Gorenstein homological dimensions

Holm, Henrik

doi:10.1016/j.jpaa.2003.11.007

Cited by 631 publications

(689 citation statements)

References 15 publications

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“…For a Gorenstein flat module G its character module G + is Gorenstein injective (by [6]), so we have that Ext i (F + , G + ) = 0 for all i ≥ 1 (because F + is injective). Thus Ext i (G, F ++ ) = 0 and therefore Ext i (G, F ) = 0 for all i ≥ 1.…”

Section: Resultsmentioning

confidence: 99%

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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Section: Resultsmentioning

confidence: 99%

Gorenstein Projective Precovers

2017

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“…This follows from the fact that the Gorenstein projective dimension of any Gorenstein flat module is at most d in view of the characterization of Gorenstein projective complexes 5.1. Therefore we conclude that the Gorenstein projective dimension of X is at most d. A complex version of [H,Theorem 2.10] implies that there exists a short exact sequence…”

Section: The Homotopy Categories K(gprj R) and K(ginj R)mentioning

confidence: 69%

Homotopy category of projective complexes and complexes of Gorenstein projective modules

2014

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Abstract. Let R be a ring with identity and C(R) denote the category of complexes of R-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over R, denoted K(Prj C(R)), is always well generated and is compactly generated provided K(Prj R) is so. Based on this result, it will be proved that the class of Gorenstein projective complexes is precovering, whenever R is a commutative noetherian ring of finite Krull dimension. Furthermore, it turns out that over such rings the inclusion functor ι : K(GPrj R) ֒→ K(R) has a right adjoint ιρ, where K(GPrj R) is the homotopy category of Gorenstein projective R modules. Similar, or rather dual, results for the injective (resp. Gorenstein injective) complexes will be provided. If R has a dualising complex, a triangle-equivalence between homotopy categories of projective and of injective complexes will be provided. As an application, we obtain an equivalence between the triangulated categories K(GPrj R) and K(GInj R), that restricts to an equivalence between K(Prj R) and K(Inj R), whenever R is commutative, noetherian and admits a dualising complex.

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“…Then [11] Covers and envelopes 395 is split and M + ∈ GI by [18,Theorem 3.6]. Here GI denotes the class of Gorenstein injective modules.…”

Section: L 27 [1 Proposition 41]mentioning

confidence: 99%

SOME COVERS AND ENVELOPES IN THE CHAIN COMPLEX CATEGORY OF R-MODULES

Wang¹,

Liu²

2011

J. Aust. Math. Soc.

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We study the existence of some covers and envelopes in the chain complex category of R-modules. Let (A, B) be a cotorsion pair in R-Mod and let E A stand for the class of all exact complexes with each term in A. We prove that (E A, E A ⊥ ) is a perfect cotorsion pair whenever A is closed under pure submodules, cokernels of pure monomorphisms and direct limits and so every complex has an E A-cover.As an application we show that every complex of R-modules over a right coherent ring R has an exact Gorenstein flat cover. In addition, the existence of A-covers and B-envelopes of special complexes is considered where A and B denote the classes of all complexes with each term in A and B, respectively. 2010 Mathematics subject classification: primary 18G35; secondary 55U15, 03C35.

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Gorenstein homological dimensions

Cited by 631 publications

References 15 publications

Gorenstein Projective Precovers

Gorenstein Projective Precovers

Homotopy category of projective complexes and complexes of Gorenstein projective modules

SOME COVERS AND ENVELOPES IN THE CHAIN COMPLEX CATEGORY OF R-MODULES

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