Covers induced by Ext
type: Item
instance of: scholarly article
title: Covers induced by Ext
zbMATH Open document ID: 0981.16010
author: Paul C. Eklof
author: Jan Trlifaj
published in: Journal of Algebra
publication date: 2001-12-11T00:00:00Z
review text: Let \({\mathbf C}\subseteq\text{Mod-}R\) and \({\mathbf B}\subseteq R\text{-Mod}\). Then \(^\bot{\mathbf C}\) is defined to be the class \(\text{Ker Ext}(-,{\mathbf C})=\{D\mid\text{Ext}(D,C)=0\) for all \(C\in{\mathbf C}\}\) and \(\text{Ker Tor}(-,{\mathbf B})=\{A\mid\text{Tor}(A,B)=0\) for all \(B\in{\mathbf B}\}\). If \(M\in\text{Mod-}R\), then \(\phi\in\text{Hom}(A,M)\) (\(A\in{\mathbf C}\)) is called a \(\mathbf C\)-pre-cover of \(M\) if the induced map \(\text{Hom}(A',A)\to\text{Hom}(A',M)\) is surjective for all \(A'\in{\mathbf C}\). If each \(\psi\in\text{Hom}(A,A)\) satisfying \(\phi=\phi\psi\) is an automorphism of \(A\), then \(\phi\) is said to be a \(\mathbf C\)-cover. It is proved that each right \(R\)-module has a \(^\bot{\mathbf C}\)-cover if \(\mathbf C\) is any class of pure-injective modules. On the other hand, for any class \(\mathbf B\) each module has a \(\text{Ker Tor}(-,{\mathbf B})\)-cover. These results therefore provide a generalization of the flat cover conjecture. \(\mathbf C\)-pre-covers and \(\mathbf C\)-covers are studied intensively for modules over hereditary Dedekind domains. For Dedekind domains \(^\bot{\mathbf C}\) is described explicitly for any class of cotorsion modules \(\mathbf C\). The paper ends with three open problems.
reviewed by: Frieda Theron
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16S90
zbMATH DE Number: 1526429
zbMATH Keywords: pre-covers
zbMATH Keywords: covers
zbMATH Keywords: Ext
zbMATH Keywords: Tor
zbMATH Keywords: pure-injective modules
zbMATH Keywords: Dedekind domains
zbMATH Keywords: flat cover conjecture
zbMATH Keywords: cotorsion modules
Wikidata QID: Q57571325
MaRDI profile type: Publication
OpenAlex ID: W2088078217
cites work: Tilting preenvelopes and cotilting precovers
cites work: Preprojective modules over Artin algebras
cites work: Whitehead Modules over Domains
cites work: ALL MODULES HAVE FLAT COVERS
cites work: Quasi-tilting modules and counter equivalences
cites work: Q3240881
cites work: Homological Algebra and Set Theory
cites work: Q3949041
cites work: On Whitehead modules
cites work: How To Make Ext Vanish
cites work: Injective and flat covers, envelopes and resolvents
cites work: Q3699812
cites work: Cotilting and a hierarchy of almost cotorsion groups
cites work: Q3999266
cites work: A Module is Flat if and Only if its Character Module is Injective
cites work: Injective modules over Noetherian rings
cites work: Q3739243
cites work: Q5600865
cites work: Q3981822
cites work: Whitehead test modules
cites work: Q4002547
cites work: Flat covers of modules
DOI: 10.1006/JABR.2000.8343
description: scientific article; zbMATH DE number 1526429