Right <i>n</i>-angulated categories arising from covariantly finite subcategories

Lin, Zengqiang

doi:10.1080/00927872.2016.1175591

Cited by 20 publications

(7 citation statements)

References 12 publications

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“…This was first considered in [38], [39] (where it would be called a co-suspended category), and later in [3,8,9]. A higher dimensional version has also been introduced in [42].…”

Section: Left Triangulated Categoriesmentioning

confidence: 99%

$$d\mathbb {Z}$$-Cluster tilting subcategories of singularity categories

Kvamme

2020

Math. Z.

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For an exact category $${{\mathcal {E}}}$$ E with enough projectives and with a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory, we show that the singularity category of $${{\mathcal {E}}}$$ E admits a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory. To do this we introduce cluster tilting subcategories of left triangulated categories, and we show that there is a correspondence between cluster tilting subcategories of $${{\mathcal {E}}}$$ E and $${\underline{{{\mathcal {E}}}}}$$ E ̲ . We also deduce that the Gorenstein projectives of $${{\mathcal {E}}}$$ E admit a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory under some assumptions. Finally, we compute the $$d\mathbb {Z}$$ d Z -cluster tilting subcategory of the singularity category for a finite-dimensional algebra which is not Iwanaga–Gorenstein.

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“…This was first considered in [38], [39] (where it would be called a co-suspended category), and later in [3,8,9]. A higher dimensional version has also been introduced in [42].…”

Section: Left Triangulated Categoriesmentioning

confidence: 99%

$$d\mathbb {Z}$$-Cluster tilting subcategories of singularity categories

Kvamme

2020

Math. Z.

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“…We recall the notion of a right (n + 2)-angulated category from [L,Definition 2.1]. Compare with [L, Definition 2.1], the condition (RN1)(a) is slightly different from that in [L], we don't assume that the class Θ is closed under direct summands.…”

Section: Right (N + 2)-angulated Categoriesmentioning

confidence: 99%

From $n$-exangulated categories to $n$-abelian categories

Liu

Zhou

2020

Preprint

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Herschend-Liu-Nakaoka introduced the notion of n-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of n-exact categories in the sense of Jasso and (n + 2)-angulated in the sense of Geiss-Keller-Oppermann. Let C be an n-exangulated category with enough projectives and enough injectives, and X a cluster tilting subcategory of C . In this article, we show that the quotient category C /X is an n-abelian category.This extends a result of Zhou-Zhu for (n + 2)-angulated categories. Moreover, it highlights new phenomena when it is applied to n-exact categories.

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“…Since the proof is similar to [L1, Theorem 3.7], we omit it. See also [L2,Remark 3.8] Theorem 3.2. Let C be an (n + 2)-angulated category with split idempotents and X an additive subcategory of C .…”

Section: N-abelian Quotient Categoriesmentioning

confidence: 99%

n-Abelian quotient categories

Zhou

Zhu

2019

Journal of Algebra

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Let C be an (n + 2)-angulated category with shift functor Σ and X be a cluster-tilting subcategory of C . Then we show that the quotient category C /X is an n-abelian category.If C has a Serre functor, then C /X is equivalent to an n-cluster tilting subcategory of an abelian category mod(Σ −1 X ). Moreover, we also prove that mod(Σ −1 X ) is Gorenstein of Gorenstein dimension at most n. As an application, we generalize recent results of Jacobsen-Jørgensen and Koenig-Zhu.Definition 1.1. Let C be an (n + 2)-angulated category with an autoequivalence Σ and X an additive subcategory of C . X is called cluster-tilting if (1) Hom C (X , ΣX ) = 0.(2) For any object C ∈ C , there exists an (n + 2)-angleAn object X is called cluster-tilting in the sense of Opperman-Thomas [OT, Definition 5.3] if add(X) is clsuter-tilting.Recently, Jacobsen-Jørgensen showed that the following result, which was obtained in a special case in the first part of [OT, Theorem 5.6].Theorem 1.2. [JJ, Theorem 0.6] Let C be a k-linear Hom-finite (n + 2)-angulated category with split idempotents, where k is an algebraically closed field and n is a positive integer.Assume that C has a Serre functor S, that is, an autoequivlence for which there are natural equivalenceswhere D(−) = Hom k (−, k) is the k-linear duality functor and Y, Z ∈ C . Assume that X is a cluster tilting object in C with Γ = End C (X). Then

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Right n-angulated categories arising from covariantly finite subcategories

Cited by 20 publications

References 12 publications

$$d\mathbb {Z}$$-Cluster tilting subcategories of singularity categories

$$d\mathbb {Z}$$-Cluster tilting subcategories of singularity categories

From $n$-exangulated categories to $n$-abelian categories

n-Abelian quotient categories

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