Relative Homological Algebra

Enochs, Edgar E.; Jenda, Overtoun M. G.

doi:10.1515/9783110803662

Cited by 901 publications

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“…The flat conjecture (every module has a flat cover) was settled in the affirmative by Bican, El Bashir and Enochs [2]. Their result shows the existence of flat resolutions (in the sense of [7], Definition 8.1.2) over arbitrary rings. Things are a little different for Gorenstein homological algebra.…”

Section: Introductionmentioning

confidence: 96%

Gorenstein projective and flat complexes over noetherian rings

Enochs

Estrada

Iacob

2012

Mathematische Nachrichten

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Key words Precover, cover, Gorenstein flat complex, Gorenstein projective complex MSC (2010) 18G35, 18G25We give sufficient conditions on a class of R-modules C in order for the class of complexes of C-modules, dwC, to be covering in the category of complexes of R-modules. More precisely, we prove that if C is precovering in R − M od and if C is closed under direct limits, direct products, and extensions, then the class dwC is covering in Ch(R).Our first application concerns the class of Gorenstein flat modules. We show that when the ring R is two sided noetherian, a complex C is Gorenstein flat if and only if each module Cn is Gorenstein flat. If moreover every direct product of Gorenstein flat modules is a Gorenstein flat module, then the class of Gorenstein flat complexes is covering.We consider Gorenstein projective complexes as well. We prove that if R is a commutative noetherian ring of finite Krull dimension, then the class of Gorenstein projective complexes coincides with that of complexes of Gorenstein projective modules. We also show that if R is commutative noetherian with a dualizing complex then every right bounded complex has a Gorenstein projective precover.

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Section: Introductionmentioning

confidence: 96%

Gorenstein projective and flat complexes over noetherian rings

Enochs

Estrada

Iacob

2012

Mathematische Nachrichten

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“…This is extended to (infinite-dimensional) Iwanaga-Gorenstein rings [30] under restriction to Cohen-Macaulay modules; a stable equivalence of Iwanaga-Gorenstein rings is a triangle equivalence between the stable categories of Cohen-Macaulay modules over those rings.…”

Section: Introductionmentioning

confidence: 99%

Singularity categories and singular equivalences for resolving subcategories

Matsui

Takahashi

2016

Math. Z.

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Abstract. Let X be a resolving subcategory of an abelian category. In this paper we investigate the singularity category Dsg(X ) = D b (mod X )/K b (proj(mod X )) of the stable category X of X . We consider when the singularity category is triangle equivalent to the stable category of Gorenstein projective objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity categories. Applying this to the module category of a Gorenstein ring, we prove that the complete intersections over which the stable categories of resolving subcategories have trivial singularity categories are the simple hypersurface singularities of type (A 1 ). We also generalize several results of Yoshino on totally reflexive modules.

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“…The G-dimension has strong parallels to the projective dimension. Enochs and Jenda [9,10] extended the ideas of Auslander and Bridger, and introduced Gorenstein projective dimensions, which have all been studied extensively by their founders and by Avramov, Christensen, Foxby, Frankild, Holm, Martsinkovsky, and Xu among others [2,7,8,[11][12][13]15] over arbitrary associative rings. Bennis and Mahdou [4] studied a particular case of Gorenstein projective modules, which they call strongly Gorenstein projective (SG -projective for short) modules.…”

Section: Introductionmentioning

confidence: 99%