Frobenius n-exangulated categories

Liu, Yu; Zhou, Panyue

doi:10.1016/j.jalgebra.2020.03.036

Cited by 29 publications

(35 citation statements)

References 7 publications

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“…Moreover, X ′ is not an injective object by the dual of [23,Lemma 3.4]. Hence we have an Auslander-Reiten n-exangle of the form by Lemma 3.7…”

Section: A Map Formmentioning

confidence: 95%

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Auslander-Reiten-Serre duality for n-exangulated categories

He¹,

He²,

Zhou³

2021

Preprint

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Let (C , E, s) be an Ext-finite, Krull-Schmidt and k-linear n-exangulated category with k a commutative artinian ring. In this note, we prove that C has Auslander-Reiten-Serre duality if and only if C has Auslander-Reiten n-exangles. Moreover, we also give an equivalent condition for the existence of Serre duality (which is a special type of Auslander-Reiten-Serre duality). Finally, assume further that C has Auslander-Reiten-Serre duality. We exploit a bijection triangle, which involves the restricted Auslander bijection and the Auslander-Reiten-Serre duality.

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“…Moreover, X ′ is not an injective object by the dual of [23,Lemma 3.4]. Hence we have an Auslander-Reiten n-exangle of the form by Lemma 3.7…”

Section: A Map Formmentioning

confidence: 95%

“…It gives a common generalization of n-exact categories (n-abelian categories are also n-exact categories) in the sense of Jasso [17] and (n + 2)-angulated categories in the sense of Geiss-Keller-Oppermann [9]. However, there are some other examples of n-exangulated categories which are neither n-exact nor (n + 2)angulated, see [13,15,23].…”

Section: Introductionmentioning

confidence: 99%

Auslander-Reiten-Serre duality for n-exangulated categories

He¹,

He²,

Zhou³

2021

Preprint

Self Cite

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“…In this subsection, we describe our results using the framework of n-exangulated categories. We refer to the readers to [39], [16] and [37] for the relevant Definitions and facts concerning n-exangulated categories. Proof.…”

Section: ]mentioning

confidence: 99%

“…We define the functor S : addA/[P] → addA/[P] such that it takes M to Q n . By [37,Proposition 3.7], the S functor is well defined and it is an auto-equivalence. It is easy to see that S(M ) is isomorphic to…”

Section: The Space Hommentioning

confidence: 99%

Relative cluster categories and Higgs categories

Wu¹

2021

Preprint

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We generalize the construction of (higher) cluster categories by Claire Amiot and by Lingyan Guo to the relative context. We prove the existence of an n-cluster tilting object in a Frobenius extriangulated category which is stably n-Calabi-Yau and Hom-finite, arising from a left (n + 1)-Calabi-Yau morphism. Our results apply in particular to relative Ginzburg dg algebras coming from ice quivers with potential and higher Auslander algebras associated to n-representation-finite algebras.

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“…Recently, Herschend-Liu-Nakaoka [10] introduced the notion of n-exangulated categories for any positive integer n. It is not only a higher dimensional analogue of extriangulated categories, but also gives a common generalization of n-exact categories (n-abelian categories are also n-exact categories) in the sense of Jasso [15] and (n + 2)-angulated in the sense of Geiss-Keller-Oppermann [8]. However, there are some other examples of n-exangulated categories which are neither n-exact nor (n + 2)-angulated, see [10,13,23].…”

Section: Introductionmentioning

confidence: 99%

Higher Auslander-Reiten sequences via morphisms determined by objects

He¹,

He²,

Zhou³

2021

Preprint

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Let (C , E, s) be an Ext-finite, Krull-Schmidt and k-linear n-exangulated category with k a commutative artinian ring. In this note, we define two additive subcategories C r and C l of C in terms of the representable functors from the stable category of C to the category of finitely generated k-modules. Moreover, we show that there exists an equivalence between the stable categories of these two full subcategories. Finally, we give some equivalent characterizations on the existence of Auslander-Reiten n-exangles via determined morphisms. These results unify and extend their works by Jiao-Le for exact categories, Zhao-Tan-Huang for extriangulated categories, Xie-Liu-Yang for n-abelian categories.

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Frobenius n-exangulated categories

Cited by 29 publications

References 7 publications

Auslander-Reiten-Serre duality for n-exangulated categories

Auslander-Reiten-Serre duality for n-exangulated categories

Relative cluster categories and Higgs categories

Higher Auslander-Reiten sequences via morphisms determined by objects

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