Hearts of twin cotorsion pairs on 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.

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Complete Cotorsion Pairs in Exact Categories

Taiwanese Journal of Mathematics ◽

10.11650/tjm/190503 ◽

2020 ◽

Vol 24 (1) ◽

pp. 19-30

Author(s):

Zhi-Wei Li

Keyword(s):

Cotorsion Pairs ◽

Exact Categories

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Relatively divisible and relatively flat objects in exact categories

Journal of Algebra and Its Applications ◽

10.1142/s0219498821500882 ◽

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.

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MODEL STRUCTURES ON TRIANGULATED CATEGORIES

Glasgow Mathematical Journal ◽

10.1017/s0017089514000299 ◽

2014 ◽

Vol 57 (2) ◽

pp. 263-284 ◽

Cited By ~ 5

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.

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New model structures and projective (injective) cotorsion pairs

Journal of Algebra and Its Applications ◽

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 ◽

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.

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