Displaying similar documents to “Gorenstein dimension of abelian categories arising from cluster tilting subcategories”

n -angulated quotient categories induced by mutation pairs

Zengqiang Lin (2015)

Czechoslovak Mathematical Journal

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Geiss, Keller and Oppermann (2013) introduced the notion of n -angulated category, which is a “higher dimensional” analogue of triangulated category, and showed that certain ( n - 2 ) -cluster tilting subcategories of triangulated categories give rise to n -angulated categories. We define mutation pairs in n -angulated categories and prove that given such a mutation pair, the corresponding quotient category carries a natural n -angulated structure. This result generalizes a theorem of Iyama-Yoshino...

One-sided n -suspended categories

Jing He, Yonggang Hu, Panyue Zhou (2024)

Czechoslovak Mathematical Journal

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For an integer n 3 , we introduce a simultaneous generalization of ( n - 2 ) -exact categories and n -angulated categories, referred to as one-sided n -suspended categories. Notably, one-sided n -angulated categories are specific instances of this structure. We establish a framework for transitioning from these generalized categories to their n -angulated counterparts. Additionally, we present a method for constructing n -angulated quotient categories from Frobenius n -prile categories. Our results unify...

Two results of n -exangulated categories

Jian He, Jing He, Panyue Zhou (2024)

Czechoslovak Mathematical Journal

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M. Herschend, Y. Liu, H. Nakaoka introduced n -exangulated categories, which are a simultaneous generalization of n -exact categories and ( n + 2 ) -angulated categories. This paper consists of two results on n -exangulated categories: (1) we give an equivalent characterization of axiom (EA2); (2) we provide a new way to construct a closed subfunctor of an n -exangulated category.

Higher-dimensional Auslander-Reiten sequences

Jiangsha Li, Jing He (2024)

Czechoslovak Mathematical Journal

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Zhou and Zhu have shown that if 𝒞 is an ( n + 2 ) -angulated category and 𝒳 is a cluster tilting subcategory of 𝒞 , then the quotient category 𝒞 / 𝒳 is an n -abelian category. We show that if 𝒞 has Auslander-Reiten ( n + 2 ) -angles, then 𝒞 / 𝒳 has Auslander-Reiten n -exact sequences.

On the structure of triangulated categories with finitely many indecomposables

Claire Amiot (2007)

Bulletin de la Société Mathématique de France

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We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field k . We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category 𝒯 is of the form Δ / G where Δ is a disjoint union of simply-laced Dynkin diagrams and G a weakly admissible group of automorphisms of Δ . Then we prove that for ‘most’ groups G , the category 𝒯 is standard, ...

On n -exact categories

Said Manjra (2019)

Czechoslovak Mathematical Journal

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An n -exact category is a pair consisting of an additive category and a class of sequences with n + 2 terms satisfying certain axioms. We introduce n -weakly idempotent complete categories. Then we prove that an additive n -weakly idempotent complete category together with the class 𝒞 n of all contractible sequences with n + 2 terms is an n -exact category. Some properties of the class 𝒞 n are also discussed.

The categories of presheaves containing any category of algebras

V. Trnková, J. Reiterman

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ContentsIntroduction.................................................................................................................................................. 5I. Preliminaries........................................................................................................................................... 6II. Main theorem.......................................................................................................................................... 8III. The...

How to construct a Hovey triple from two cotorsion pairs

James Gillespie (2015)

Fundamenta Mathematicae

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Let be an abelian category, or more generally a weakly idempotent complete exact category, and suppose we have two complete hereditary cotorsion pairs ( , ˜ ) and ( ˜ , ) in satisfying ˜ and ˜ = ˜ . We show how to construct a (necessarily unique) abelian model structure on with (resp. ˜ ) as the class of cofibrant (resp. trivially cofibrant) objects, and (resp. ˜ ) as the class of fibrant (resp. trivially fibrant) objects.

Hall algebras of two equivalent extriangulated categories

Shiquan Ruan, Li Wang, Haicheng Zhang (2024)

Czechoslovak Mathematical Journal

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For any positive integer n , let A n be a linearly oriented quiver of type A with n vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories n + 1 and n , where n + 1 and n are the two extriangulated categories corresponding to the representation category of A n + 1 and the morphism category of projective representations of A n , respectively. As a...

T 2 and T 3 objects at p in the category of proximity spaces

Muammer Kula, Samed Özkan (2020)

Mathematica Bohemica

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In previous papers, various notions of pre-Hausdorff, Hausdorff and regular objects at a point p in a topological category were introduced and compared. The main objective of this paper is to characterize each of these notions of pre-Hausdorff, Hausdorff and regular objects locally in the category of proximity spaces. Furthermore, the relationships that arise among the various Pre T 2 , T i , i = 0 , 1 , 2 , 3 , structures at a point p are investigated. Finally, we examine the relationships between the generalized...

Yetter-Drinfeld-Long bimodules are modules

Daowei Lu, Shuan Hong Wang (2017)

Czechoslovak Mathematical Journal

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Let H be a finite-dimensional bialgebra. In this paper, we prove that the category ℒℛ ( H ) of Yetter-Drinfeld-Long bimodules, introduced by F. Panaite, F. Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category H H * H H * 𝒴𝒟 over the tensor product bialgebra H H * as monoidal categories. Moreover if H is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results.

On category 𝒪 for cyclotomic rational Cherednik algebras

Iain G. Gordon, Ivan Losev (2014)

Journal of the European Mathematical Society

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We study equivalences for category 𝒪 p of the rational Cherednik algebras 𝐇 p of type G ( n ) = ( μ ) n 𝔖 n : a highest weight equivalence between 𝒪 p and 𝒪 σ ( p ) for σ 𝔖 and an action of 𝔖 on an explicit non-empty Zariski open set of parameters p ; a derived equivalence between 𝒪 p and 𝒪 p ' whenever p and p ' have integral difference; a highest weight equivalence between 𝒪 p and a parabolic category 𝒪 for the general linear group, under a non-rationality assumption on the parameter p . As a consequence, we confirm special cases...

Limits and colimits in certain categories of spaces of continuous functions

Marvin W. Grossman

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CONTENTSIntroduction................................................................................................................................................................................5§ 1. Notation and preliminaries.............................................................................................................................................6§ 2. Epimorphisms and monomorphisms.........................................................................................................................7§...

One-sided Gorenstein subcategories

Weiling Song, Tiwei Zhao, Zhaoyong Huang (2020)

Czechoslovak Mathematical Journal

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We introduce the right (left) Gorenstein subcategory relative to an additive subcategory 𝒞 of an abelian category 𝒜 , and prove that the right Gorenstein subcategory r 𝒢 ( 𝒞 ) is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When 𝒞 is self-orthogonal, we give a characterization for objects in r 𝒢 ( 𝒞 ) , and prove that any object in 𝒜 with finite r 𝒢 ( 𝒞 ) -projective dimension is isomorphic to a kernel (or a cokernel) of a morphism from an object in 𝒜 with finite 𝒞 -projective...

Derived endo-discrete artin algebras

Raymundo Bautista (2006)

Colloquium Mathematicae

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Let Λ be an artin algebra. We prove that for each sequence ( h i ) i of non-negative integers there are only a finite number of isomorphism classes of indecomposables X b ( Λ ) , the bounded derived category of Λ, with l e n g t h E ( X ) H i ( X ) = h i for all i ∈ ℤ and E(X) the endomorphism ring of X in b ( Λ ) if and only if b ( M o d Λ ) , the bounded derived category of the category M o d Λ of all left Λ-modules, has no generic objects in the sense of [4].