(n + 2)-Angulated Quotient Categories

2019 ◽  
Vol 26 (04) ◽  
pp. 689-720 ◽  
Author(s):  
Qilian Zheng ◽  
Jiaqun Wei
Keyword(s):  
Quotient Category

The notion of [Formula: see text]-mutation pairs of subcategories in an n-exangulated category is defined in this article. When (Ƶ, Ƶ) is a [Formula: see text]-mutation pair in an n-exangulated category (C, [Formula: see text]), the quotient category Ƶ/[Formula: see text] carries naturally an (n+2)-angulated structure. This result generalizes a theorem of Zhou and Zhu for extriangulated categories.

Decomposing the complexity quotient category

1996 ◽  
Vol 120 (4) ◽  
pp. 589-595
Author(s):  
D. J. Benson
Keyword(s):  
Representation Theory ◽  
Finite Groups ◽  
Modular Representation ◽  
Recent Effort ◽  
Finitely Generated ◽  
Modular Representation Theory ◽  
Joint Work ◽  
Quotient Category ◽  
Finitely Generated Modules

In the modular representation theory of finite groups, much recent effort has gone into describing cohomological properties of the category of finitely generated modules. In recent joint work of the author with Jon Carlson and Jeremy Rickard[3], it has become clear that for some purposes the finiteness restriction is undesirable. In particular, in the quotient category of kG-modules by the subcategory of modules of less than maximal complexity, it turns out that finitely generated modules can have infinitely generated summands, and that including these summands in the category repairs the lack of Krull–Schmidt property.

On a characterization of finitely presented functors

1989 ◽  
Vol 425 (1869) ◽  
pp. 329-339
Keyword(s):  
Representation Type ◽  
Equivalence Of Categories ◽  
Quotient Category ◽  
Auslander Algebra ◽  
Finitely Presented

The concept of finitely presented functor was introduced by Auslander. Proposition 3.1 of Auslander & Reiten provides a way of dealing with the category of finitely presented functors, that seems concrete and easy to use, at least in some examples. The study of this category, using this particular line of thought, is the main purpose of this work. In §1 I recall some basic definitions and give the required notation. In §2 I state the theorem of Auslander & Reiten referred to above and give a new proof of this result. The first part of this proof is an immediate consequence of the theory developed by Green. In §3 I state and prove an unpublished theorem by J. A. Green and I introduce a new category I such that the category of finitely presented functors. mmod A , is equivalent to a quotient category I / J , where J is an ideal of I . In §4 I give some examples of properties of mmod A , stated and proved in terms of the category I , by using the equivalence of categories referred to in §3. In §5 I consider the particular case where A = A q = k -alg < z : z q = 0>, apply the results of previous sections to study mmod A q and make conclusions about the representation type of the Auslander algebra of A q .

scholarly journals Recollement of homotopy categories and Cohen-Macaulay modules

2011 ◽  
Vol 8 (3) ◽  
pp. 507-542 ◽  
Author(s):  
Osamu Iyama ◽  
Kiriko Kato ◽  
Jun-ichi Miyachi
Keyword(s):  
Homotopy Category ◽  
Module Category ◽  
Gorenstein Ring ◽  
Projective Modules ◽  
Finitely Generated ◽  
Stable Module Category ◽  
Quotient Category ◽  
Stable Module

AbstractWe study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show the existence of a new structure in the above quotient category, which we call a triangle of recollements. Moreover, we show that this quotient category is triangle equivalent to the stable module category of Cohen-Macaulay T2(R)-modules.

scholarly journals A Diagrammatic Temperley-Lieb Categorification

2010 ◽  
Vol 2010 ◽  
pp. 1-47 ◽  
Author(s):  
Ben Elias
Keyword(s):  
Symmetric Group ◽  
Weyl Group ◽  
Planar Graphs ◽  
Monoidal Category ◽  
Hecke Algebra ◽  
Coxeter Complex ◽  
Quotient Category ◽  
Irreducible Modules ◽  
Soergel Bimodules

The monoidal category of Soergel bimodules categorifies the Hecke algebra of a finite Weyl group. In the case of the symmetric group, morphisms in this category can be drawn as graphs in the plane. We define a quotient category, also given in terms of planar graphs, which categorifies the Temperley-Lieb algebra. Certain ideals appearing in this quotient are related both to the 1-skeleton of the Coxeter complex and to the topology of 2D cobordisms. We demonstrate how further subquotients of this category will categorify the irreducible modules of the Temperley-Lieb algebra.

QUOTIENT CATEGORIES OF n-ABELIAN CATEGORIES

2019 ◽  
Vol 62 (3) ◽  
pp. 673-705
Author(s):  
QILIAN ZHENG ◽  
JIAQUN WEI
Keyword(s):  
Abelian Category ◽  
Quotient Category ◽  
Abelian Categories

AbstractThe notion of mutation pairs of subcategories in an n-abelian category is defined in this paper. Let ${\cal D} \subseteq {\cal Z}$ be subcategories of an n-abelian category ${\cal A}$. Then the quotient category ${\cal Z}/{\cal D}$ carries naturally an (n + 2) -angulated structure whenever $ ({\cal Z},{\cal Z}) $ forms a ${\cal D} \subseteq {\cal Z}$-mutation pair and ${\cal Z}$ is extension-closed. Moreover, we introduce strongly functorially finite subcategories of n-abelian categories and show that the corresponding quotient categories are one-sided (n + 2)-angulated categories. Finally, we study homological finiteness of subcategories in a mutation pair.

Higher-dimensional Auslander–Reiten theory on (d+2)-angulated categories

2021 ◽  
pp. 1-21
Author(s):  
Panyue Zhou
Keyword(s):  
Quotient Category ◽  
Higher Dimensional ◽  
Generalize Work ◽  
Serre Functor

Abstract Let $\mathscr{C}$ be a $(d+2)$ -angulated category with d-suspension functor $\Sigma^d$ . Our main results show that every Serre functor on $\mathscr{C}$ is a $(d+2)$ -angulated functor. We also show that $\mathscr{C}$ has a Serre functor $\mathbb{S}$ if and only if $\mathscr{C}$ has Auslander–Reiten $(d+2)$ -angles. Moreover, $\tau_d=\mathbb{S}\Sigma^{-d}$ where $\tau_d$ is d-Auslander–Reiten translation. These results generalize work by Bondal–Kapranov and Reiten–Van den Bergh. As an application, we prove that for a strongly functorially finite subcategory $\mathscr{X}$ of $\mathscr{C}$ , the quotient category $\mathscr{C}/\mathscr{X}$ is a $(d+2)$ -angulated category if and only if $(\mathscr{C},\mathscr{C})$ is an $\mathscr{X}$ -mutation pair, and if and only if $\tau_d\mathscr{X} =\mathscr{X}$ .

Triangulated equivalence between a homotopy category and a triangulated quotient category

2018 ◽  
Vol 506 ◽  
pp. 297-321 ◽  
Author(s):  
Zhenxing Di ◽  
Zhongkui Liu ◽  
Xiaoyan Yang ◽  
Xiaoxiang Zhang
Keyword(s):  
Homotopy Category ◽  
Quotient Category

scholarly journals The quotient category of a Morita context

1974 ◽  
Vol 28 (3) ◽  
pp. 389-407 ◽  
Author(s):  
Bruno J Müller
Keyword(s):  
Morita Context ◽  
Quotient Category

Triples and Localizations(1)

1971 ◽  
Vol 14 (3) ◽  
pp. 333-339 ◽  
Author(s):  
A. G. Heinicke
Keyword(s):  
Abelian Category ◽  
Exact Functor ◽  
Quotient Category

Let A be a ring (associative) with unity, and let denote the category of unital left A-modules. If is a strongly complete Serre class in then (see [3], and also [6]) there is an exact functor S: , where is the quotient category , and is an abelian category.

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