Ext and inverse limits.
In 2001, Eklof and Trlifaj showed that given a set \(\mathcal S\) of modules there is always a special \({\mathcal S}^\perp(=\text{Ker\,Ext}^1_R({\mathcal C},-))\)-preenvelope, \(\mu_M\), for any module \(M\). Moreover, \(\text{Coker\,}\mu_M\) is \(\mathcal S\)-filtered. This fact was used by Enochs to prove the Flat Cover Conjecture and has also been used in work on approximation theory, tilting theory, and the Finitistic Dimension Conjectures. It is natural to ask about the dualization, that is, for a possible construction of special \(^\perp{\mathcal S}(=\text{ Ker\,Ext}^1_R(-,{\mathcal C}))\)-precovers whose kernels are \(\mathcal S\)-cofiltered. The main goal of this paper is to give a construction of this sort. Unfortunately, by a recent result of Eklof and Shelah (2003), it is not possible to obtain in ZFC a complete dual of Eklof and Trlifaj's result for any module over any ring. Thus the results in the current paper, proved for any ring and any module, provide only weak \(\mathcal S\)-precovers. The author remarks that when defined appropriately, the required duals of the homological and category theoretic properties hold. The general dual result fails for set-theoretic reasons.
scientific article; zbMATH DE number 1993651