Tilting objects in triangulated categories

Hu, Yonggang; Yao, Huiqin; Fu, Xuerong

doi:10.1080/00927872.2019.1648646

Cited by 3 publications

(6 citation statements)

References 19 publications

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“…In this article, we introduce and study the notion of ξ-tilting object in an extriangulated category, where ξ is a proper class of E-triangles. This extends results in [11].…”

supporting

confidence: 90%

“…This definition generalizes the classical ones for artin algebras given by Happel and Ringel [6] and Miyashita [14], as well as the one given by Y.G. Hu [11] for ξ-tilting object in a triangulated category.…”

Section: Resultssupporting

confidence: 54%

“…The aim of this paper is to extend the results of Y.G. Hu et al [11] to extriangulated categories. We introduce the notion of ξ-tilting object in a triangulated category and study their properties.…”

Section: Introductionmentioning

confidence: 81%

“…In section 3, we extend Y.G. Hu's notion of ξ-tilting object in a triangulated category [11] to ξ-tilting object in an extriangulated category and we state and prove our main result. We get the Bazzoni characterization of ξ-tilting object in extriangulated category as follows.…”

Section: Introductionmentioning

confidence: 97%

“…Recently, Y.G. Hu et al [11] introduced the notion of ξ-tilting object in a triangulated category by means of the homological theory in the triangulated category.…”

Section: Introductionmentioning

confidence: 99%

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$ξ$-tilting objects in extriangulated categories

Mei¹,

Wei

2020

Preprint

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“…In this article, we introduce and study the notion of ξ-tilting object in an extriangulated category, where ξ is a proper class of E-triangles. This extends results in [11].…”

supporting

confidence: 90%

Section: Resultssupporting

confidence: 54%

Section: Introductionmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 97%

“…Recently, Y.G. Hu et al [11] introduced the notion of ξ-tilting object in a triangulated category by means of the homological theory in the triangulated category.…”

Section: Introductionmentioning

confidence: 99%

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$ξ$-tilting objects in extriangulated categories

Mei¹,

Wei

2020

Preprint

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Relative homological dimensions in recollements of extriangulated categories

Zhang

Zhao

2023

J. Algebra Appl.

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Let [Formula: see text] be a recollement of extriangulated categories with a negative first extension and [Formula: see text] be a generative subcategory of [Formula: see text]. We prove that [Formula: see text] and [Formula: see text] have a generative subcategory, respectively. Using these subcategories, we obtain proper classes of [Formula: see text]-triangles. Moreover, we compare [Formula: see text] and [Formula: see text] with respect to the [Formula: see text]-global dimensions of [Formula: see text] under mild conditions. In particular, we show that the finiteness of the [Formula: see text]-global dimensions of extriangulated categories is invariant with respect to the recollements of extriangulated categories. As an application, these results generalize the work by Hu-Yao in triangulated categories.

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Balanced pairs and recollements in extriangulated categories with negative first extensions

He¹,

Zhou²

2021

Preprint

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A notion of balanced pairs in an extriangulated category with a negative first extension is defined in this article. We prove that there exists a bijective correspondence between balanced pairs and proper classes ξ with enough ξ-projectives and enough ξ-injectives.It can be regarded as a simultaneous generalization of Fu-Hu-Zhang-Zhu and Wang-Li-Huang. Besides, we show that if (A, B, C) is a recollement of extriangulated categories, then balanced pairs in B can induce balanced pairs in A and C under natural assumptions. As a application, this result gengralizes a result by Fu-Hu-Yao in abelian categories. Moreover, it highlights a new phenomena when it applied to triangulated categories.

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Tilting objects in triangulated categories

Cited by 3 publications

References 19 publications

$ξ$-tilting objects in extriangulated categories

$ξ$-tilting objects in extriangulated categories

Relative homological dimensions in recollements of extriangulated categories

Balanced pairs and recollements in extriangulated categories with negative first extensions

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