Projectively Coresolved Gorenstein Flat Modules and Gorenstein AC Projective Modules

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.

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Model structures, recollements and duality pairs

Journal of Algebra and Its Applications ◽

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 ◽

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.

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STRONGLY GORENSTEIN FLAT MODULES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788708000761 ◽

2009 ◽

Vol 86 (3) ◽

pp. 323-338 ◽

Cited By ~ 44

Author(s):

NANQING DING ◽

YUANLIN LI ◽

LIXIN MAO

Keyword(s):

Exact Sequence ◽

Projective Modules ◽

Flat Dimension ◽

Flat Modules ◽

Coherent Rings ◽

Gorenstein Flat ◽

Strongly Gorenstein Flat Modules

AbstractIn this paper, strongly Gorenstein flat modules are introduced and investigated. An R-module M is called strongly Gorenstein flat if there is an exact sequence ⋯→P1→P0→P0→P1→⋯ of projective R-modules with M=ker (P0→P1) such that Hom(−,F) leaves the sequence exact whenever F is a flat R-module. Several well-known classes of rings are characterized in terms of strongly Gorenstein flat modules. Some examples are given to show that strongly Gorenstein flat modules over coherent rings lie strictly between projective modules and Gorenstein flat modules. The strongly Gorenstein flat dimension and the existence of strongly Gorenstein flat precovers and pre-envelopes are also studied.

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STABILITY OF GORENSTEIN FLAT MODULES

Glasgow Mathematical Journal ◽

10.1017/s0017089511000516 ◽

2011 ◽

Vol 54 (1) ◽

pp. 169-175 ◽

Cited By ~ 6

Author(s):

SAMIR BOUCHIBA ◽

MOSTAFA KHALOUI

Keyword(s):

Projective Modules ◽

Natural Question ◽

Flat Module ◽

Flat Modules ◽

Closed Ring ◽

Gorenstein Projective Modules ◽

Gorenstein Categories ◽

Gorenstein Flat ◽

Gorenstein Injective ◽

Gorenstein Flat Module

AbstractSather-Wagstaff et al. proved in [8] (S. Sather-Wagsta, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc.(2), 77(2) (2008), 481–502) that iterating the process used to define Gorenstein projective modules exactly leads to the Gorenstein projective modules. Also, they established in [9] (S. Sather-Wagsta, T. Sharif and D. White, AB-contexts and stability for Goren-stein at modules with respect to semi-dualizing modules, Algebra Represent. Theory14(3) (2011), 403–428) a stability of the subcategory of Gorenstein flat modules under a procedure to build R-modules from complete resolutions. In this paper we are concerned with another kind of stability of the class of Gorenstein flat modules via-à-vis the very Gorenstein process used to define Gorenstein flat modules. We settle in affirmative the following natural question in the setting of a left GF-closed ring R: Given an exact sequence of Gorenstein flat R-modules G = ⋅⋅⋅ G2G1G0G−1G−2 ⋅⋅⋅ such that the complex H ⊗RG is exact for each Gorenstein injective right R-module H, is the module M:= Im(G0 → G−1) a Gorenstein flat module?

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Notes on Gorenstein Flat Modules

Bulletin of the Iranian Mathematical Society ◽

10.1007/s41980-018-0135-5 ◽

2018 ◽

Vol 45 (2) ◽

pp. 337-344

Author(s):

Yanjiong Yang ◽

Xiaoguang Yan

Keyword(s):

Flat Modules ◽

Gorenstein Flat

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Gorenstein flat modules relative to injectively resolving subcategories

Journal of Algebra and Its Applications ◽

10.1142/s0219498822500992 ◽

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.

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QUILLEN EQUIVALENT MODELS FOR THE DERIVED CATEGORY OF FLATS AND THE RESOLUTION PROPERTY

Journal of the Australian Mathematical Society ◽

10.1017/s1446788720000075 ◽

2020 ◽

pp. 1-19

Author(s):

SERGIO ESTRADA ◽

ALEXANDER SLÁVIK

Keyword(s):

Vector Bundles ◽

Derived Categories ◽

Derived Category ◽

Projective Modules ◽

Dimensional Vector ◽

Infinite Dimensional ◽

Equivalent Models ◽

Flat Modules ◽

Quillen Equivalent ◽

Resolution Property

We investigate the assumptions under which a subclass of flat quasicoherent sheaves on a quasicompact and semiseparated scheme allows us to ‘mock’ the homotopy category of projective modules. Our methods are based on module-theoretic properties of the subclass of flat modules involved as well as their behaviour with respect to Zariski localizations. As a consequence we get that, for such schemes, the derived category of flat quasicoherent sheaves is equivalent to the derived category of very flat quasicoherent sheaves. If, in addition, the scheme satisfies the resolution property then both derived categories are equivalent to the derived category of infinite-dimensional vector bundles. The equivalences are inferred from a Quillen equivalence between the corresponding models.

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