scholarly journals AB-Contexts and Stability for Gorenstein Flat Modules with Respect to Semidualizing Modules

2009 ◽  
Vol 14 (3) ◽  
pp. 403-428 ◽  
Author(s):  
Sean Sather-Wagstaff ◽  
Tirdad Sharif ◽  
Diana White
Keyword(s):  
Flat Modules ◽  
Semidualizing Modules ◽  
Gorenstein Flat

Buchweitz’s equivalences for Gorenstein flat modules with respect to semidualizing modules

2019 ◽  
pp. 2150006
Author(s):  
Jiangsheng Hu ◽  
Yuxian Geng ◽  
Jinyong Wu ◽  
Huanhuan Li
Keyword(s):  
Noetherian Ring ◽  
Full Subcategory ◽  
Stable Category ◽  
Commutative Noetherian Ring ◽  
Singularity Category ◽  
Flat Modules ◽  
Frobenius Category ◽  
Semidualizing Modules ◽  
Structure Formula ◽  
Gorenstein Flat

Let [Formula: see text] be a commutative Noetherian ring and [Formula: see text] a semidualizing [Formula: see text]-module. We obtain an exact structure [Formula: see text] and prove that the full subcategory [Formula: see text] of [Formula: see text] is a Frobenius category with [Formula: see text] the subcategory of projective and injective objects, where [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]) is the subcategory of [Formula: see text]-Gorenstein flat (respectively, [Formula: see text]-flat [Formula: see text]-cotorsion) [Formula: see text]-modules. Then the stable category [Formula: see text] of [Formula: see text] and the singularity category [Formula: see text] of [Formula: see text] are also considered. As a consequence, we get that there is a Buchweitz’s equivalence [Formula: see text] if and only if [Formula: see text] is a part of some AB-context.

Noncommutative G-semihereditary rings

2018 ◽  
Vol 17 (01) ◽  
pp. 1850014 ◽  
Author(s):  
Jian Wang ◽  
Yunxia Li ◽  
Jiangsheng Hu
Keyword(s):  
Associative Ring ◽  
Sufficient Condition ◽  
Projective Modules ◽  
The Third ◽  
Flat Modules ◽  
Gorenstein Projective Modules ◽  
Direct Limits ◽  
Gorenstein Flat ◽  
Finitely Presented ◽  
Semihereditary Rings

In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. Some new characterizations of left [Formula: see text]-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right [Formula: see text]-module is Gorenstein flat if and only if it is Gorenstein projective provided that [Formula: see text] is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, [Formula: see text]-[Formula: see text] rings and [Formula: see text] rings.

Notes on Gorenstein Flat Modules

2018 ◽  
Vol 45 (2) ◽  
pp. 337-344
Author(s):  
Yanjiong Yang ◽  
Xiaoguang Yan
Keyword(s):  
Flat Modules ◽  
Gorenstein Flat

Gorenstein flat modules relative to injectively resolving subcategories

2021 ◽  
pp. 2250099
Author(s):  
Zenghui Gao ◽  
Wan Wu
Keyword(s):  
Homological Dimension ◽  
Common Generalization ◽  
Flat Modules ◽  
Resolving Subcategory ◽  
Homological Dimensions ◽  
Gorenstein Flat

Let [Formula: see text] be an injectively resolving subcategory of left [Formula: see text]-modules. We introduce and study [Formula: see text]-Gorenstein flat modules as a common generalization of some known modules such as Gorenstein flat modules (Enochs, Jenda and Torrecillas, 1993), Gorenstein AC-flat modules (Bravo, Estrada and Iacob, 2018). Then we define a resolution dimension relative to the [Formula: see text]-Gorensteinflat modules, investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, stability of the category of [Formula: see text]-Gorensteinflat modules is discussed, and some known results are obtained as applications.

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.

scholarly journals Resolutions and stability of $C$-Gorenstein flat modules

2016 ◽  
Vol 46 (5) ◽  
pp. 1739-1753 ◽  
Author(s):  
Guoqiang Zhao ◽  
Xiaoguang Yan
Keyword(s):  
Flat Modules ◽  
Gorenstein Flat

scholarly journals Acyclic Complexes and Gorenstein Rings

2020 ◽  
Vol 27 (03) ◽  
pp. 575-586
Author(s):  
Sergio Estrada ◽  
Alina Iacob ◽  
Holly Zolt
Keyword(s):  
Analogous Result ◽  
Gorenstein Ring ◽  
Gorenstein Rings ◽  
Injective Modules ◽  
Flat Modules ◽  
Coherent Ring ◽  
Acyclic Complexes ◽  
Gorenstein Flat ◽  
Gorenstein Injective

For a given class of modules [Formula: see text], let [Formula: see text] be the class of exact complexes having all cycles in [Formula: see text], and dw([Formula: see text]) the class of complexes with all components in [Formula: see text]. Denote by [Formula: see text][Formula: see text] the class of Gorenstein injective R-modules. We prove that the following are equivalent over any ring R: every exact complex of injective modules is totally acyclic; every exact complex of Gorenstein injective modules is in [Formula: see text]; every complex in dw([Formula: see text][Formula: see text]) is dg-Gorenstein injective. The analogous result for complexes of flat and Gorenstein flat modules also holds over arbitrary rings. If the ring is n-perfect for some integer n ≥ 0, the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules. We also prove the following characterization of Gorenstein rings. Let R be a commutative coherent ring; then the following are equivalent: (1) every exact complex of FP-injective modules has all its cycles Ding injective modules; (2) every exact complex of flat modules is F-totally acyclic, and every R-module M such that M+ is Gorenstein flat is Ding injective; (3) every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that M+ is Gorenstein flat is Ding injective. If R has finite Krull dimension, statements (1)–(3) are equivalent to (4) R is a Gorenstein ring (in the sense of Iwanaga).

Gorenstein Flat Complexes with Respect to a Semidualizing Module

2015 ◽  
Vol 22 (02) ◽  
pp. 259-270
Author(s):  
Li Liang ◽  
Chunhua Yang
Keyword(s):  
Noetherian Ring ◽  
Commutative Noetherian Ring ◽  
Semidualizing Module ◽  
Flat Modules ◽  
Gorenstein Flat

In this paper, we introduce and study G C-flat complexes over a commutative Noetherian ring, where C is a semidualizing module. We prove that G C-flat complexes are actually the complexes of G C-flat modules. This complements a result of Yang and Liang. As an application, we get that every complex has a [Formula: see text]-cover, where [Formula: see text] is the class of G C-flat complexes. We also give a characterization of complexes of modules in [Formula: see text] that are defined by Sather-Wagstaff, Sharif and White.

scholarly journals Stability of Strongly Gorenstein Flat Modules

2013 ◽  
Vol 42 (2) ◽  
pp. 171-178 ◽  
Author(s):  
Zhanping Wang ◽  
Zhongkui Liu
Keyword(s):  
Flat Modules ◽  
Gorenstein Flat ◽  
Strongly Gorenstein Flat Modules
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