Cotorsion pairs in \(\mathbf{C}(R-\text{Mod})\)
type: Item
instance of: scholarly article
title: Cotorsion pairs in \(\mathbf{C}(R-\text{Mod})\)
zbMATH Open document ID: 1268.18011
DOI: 10.1216/RMJ-2012-42-6-1787
author: Diego Bravo
author: E. Enochs
author: A. Iacob
author: Overtoun M. G. Jenda
author: Juan Rada
published in: Rocky Mountain Journal of Mathematics
publication date: 2013-04-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1361800606
review text: For any ring \(R\), the authors consider complete cotorsion pairs \((\mathcal{C},\mathcal{D})\) in the category \(\mathbf{C}(R-\mathrm{Mod})\) of complexes of left \(R\)-modules. If \(\mathcal{C}\) is closed under taking suspensions and \(\mathbf{K}(R-\mathrm{Mod})\) denotes the homotopy category, they prove that the embedding functors \(\mathbf{K}(\mathcal{C})\rightarrow \mathbf{K}(R-\mathrm{Mod})\) and \(\mathbf{K}(\mathcal{D})\rightarrow \mathbf{K}(R-\mathrm{Mod})\) have right and left adjoints, respectively. Examples of such cotorsion pairs and the corresponding induced adjoint situations are given in Section 4. Using results by \textit{M. Hovey} in [Math. Z. 241, No. 3, 553--592 (2002; Zbl 1016.55010)] that assure when a pair of classes of complexes of left \(R\)-modules gives rise to a model structure on \(\mathbf{C}(R-\mathrm{Mod})\), the authors finally describe a new model structure on \(\mathbf{C}(R-\mathrm{Mod})\) where the cofibrant objects are the projective ones.
reviewed by: Antonio R. Garzón
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 6152186
zbMATH Keywords: cotorsion pairs
zbMATH Keywords: complexes of modules
zbMATH Keywords: adjoint functors
zbMATH Keywords: model structures
zbMATH Keywords: homotopy category
MaRDI profile type: Publication
cites work: Covers and envelopes in Grothendieck categories: Flat covers of complexes with applications
cites work: Q3836846
cites work: Cotorsion pairs and degreewise homological model structures
cites work: Approximations and endomorphism algebras of modules.
cites work: Cotorsion pairs, model category structures, and representation theory
cites work: Projective modules
cites work: Some adjoints in homotopy categories
cites work: Q3861607
Recommended article: Cotorsion Pairs in 𝒞N(A)
Recommended article: Q5074623
Recommended article: Cotorsion pairs, torsion pairs, and \(\varSigma \)-pure-injective cotilting modules
Recommended article: On the Cotorsion Pair (𝒫1, 𝒟)
Recommended article: \(n\)-cotorsion pairs
Recommended article: Q3144547
Recommended article: On the cogeneration of cotorsion pairs
Recommended article: Q3193960
Recommended article: Cotorsion and Tor Pairs and Finitistic Dimensions over Commutative Rings
Recommended article: Q4892990
description: scientific article; zbMATH DE number 6152186