Singular compactness and definability for $$\Sigma $$-cotorsion and Gorenstein modules

Šaroch, Jan; Šťovíček, Jan

doi:10.1007/s00029-020-0543-2

Cited by 86 publications

(124 citation statements)

References 38 publications

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“…By the above we have that DP = PGF . Also, by [13], Corollary 3.5, we have that over any coherent ring R, PGF = GP ac .…”

Section: Resultsmentioning

confidence: 91%

“…The class of projectively coresolved Gorenstein flat modules is denoted PGF . As noted in [13], these modules are both Gorenstein projective and Gorenstein flat. Also, any projectively coresolved Gorenstein flat module is Ding projective (see section 2).…”

Section: Recently Saroch Andmentioning

confidence: 80%

“…Over any ring R, we have that PGF ⊆ GF (by definition). By [13] we have GP ac ⊆ PGF ⊆ GP over any ring R (see [13], page 15 and Theorem 3.4).…”

Section: Resultsmentioning

confidence: 99%

“…If M ′ is a Gorenstein projective module then M ′ is a direct summand of a strongly Gorenstein projective module M (by[1]). By the above M ∈ PGF , and by[13] Theorem 3.9, this class is closed under direct summands. Thus M ′ is projectively coresolved Gorenstein flat.…”

mentioning

confidence: 88%

“…Stovicek introduced a new class of modules (in [13]) -the projectively coresolved Gorenstein flat modules, or P GFmodules for short. They are the cycles of exact complexes of projective modules that remain exact when tensored with any injective module.…”

Section: Recently Saroch Andmentioning

confidence: 99%

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Projectively coresolved Gorenstein flat and ding projective modules

Iacob

2020

Communications in Algebra

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We give necessary and sufficient conditions in order for the class of projectively coresolved Gorenstein flat modules, PGF , (respectively that of projectively coresolved Gorenstein B flat modules, PGF B ) to coincide with the class of Ding projective modules (DP). We show that PGF = DP if and only if every Ding projective module is Gorenstein flat. This is the case if the ring R is coherent for example. We include an example to show that the coherence is a sufficient, but not a necessary condition in order to have PGF = DP. We also show that PGF = DP over any ring R of finite weak Gorenstein global dimension (this condition is also sufficient, but not necessary). We prove that if the class of Ding projective modules, DP, is covering then the ring R is perfect. And we show that, over a coherent ring R, the converse also holds. We also give necessary and sufficient conditions in order to have PGF = GP, where GP is the class of Gorenstein projective modules.2010 MSC: 16E05, 16E10.

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“…By the above we have that DP = PGF . Also, by [13], Corollary 3.5, we have that over any coherent ring R, PGF = GP ac .…”

Section: Resultsmentioning

confidence: 91%

Section: Recently Saroch Andmentioning

confidence: 80%

“…Over any ring R, we have that PGF ⊆ GF (by definition). By [13] we have GP ac ⊆ PGF ⊆ GP over any ring R (see [13], page 15 and Theorem 3.4).…”

Section: Resultsmentioning

confidence: 99%

mentioning

confidence: 88%

Section: Recently Saroch Andmentioning

confidence: 99%

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Projectively coresolved Gorenstein flat and ding projective modules

Iacob

2020

Communications in Algebra

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Gorenstein weak global dimension is symmetric

Christensen

Estrada

Thompson

2021

Mathematische Nachrichten

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Modules of finite Gorenstein flat dimension and approximations

Emmanouil

2023

Mathematische Nachrichten

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We study approximations of modules of finite Gorenstein flat dimension by (projectively coresolved) Gorenstein flat modules and modules of finite flat dimension. These approximations determine the Gorenstein flat dimension and lead to descriptions of the corresponding relative homological dimensions, for such modules, in more classical terms. We also describe two hereditary Hovey triples on the category of modules of finite Gorenstein flat dimension, whose associated exact structures have homotopy categories equivalent to the stable category of projectively coresolved Gorenstein flat modules and the stable category of cotorsion Gorenstein flat modules, respectively.

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Singular compactness and definability for $$\Sigma $$-cotorsion and Gorenstein modules

Cited by 86 publications

References 38 publications

Projectively coresolved Gorenstein flat and ding projective modules

Projectively coresolved Gorenstein flat and ding projective modules

Gorenstein weak global dimension is symmetric

Modules of finite Gorenstein flat dimension and approximations

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