Gorenstein flat and projective (pre)covers

Estrada, Sergio; Iacob, Alina; Odabasi, Sinem

doi:10.5486/pmd.2017.7689

Cited by 13 publications

(19 citation statements)

References 7 publications

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“…ln particular, (3) says that if R is a (right) coherent ring in which all flat (left) modules have finite projective dimension (called left n-perfect ), then every chain complex has a special Gorenstein projective precover. This was also recently established in [EIO15] and [Gil16b]. The same results of Corollary 5.6, but for Rmodules, are proved in [BGH14].…”

Section: The Gorenstein Ac-projective Model Structure On Complexessupporting

confidence: 61%

Gorenstein AC-projective complexes

Gillespie

2018

J. Homotopy Relat. Struct.

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Let R be any ring with identity and Ch(R) the category of chain complexes of (left) R-modules. We show that the Gorenstein AC-projective chain complexes of [BG16] are the cofibrant objects of an abelian model structure on Ch(R). The model structure is cofibrantly generated and is projective in the sense that the trivially cofibrant objects are the categorically projective chain complexes. We show that when R is a Ding-Chen ring, that is, a two-sided coherent ring with finite self FP-injective dimension, then the model structure is finitely generated, and so its homotopy category is compactly generated. Constructing this model structure also shows that every chain complex over any ring has a Gorenstein AC-projective precover. These are precisely Gorenstein projective (in the usual sense) precovers whenever R is either a Ding-Chen ring, or, a ring for which all level (left) R-modules have finite projective dimension. For a general (right) coherent ring R, the Gorenstein AC-projective complexes coincide with the Ding projective complexes of [YLL13] and so provide such precovers in this case.

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Section: The Gorenstein Ac-projective Model Structure On Complexessupporting

confidence: 61%

Gorenstein AC-projective complexes

Gillespie

2018

J. Homotopy Relat. Struct.

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“…We already proved in [4] that the class GP is special precovering over any right coherent and left n-perfect ring R. For completeness, we included a different proof here.…”

Section: Resultsmentioning

confidence: 99%

“…More recently (2011), Murfet and Salarian proved the existence of the Gorenstein projective precovers over commutative noetherian rings of finite Krull dimension. In [4] we extended their result: we proved that if R is a right coherent and left n-perfect ring, then the class of Gorenstein projective complexes is special precovering in the category of unbounded complexes, Ch(R). As a corollary we obtained the existence of the special Gorenstein projective precovers in R −Mod over the same type of rings.…”

Section: Introductionmentioning

confidence: 87%

Gorenstein Projective Precovers

2017

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Abstract. We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In section 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Gorenstein projective dimension) and that are not right coherent.

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“…It is also proved in[12] that the pair (GProj(R), GProj(R) ⊥ ) is hereditary projective cotorsion pair over right coherent and left n-perfect rings 3. This holds for example if R is Iwanaga-Gorenstein, since in this case GProj(R) ⊥ is the class of modules of finite projective dimension[20, Thm.…”

mentioning

confidence: 95%

Abelian model structures on categories of quiver representations

Dalezios

2019

J. Algebra Appl.

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Let M be an abelian model category (in the sense of Hovey). For a large class of quivers, we describe associated abelian model structures on categories of quiver representations with values in M. This is based on recent work of Holm and Jørgensen on cotorsion pairs in categories of quiver representations. An application on Ding projective and Ding injective representations of quivers over Ding-Chen rings is given.

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Gorenstein flat and projective (pre)covers

Abstract: We consider a right coherent ring R. We prove that the class of Gorenstein flat complexes is covering in the category of complexes of left R-modules Ch(R).

Cited by 13 publications

References 7 publications

Gorenstein AC-projective complexes

Gorenstein AC-projective complexes

Gorenstein Projective Precovers

Abelian model structures on categories of quiver representations

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