Cotorsion Pairs in 𝒞N(A)

2017 ◽  
Vol 24 (04) ◽  
pp. 577-602 ◽  
Author(s):  
Xiaoyan Yang ◽  
Tianya Cao
Keyword(s):  
Abelian Category ◽  
Cotorsion Pair ◽  
Cotorsion Pairs ◽  
Model Structures

Given a cotorsion pair ([Formula: see text], [Formula: see text]) in an abelian category [Formula: see text] , we define cotorsion pairs ([Formula: see text], dg[Formula: see text]) and (dg[Formula: see text], [Formula: see text]) in the category [Formula: see text]N([Formula: see text]) of N-complexes on [Formula: see text]. We prove that if the cotorsion pair ([Formula: see text], [Formula: see text]) is complete and hereditary in a bicomplete abelian category, then both of the induced cotorsion pairs are complete, compatible and hereditary. We also create complete cotorsion pairs (dw[Formula: see text], (dw[Formula: see text])⊥), (ex[Formula: see text], (ex[Formula: see text])⊥) and (⊥(dw[Formula: see text]), dw[Formula: see text]), (⊥(ex[Formula: see text]); ex[Formula: see text]) in a termwise manner by starting with a cotorsion pair ([Formula: see text], [Formula: see text]) that is cogenerated by a set. As applications of these results, we obtain more abelian model structures from the cotorsion pairs.

New model structures and projective (injective) cotorsion pairs

2021 ◽  
Author(s):  
Aimin Xu
Keyword(s):  
Positive Integer ◽  
Triangulated Categories ◽  
Model Structure ◽  
Cotorsion Pair ◽  
Flat Dimension ◽  
Gorenstein Projective Module ◽  
Chain Complexes ◽  
Cotorsion Pairs ◽  
Gorenstein Injective ◽  
Model Structures

Let [Formula: see text] be either the category of [Formula: see text]-modules or the category of chain complexes of [Formula: see text]-modules and [Formula: see text] a cofibrantly generated hereditary abelian model structure on [Formula: see text]. First, we get a new cofibrantly generated model structure on [Formula: see text] related to [Formula: see text] for any positive integer [Formula: see text], and hence, one can get new algebraic triangulated categories. Second, it is shown that any [Formula: see text]-strongly Gorenstein projective module gives rise to a projective cotorsion pair cogenerated by a set. Finally, let [Formula: see text] be an [Formula: see text]-module with finite flat dimension and [Formula: see text] a positive integer, if [Formula: see text] is an exact sequence of [Formula: see text]-modules with every [Formula: see text] Gorenstein injective, then [Formula: see text] is injective.

Model structures, recollements and duality pairs

2021 ◽  
Author(s):  
Wenjing Chen ◽  
Zhongkui Liu
Keyword(s):  
Projective Modules ◽  
Cotorsion Pair ◽  
Gorenstein Rings ◽  
Injective Modules ◽  
Flat Modules ◽  
Duality Pairs ◽  
Cotorsion Pairs ◽  
Gorenstein Flat ◽  
Model Structures

In this paper, we construct some model structures corresponding Gorenstein [Formula: see text]-modules and relative Gorenstein flat modules associated to duality pairs, Frobenius pairs and cotorsion pairs. By investigating homological properties of Gorenstein [Formula: see text]-modules and some known complete hereditary cotorsion pairs, we describe several types of complexes and obtain some characterizations of Iwanaga–Gorenstein rings. Based on some facts given in this paper, we find new duality pairs and show that [Formula: see text] is covering as well as enveloping and [Formula: see text] is preenveloping under certain conditions, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-injective modules and [Formula: see text] denotes the class of Gorenstein [Formula: see text]-flat modules. We give some recollements via projective cotorsion pair [Formula: see text] cogenerated by a set, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-projective modules. Also, many recollements are immediately displayed through setting specific complete duality pairs.

Cotorsion pairs and model structures on Ch(R)

2011 ◽  
Vol 54 (3) ◽  
pp. 783-797 ◽  
Author(s):  
Gang Yang ◽  
Zhongkui Liu
Keyword(s):  
Model Structure ◽  
Cotorsion Pair ◽  
Projective Model ◽  
Cotorsion Pairs ◽  
Category Of Modules ◽  
The Given ◽  
Model Structures

AbstractWe show that if the given cotorsion pair $(\mathcal{A},\mathcal{B})$ in the category of modules is complete and hereditary, then both of the induced cotorsion pairs in the category of complexes are complete. We also give a cofibrantly generated model structure that can be regarded as a generalization of the projective model structure.

On a question of Gillespie

2015 ◽  
Vol 27 (6) ◽  
Author(s):  
Xiaoyan Yang ◽  
Nanqing Ding
Keyword(s):  
General Setting ◽  
Abelian Category ◽  
Cotorsion Pair ◽  
Cotorsion Pairs

AbstractA question of Gillespie concerning the completeness of the induced cotorsion pairs is settled in the general setting of any bicomplete abelian category. That is, given a complete hereditary cotorsion pair

Compactly generated triangulated subcategories of homotopy categories induced by cotorsion pairs

2018 ◽  
Vol 17 (09) ◽  
pp. 1850180 ◽  
Author(s):  
Wenjing Chen ◽  
Zhongkui Liu ◽  
Xiaoyan Yang
Keyword(s):  
Abelian Category ◽  
Cotorsion Pair ◽  
Cotorsion Pairs ◽  
Compactly Generated

In this paper, we investigate the homotopy categories [Formula: see text] and [Formula: see text] with respect to a complete and hereditary cotorsion pair [Formula: see text] in a bicomplete abelian category. We prove that [Formula: see text] is compactly generated provided that [Formula: see text] is compactly generated. We introduce and characterize Gorenstein [Formula: see text]-complexes with respect to [Formula: see text] and show that a complex is a Gorenstein [Formula: see text]-complex if and only if its every term is a Gorenstein [Formula: see text]-object. We also show that the inclusion functors [Formula: see text] and [Formula: see text] have right adjoints respectively when [Formula: see text] is compactly generated.

scholarly journals MODEL STRUCTURES ON TRIANGULATED CATEGORIES

2014 ◽  
Vol 57 (2) ◽  
pp. 263-284 ◽  
Author(s):  
XIAOYAN YANG
Keyword(s):  
Proper Class ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
General Study ◽  
Proper Class Of Triangles ◽  
Cotorsion Pairs ◽  
One To One ◽  
The Relationship ◽  
Model Structures

AbstractWe define model structures on a triangulated category with respect to some proper classes of triangles and give a general study of triangulated model structures. We look at the relationship between these model structures and cotorsion pairs with respect to a proper class of triangles on the triangulated category. In particular, we get Hovey's one-to-one correspondence between triangulated model structures and complete cotorsion pairs with respect to a proper class of triangles. Some applications are given.

One-sided triangulated categories induced by concentric twin cotorsion pairs

2019 ◽  
Vol 19 (08) ◽  
pp. 2050142
Author(s):  
Qilian Zheng ◽  
Jiaqun Wei
Keyword(s):  
Triangulated Categories ◽  
Cotorsion Pair ◽  
Cotorsion Pairs ◽  
Exact Categories

Extriangulated categories were introduced by Nakaoka and Palu by extracting the similarities between exact categories and triangulated categories. Nakaoka and Palu introduced the notion of concentric twin cotorsion pairs in extriangulated categories. In this paper, let [Formula: see text] be a concentric twin cotorsion pair in an extriangulated category and [Formula: see text], [Formula: see text], we prove that [Formula: see text] has one-sided triangulated structure.

Cut cotorsion pairs

2021 ◽  
pp. 1-38
Author(s):  
Mindy Huerta ◽  
Octavio Mendoza ◽  
Marco A. Pérez
Keyword(s):  
General Framework ◽  
Abelian Category ◽  
Triangulated Categories ◽  
Finitistic Dimension ◽  
Coherent Sheaves ◽  
Chain Complexes ◽  
Cotorsion Pairs ◽  
Gorenstein Homological Algebra ◽  
Serre Subcategories ◽  
Dimension Conjecture

Abstract We present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander–Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander–Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes, and quasi-coherent sheaves, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

scholarly journals Quillen model structures for relative homological algebra

2002 ◽  
Vol 133 (2) ◽  
pp. 261-293 ◽  
Author(s):  
J. DANIEL CHRISTENSEN ◽  
MARK HOVEY
Keyword(s):  
Homotopy Theory ◽  
Homological Algebra ◽  
Abelian Category ◽  
Derived Categories ◽  
Derived Category ◽  
Category Structure ◽  
Model Category ◽  
Projective Class ◽  
Chain Complexes ◽  
Model Structures

An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category [Ascr ] is exactly the information needed to do homological algebra in [Ascr ]. The main result is that, under weak hypotheses, the category of chain complexes of objects of [Ascr ] has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the ‘pure derived category’ of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and co-homology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories.

GORENSTEIN MODULES AND GORENSTEIN MODEL STRUCTURES

2017 ◽  
Vol 59 (3) ◽  
pp. 685-703 ◽  
Author(s):  
AIMIN XU
Keyword(s):  
Projective Dimension ◽  
Model Structure ◽  
Projective Modules ◽  
Cotorsion Pair ◽  
Chain Complexes ◽  
Gorenstein Projective Modules ◽  
Coherent Ring ◽  
Gorenstein Injective Dimension ◽  
Gorenstein Modules ◽  
Model Structures

AbstractGiven a complete hereditary cotorsion pair$(\mathcal{X}, \mathcal{Y})$, we introduce the concept of$(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integern, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤nover a left Noetherian ringR. Similarly, ifRis a left coherent ring in which all flat leftR-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤n. These structures have their analogous in the category of chain complexes.

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