Derived functors of Hom relative to \(n\)-flat covers.
type: Item
instance of: scholarly article
title: Derived functors of Hom relative to \(n\)-flat covers.
zbMATH Open document ID: 1332.16001
published in: Vietnam Journal of Mathematics
publication date: 2015-09-22T00:00:00Z
review text: A left \(R\)-module \(M\) is called \(n\)-flat if \(\text{Tor}_1(N,M)=0\) for all finitely presented right \(R\)-modules \(N\) with projective dimension \(\leq n\) [\textit{S. B. Lee}, Commun. Algebra 30, No. 3, 1119-1126 (2002; Zbl 1022.16001)]. The category of all \(n\)-flat left \(R\)-modules is denoted by \(\mathfrak F_n\). It is known that every module over any ring has a flat cover [\textit{L. Bican} et al., Bull. Lond. Math. Soc. 33, No. 4, 385-390 (2001; Zbl 1029.16002)]. One of the main aims of this paper is to show that a similar result holds with respect to \(n\)-flat covers. This is done via the concept of the \(n\)-flat \(n\)-cotorsion theory. The construction of an \(n\)-flat resolution of an \(R\)-module \(M\) is subsequently described, for any right \(n\)-coherent ring \(R\). This leads to new derived functors, denoted \(F_n\text{ext}^i(N,M)\). It is shown that these functors are well defined. A canonical map \(\sigma\colon F_n\text{ext}^0(N,M)\to\Hom(N,M)\) is found and \(n\)-flat modules as well as modules with \(\mathfrak F_n\)-\(\dim(M)\leq 1\) and with \(\mathfrak F_n\)-\(\dim(M)\leq m\) (\(m\geq 2\)) are characterized in terms of this canonical map.
reviewed by: Frieda Theron
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16E05
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 18G10
zbMATH DE Number: 6484013
zbMATH Keywords: \(n\)-flat modules
zbMATH Keywords: \(n\)-flat covers
zbMATH Keywords: flat torsion theories
zbMATH Keywords: \(n\)-flat resolutions
zbMATH Keywords: derived functors
zbMATH Keywords: projective dimension
zbMATH Keywords: right coherent rings
author: C. Selvaraj
author: R. Udhayakumar
MaRDI profile type: Publication
full work available at URL: s10013-014-0097-8
OpenAlex ID: W2082109315
cites work: ALL MODULES HAVE FLAT COVERS
cites work: Injective and flat covers, envelopes and resolvents
cites work: Relative homological algebra
cites work: Approximations and endomorphism algebras of modules.
cites work: n-COHERENT RINGS
cites work: Q3882615
cites work: Flat covers of modules
cites work: n-Flat and n-FP-injective modules
DOI: 10.1007/S10013-014-0097-8
description: scientific article; zbMATH DE number 6484013