Hereditary abelian model categories

Gillespie, James

doi:10.1112/blms/bdw051

Cited by 42 publications

(26 citation statements)

References 60 publications

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“…So a good way to both construct and model an algebraic triangulated category is to construct a hereditary abelian model structure. For a recent survey, see [Gil16].…”

Section: Introductionmentioning

confidence: 99%

The flat stable module category of a coherent ring

Gillespie

2017

Journal of Pure and Applied Algebra

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“…So a good way to both construct and model an algebraic triangulated category is to construct a hereditary abelian model structure. For a recent survey, see [Gil16].…”

Section: Introductionmentioning

confidence: 99%

The flat stable module category of a coherent ring

Gillespie

2017

Journal of Pure and Applied Algebra

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“…We say that M is hereditary if both of these associated cotorsion pairs are hereditary. We refer to [13,17,18] for a more detailed discussion on model structures.…”

Section: The Gorenstein Flat Model Structures With Respect To Dualitymentioning

confidence: 99%

Gorenstein flat modules with respect to duality pairs

Wang

Yang

Zhu

2019

Communications in Algebra

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Let X be a class of left R-modules, Y be a class of right R-modules. In this paper, we introduce and study Gorenstein (X , Y)-flat modules as a common generalization of some known modules such as Gorenstein flat modules [9], Gorenstein n-flat modules [22], Gorenstein B-flat modules [7], Gorenstein AC-flat modules [2], Ω-Gorenstein flat modules [10] and so on. We show that the class of all Gorenstein (X , Y)-flat modules have a strong stability. In particular, when (X , Y) is a perfect (symmetric) duality pair, we give some functorial descriptions of Gorenstein (X , Y)-flat dimension, and construct a hereditary abelian model structure on R-Mod whose cofibrant objects are exactly the Gorenstein (X , Y)-flat modules. These results unify the corresponding results of the aforementioned modules.

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“…Standard references include [EJ01] and [GT06] and connections to abelian model categories can be found in [MH02] and [JG16b].…”

Section: Preliminariesmentioning

confidence: 99%

Ac-Gorenstein Rings and Their Stable Module categories

Gillespie¹

2018

J. Aust. Math. Soc.

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We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring which is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo-Gillespie-Hovey. It is also compatible with the notion of n-coherent rings introduced by Bravo-Perez: So a 0-coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a 1-coherent AC-Gorenstein ring is precisely a Ding-Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of two Quillen equivalent abelian model category structures. One is projective with cofibrant objects the Gorenstein AC-projective modules while the other is an injective model structure with fibrant objects the Gorenstein AC-injectives.

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Hereditary abelian model categories

Abstract: We discuss some recent developments in the theory of abelian model categories. The emphasis is on the hereditary condition and applications to homotopy categories of chain complexes and stable module categories.

Cited by 42 publications

References 60 publications

The flat stable module category of a coherent ring

The flat stable module category of a coherent ring

Gorenstein flat modules with respect to duality pairs

Ac-Gorenstein Rings and Their Stable Module categories

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