scholarly journals Relatively divisible and relatively flat objects in exact categories

2020 ◽  
pp. 2150088
Author(s):  
Septimiu Crivei ◽  
Derya Keski̇n Tütüncü
Keyword(s):  
Approximation Theory ◽  
Closure Properties ◽  
Cotorsion Pair ◽  
Exact Category ◽  
Galois Connections ◽  
Cotorsion Pairs ◽  
Exact Categories

We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair [Formula: see text] in an exact category [Formula: see text], [Formula: see text] coincides with the class of relatively flat objects of [Formula: see text] for some relative projectively generated exact structure, while [Formula: see text] coincides with the class of relatively divisible objects of [Formula: see text] for some relative injectively cogenerated exact structure. We exhibit Galois connections between relative cotorsion pairs in exact categories, relative projectively generated exact structures and relative injectively cogenerated exact structures in additive categories. We establish closure properties and characterizations in terms of the approximation theory.

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.

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.

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.

scholarly journals Covering classes and 1-tilting cotorsion pairs over commutative rings

2021 ◽  
Vol 33 (3) ◽  
pp. 601-629
Author(s):  
Silvana Bazzoni ◽  
Giovanna Le Gros
Keyword(s):  
New York ◽  
Projective Dimension ◽  
Commutative Rings ◽  
Cotorsion Pair ◽  
Bijective Correspondence ◽  
Ring Of Quotients ◽  
Cotorsion Pairs ◽  
Rings Of Quotients ◽  
Covering Class ◽  
Quotient Rings

Abstract We are interested in characterising the commutative rings for which a 1-tilting cotorsion pair ( 𝒜 , 𝒯 ) {(\mathcal{A},\mathcal{T})} provides for covers, that is when the class 𝒜 {\mathcal{A}} is a covering class. We use Hrbek’s bijective correspondence between the 1-tilting cotorsion pairs over a commutative ring R and the faithful finitely generated Gabriel topologies on R. Moreover, we use results of Bazzoni–Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for 1-tilting cotorsion pairs arising from flat injective ring epimorphisms. Explicitly, if 𝒢 {\mathcal{G}} is the Gabriel topology associated to the 1-tilting cotorsion pair ( 𝒜 , 𝒯 ) {(\mathcal{A},\mathcal{T})} , and R 𝒢 {R_{\mathcal{G}}} is the ring of quotients with respect to 𝒢 {\mathcal{G}} , we show that if 𝒜 {\mathcal{A}} is covering, then 𝒢 {\mathcal{G}} is a perfect localisation (in Stenström’s sense [B. Stenström, Rings of Quotients, Springer, New York, 1975]) and the localisation R 𝒢 {R_{\mathcal{G}}} has projective dimension at most one as an R-module. Moreover, we show that 𝒜 {\mathcal{A}} is covering if and only if both the localisation R 𝒢 {R_{\mathcal{G}}} and the quotient rings R / J {R/J} are perfect rings for every J ∈ 𝒢 {J\in\mathcal{G}} . Rings satisfying the latter two conditions are called 𝒢 {\mathcal{G}} -almost perfect.

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

scholarly journals LOCALLY COMPACT OBJECTS IN EXACT CATEGORIES

2011 ◽  
Vol 22 (12) ◽  
pp. 1787-1821 ◽  
Author(s):  
LUIGI PREVIDI
Keyword(s):  
Compact Objects ◽  
Locally Compact ◽  
Exact Category ◽  
Mutual Relations ◽  
Exact Categories

We identify two categories of locally compact objects on an exact category [Formula: see text]. They correspond to the well-known constructions of the Beilinson category [Formula: see text] and the Kato category [Formula: see text]. We study their mutual relations and compare the two constructions. We prove that [Formula: see text] is an exact category, which gives to this category a very convenient feature when dealing with K-theoretical invariants, and study the exact structure of the category [Formula: see text] of Tate spaces. It is natural therefore to consider the Beilinson category [Formula: see text] as the most convenient candidate to the role of the category of locally compact objects over an exact category. We also show that the categories [Formula: see text], [Formula: see text] of countably indexed ind/pro-objects over any category [Formula: see text] can be described as localizations of categories of diagrams over [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document