On the existence of precovers. (English) Zbl 1033.20065

Eklof, Paul C.; Shelah, Saharon

Ill. J. Math. 47, No. 1-2, 173-188 (2003).

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.

Reviewer: Rüdiger Göbel (Essen)

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

Keywords:

Whitehead groups; covers of modules; consistency; undecidability; precovers of modules