Proper resolutions and Gorenstein categories

Huang, Zhaoyong

doi:10.1016/j.jalgebra.2013.07.008

Cited by 68 publications

(54 citation statements)

References 21 publications

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“…On the other hand, the middle column is Hom A (−, C )-exact by Proposition 2.7 (2). So G ′ ∈ G(C ) by [11,Proposition 4.7(5)], and hence the upper row is a special G(C )-precover of M ′ .…”

Section: The Special Precovered Category Of G(c )mentioning

confidence: 95%

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Special precovered categories of Gorenstein categories

Zhao

Huang

2018

Sci. China Math.

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Let A be an abelian category and P(A ) the subcategory of A consisting of projective objects. Let C be a full, additive and self-orthogonal subcategory of A with P(A ) a generator, and let G(C ) be the Gorenstein subcategory of A . Then the right 1-orthogonal category G(C ) ⊥ As a generalization of finitely generated projective modules, Auslander and Bridger introduced in [3] the notion of finitely generated modules of Gorenstein dimension zero over commutative Noetherian rings.Then Enochs and Jenda generalized it in [7] to arbitrary modules over a general ring and introduced the notion of Gorenstein projective modules and its dual (that is, the notion of Gorenstein injective modules). Let A be an abelian category and C an additive and full subcategory of A . Recently Sather-Wagstaff, Sharif and White introduced in [14] the notion of the Gorenstein subcategory G(C ) of A , which is a common generalization of the notions of modules of Gorenstein dimension zero [3], Gorenstein projective modules, Gorenstein injective modules [7], V -Gorenstein projective modules and V -Gorenstein injective modules [9], and so on. Let R be an associative ring with identity, and let Mod R be the category of left R-modules and G(P(Mod R) the subcategory of Mod R consisting of Gorenstein projective modules. Let PC(G(P(Mod R))and SPC(G(P(Mod R)) be the subcategories of Mod R consisting of modules admitting a G(P(Mod R))precover and admitting a special G(P(Mod R))-precover respectively. The following question in relative homological algebra remains still open: does PC(G(P(Mod R)) = Mod R always hold true? Several authors have gave some partially positive answers to this question, see [2,4,5,16]. Note that in these references, PC(G(P(Mod R)) = SPC(G(P(Mod R)), see Example 4.8 below for details. In particular, any module in Mod R with finite Gorenstein projective dimension admits a G(P(Mod R))-precover which is * 2010 Mathematics Subject Classification: 18G25, 18E10. †

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Section: The Special Precovered Category Of G(c )mentioning

confidence: 95%

“…By [11,Lemma 2.5], the middle row is both Hom A (C , −)-exact and Hom A (−, C )-exact, and hence G ′ ∈ G(C ) by [11,Proposition 4.7], that is, the middle row is the desired sequence.…”

Section: Preliminariesmentioning

confidence: 99%

Special precovered categories of Gorenstein categories

Zhao

Huang

2018

Sci. China Math.

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“…From the proof of [Hu,Theorem 3.2] we see that if (2.1) is Hom A (E , −)-exact, then so is (2.2). Because C is closed under kernels of (E -proper) epimorphisms and A 3 is an object in C by assumption, C is an object in C and C -dim A 1 ≤ n.…”

Section: Preliminariesmentioning

confidence: 99%

“…be an exact sequence in A with all C i objects in C . By [Hu,Theorem 3.2], there exist exact sequences:…”

Section: Preliminariesmentioning

confidence: 99%

Homological dimensions relative to preresolving subcategories

Huang¹

2014

Kyoto J. Math.

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We introduce relative preresolving subcategories and precoresolving subcategories of an abelian category and define homological dimensions and codimensions relative to these subcategories respectively. We study the properties of these homological dimensions and codimensions and unify some important properties possessed by some known homological dimensions. Then we apply the obtained properties to special subcategories and in particular to module categories. Finally we propose some open questions and conjectures, which are closely related to the generalized Nakayama conjecture and the strong Nakayama conjecture.

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“…Let A be an abelian category and W an additive full subcategory of A. Huang [3] provided a method for constructing a proper W-resolution (respectively, coproper W-coresolution) of one term in a short exact sequence in A from those of the other two terms. By using these, he affirmatively answered an open question on the stability of the Gorenstein category G(W) posed by Sather-Wagstaff, Sharif and White [7] and also proved that G(W) is closed under direct summands.…”

Section: Introductionmentioning

confidence: 99%

Proper classes and Gorensteinness in extriangulated categories

2020

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Let (C, E, s) be an extriangulated category with a proper class ξ of E-triangles, and W an additive full subcategory of C. We provide a method for constructing a proper W(ξ)-resolution (respectively, coproper W(ξ)-coresolution) of one term in an E-triangle in ξ from that of the other two terms. By using this way, we establish the stability of the Gorenstein category GW(ξ) in extriangulated categories. These results generalise their work by Huang and Yang-Wang, but the proof is not too far from their case. Finally, we give some applications about our main results.

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Proper resolutions and Gorenstein categories

Cited by 68 publications

References 21 publications

Special precovered categories of Gorenstein categories

Special precovered categories of Gorenstein categories

Homological dimensions relative to preresolving subcategories

Proper classes and Gorensteinness in extriangulated categories

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