Ext and inverse limits.
type: Item
instance of: scholarly article
title: Ext and inverse limits.
zbMATH Open document ID: 1035.16006
author: Jan Trlifaj
published in: Illinois Journal of Mathematics
publication date: 2003-10-16T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18G25
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 1993651
zbMATH Keywords: Ext
zbMATH Keywords: preenvelopes
zbMATH Keywords: special precovers
reviewed by: Ross P. Abraham
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1993651