuming ZFC + GCH the authors show that it is consistent that for every Whitehead group $$A$$ of infinite rank there is another one $$B$$ with $$\text{Ext}(B,A)\neq 0$$. Note that often it is equally hard to show non-splitting as splitting of short exact sequences. In this case $$A$$ would “like to split”. It follows (essentially rephrasing the last statement) the answer to an open problem for approximations of modules: Under the same set theoretic hypothesis it is undecidable whether every Abelian group has a $$^\bot\{\mathbb{Z}\}$$-precover. For many related results, also those concerning the existence of precovers we refer to the paper which is dedicated to the memory of Reinhold Baer.

### MSC:

 20K40 Homological and categorical methods for abelian groups 03E35 Consistency and independence results 16D40 Free, projective, and flat modules and ideals in associative algebras
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