Triangulated equivalence between a homotopy category and a triangulated quotient category
type: Item
instance of: scholarly article
title: Triangulated equivalence between a homotopy category and a triangulated quotient category
zbMATH Open document ID: 1391.18017
published in: Journal of Algebra
publication date: 2018-06-26T00:00:00Z
review text: \textit{M. Hovey} [Math. Z. 241, No. 3, 553--592 (2002; Zbl 1016.55010)] introduced the notion of abelian model categories as bicomplete abelian categories possessing a compatible model structure. His most wonderful result, which is now known as \textit{Hovey's correspondence}, says that there is a one-to-one correspondence between abelian model structures and complete cotorsion pairs. Given two complete hereditary cotorsion pairs \((\mathcal{Q}, \mathcal{R})\) and \((\mathcal{Q}', \mathcal{R}')\) in a bicomplete abelian category \(\mathcal{G}\) such that \(\mathcal{Q}'\subseteq\mathcal{Q}\) and \(\mathcal{Q}\cap\mathcal{R} = \mathcal{Q}'\cap\mathcal{R}'\), \textit{H. Becker} [Adv. Math. 254, 187--232 (2014; Zbl 1348.16009)] showed that there exists a hereditary abelian model structure \(\mathcal{M}=(\mathcal{Q},\mathcal{W},\mathcal{R}')\) on \(\mathcal{G}\), where \(\mathcal{W}\) is a thick subcategory of \(\mathcal{G}\). The authors prove that the homotopy category \(\text{Ho}(\mathcal{M})\) of \(\mathcal{M}\) is triangulated equivalent to the triangulated quotient category \(D^b(\mathcal{G})_{\widehat{[\mathcal{Q},\mathcal{R}']}}/K^b(\mathcal{Q}\cap\mathcal{R}')\), where \(D^b(\mathcal{G})_{\widehat{[\mathcal{Q},\mathcal{R}]}}\) is the subcategory of \(D^b(\mathcal{G})\) consisting of all homology bounded complexes with both finite \(\mathcal{Q}\) dimension and \(\mathcal{R}'\) dimension and \(K^b(\mathcal{Q}'\cap\mathcal{R})\) is the bounded homotopy category of \(\mathcal{Q}\cap\mathcal{R}\) (core) objects. The authors focus their attention on the category of modules for giving some applications of main results. It is shown that the homotopy category of the Gorenstein flat (resp., Ding projective and Gorenstein \(AC\)-projective) model structure on the category of modules established by \textit{J. Gillespie} [Homology Homotopy Appl. 12, No. 1, 61--73 (2010; Zbl 1231.16005)] can be realized as a certain triangulated quotient category.
reviewed by: Marek Golasiński
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 18E10
zbMATH DE Number: 6894500
zbMATH Keywords: hereditary abelian model structure
zbMATH Keywords: homotopy category
zbMATH Keywords: triangulated equivalence
zbMATH Keywords: singularity category
author: Zhenxing Di
author: Zhongkui Liu
author: Xiaoyan Yang
author: Xiaoxiang Zhang
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2018.04.002
OpenAlex ID: W2796013089
cites work: Stable module theory
cites work: Homological dimensions of unbounded complexes
cites work: The homological theory of contravariantly finite subcategories:auslander-buchweitz contexts, gorenstein categories and (co-)stabilization
cites work: Homotopy theory of modules and Gorenstein rings
cites work: Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras.
cites work: Models for singularity categories.
cites work: Rings Over Which the Class of Gorenstein Flat Modules is Closed Under Extensions
cites work: Absolutely Clean, Level, and Gorenstein AC-Injective Complexes
cites work: Relative singularity categories and Gorenstein-projective modules
cites work: Gorenstein dimensions
cites work: The flat dimensions of injective modules
cites work: Coherent rings with finite self-FP-injective dimension
cites work: STRONGLY GORENSTEIN FLAT MODULES
cites work: Relative homological algebra
cites work: Q4271869
cites work: Gorenstein flat covers of modules over Gorenstein rings
cites work: The existence of Gorenstein flat covers
cites work: On the finiteness of Gorenstein homological dimensions
cites work: Q4783713
cites work: Model structures on modules over Ding-Chen rings.
cites work: Hereditary abelian model categories
cites work: How to construct a Hovey triple from two cotorsion pairs
cites work: The flat model structure on 𝐂𝐡(𝐑)
cites work: The flat stable module category of a coherent ring
cites work: On Ding injective, Ding projective and Ding flat modules and complexes
cites work: Q4012108
cites work: Q3994341
cites work: Gorenstein homological dimensions.
cites work: Cotorsion pairs, model category structures, and representation theory
cites work: A model structure approach to the Tate-Vogel cohomology
cites work: Gorenstein flat dimension of complexes.
cites work: Strongly Gorenstein flat modules and dimensions.
cites work: Q5483548
cites work: Q3409132
cites work: The global dimension of Boolean rings
cites work: Cotorsion Dimension of Unbounded Complexes
cites work: Derived categories and stable equivalence
cites work: Gorenstein projective dimension for complexes
cites work: Q4968690
cites work: All Modules Have Gorenstein Flat Precovers
cites work: On a question of Gillespie
DOI: 10.1016/J.JALGEBRA.2018.04.002
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description: scientific article; zbMATH DE number 6894500