On a new cotorsion pair

10.1142/s0219498823500366 ◽

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 ◽

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

10.1142/s0219498823500172 ◽

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 ◽

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)

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091510000489 ◽

2011 ◽

Vol 54 (3) ◽

pp. 783-797 ◽

Cited By ~ 26

Author(s):

Gang Yang ◽

Zhongkui Liu

Keyword(s):

Model Structure ◽

Cotorsion Pair ◽

Projective Model ◽

Cotorsion Pairs ◽

Category Of Modules ◽

The Given ◽

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.

PURE-INJECTIVES RELATIVE TO A COTORSION PAIR: APPLICATIONS

Glasgow Mathematical Journal ◽

10.1017/s001708951200033x ◽

2012 ◽

Vol 55 (1) ◽

pp. 59-68

Author(s):

SERGIO ESTRADA ◽

PEDRO A. GUIL ASENSIO

Keyword(s):

Classical Theory ◽

General Class ◽

Cotorsion Pair ◽

Coherent Sheaves ◽

Accessible Categories

AbstractFinitely accessible categories naturally arise in the context of the classical theory of purity. In this paper we generalise the notion of purity for a more general class and introduce techniques to study such classes in terms of indecomposable pure injectives related to a new notion of purity. We apply our results in the study of the class of flat quasi-coherent sheaves on an arbitrary scheme.

Gorenstein category with respect to twin cotorsion pair

International Journal of Algebra ◽

10.12988/ija.2020.91281 ◽

2020 ◽

Vol 14 (4) ◽

pp. 287-296

Author(s):

Yunyun Zhang ◽

Jiafeng Lü

Keyword(s):

Cotorsion Pair

Complexes of FP-injective Modules

Algebra Colloquium ◽

10.1142/s1005386710000647 ◽

2010 ◽

Vol 17 (04) ◽

pp. 667-684

Author(s):

Yuxian Geng

Keyword(s):

Injective Dimension ◽

Cotorsion Pair ◽

Injective Modules ◽

Coherent Ring

In this paper, we study the class [Formula: see text] of complexes of FP-injective left R-modules. It is shown that the pair [Formula: see text] is a complete cotorsion pair. If R is a left coherent ring, it is proved that every complex has an [Formula: see text]-cover. We also introduce the FP-injective dimension of complexes. A special attention is paid to the dimension of homologically bounded above R-complexes over a left coherent ring which has a nice functorial description.

Cotorsion Pairs in 𝒞N(A)

Algebra Colloquium ◽

10.1142/s1005386717000384 ◽

2017 ◽

Vol 24 (04) ◽

pp. 577-602 ◽

Cited By ~ 2

Author(s):

Xiaoyan Yang ◽

Tianya Cao

Keyword(s):

Abelian Category ◽

Cotorsion Pair ◽

Cotorsion Pairs ◽

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.

SOME RESULTS ON TORSIONFREE MODULES

10.1142/s0219498812501381 ◽

2012 ◽

Vol 12 (01) ◽

pp. 1250138 ◽

Cited By ~ 1

Author(s):

JIANGSHENG HU ◽

NANQING DING

Keyword(s):

Cotorsion Pair ◽

Von Neumann ◽

Artinian Rings ◽

Von Neumann Regular Rings ◽

Von Neumann Regular ◽

Derived Functors ◽

Regular Rings

We study torsionfree and divisible dimensions in terms of right derived functors of -⊗-. We also investigate the cotorsion pair cogenerated by the class of cyclic torsionfree right R-modules. As applications, some new characterizations of von Neumann regular rings, F-rings and semisimple Artinian rings are given.

One-sided triangulated categories induced by concentric twin cotorsion pairs

10.1142/s021949882050142x ◽

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.

GORENSTEIN MODULES AND GORENSTEIN MODEL STRUCTURES

Glasgow Mathematical Journal ◽

10.1017/s0017089516000483 ◽

2017 ◽

Vol 59 (3) ◽

pp. 685-703 ◽

Cited By ~ 2

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 ◽

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.