STRONGLY QUASI-HEREDITARY ALGEBRAS AND REJECTIVE SUBCATEGORIES

2018 ◽  
Vol 237 ◽  
pp. 10-38 ◽  
Author(s):  
MAYU TSUKAMOTO
Keyword(s):  
Finite Algebra ◽  
Global Dimension ◽  
Artin Algebra ◽  
Hereditary Algebras ◽  
Nakayama Algebra ◽  
Auslander Algebra

Ringel’s right-strongly quasi-hereditary algebras are a distinguished class of quasi-hereditary algebras of Cline–Parshall–Scott. We give characterizations of these algebras in terms of heredity chains and right rejective subcategories. We prove that any artin algebra of global dimension at most two is right-strongly quasi-hereditary. Moreover we show that the Auslander algebra of a representation-finite algebra $A$ is strongly quasi-hereditary if and only if $A$ is a Nakayama algebra.

scholarly journals The strong global dimension of piecewise hereditary algebras

2017 ◽  
Vol 481 ◽  
pp. 36-67 ◽  
Author(s):  
Edson Ribeiro Alvares ◽  
Patrick Le Meur ◽  
Eduardo N. Marcos
Keyword(s):  
Global Dimension ◽  
Hereditary Algebras

The Global Dimension of the Endomorphism Algebra of a Generator-cogenerator for a Subcategory

2013 ◽  
Vol 20 (03) ◽  
pp. 443-456
Author(s):  
Jingjing Guo
Keyword(s):  
Global Dimension ◽  
Endomorphism Algebra ◽  
Artin Algebra ◽  
Hereditary Artin Algebra

Let A be a hereditary Artin algebra and T a tilting A-module. The possibilities for the global dimension of the endomorphism algebra of a generator-cogenerator for the subcategory T⊥ in A-mod are determined in terms of relative Auslander-Reiten orbits of indecomposable A-modules in T⊥.

scholarly journals From subcategories to the entire module categories

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Rasool Hafezi
Keyword(s):  
Representation Theory ◽  
Triangular Matrix ◽  
Projective Modules ◽  
Artin Algebra ◽  
Finite Representation ◽  
Triangular Matrix Algebra ◽  
Gorenstein Projective Modules ◽  
Category Of Modules ◽  
Almost Split Sequences ◽  
Auslander Algebra

AbstractIn this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander–Reiten quiver of the subcategories is obtained only by the Auslander–Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, when Λ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the {2\times 2} lower triangular matrix algebra over Λ and the stable Auslander algebra of Λ can be estimated by the category of modules over the stable Cohen–Macaulay Auslander algebra of Λ.

scholarly journals Classifying tilting modules over the Auslander algebras of radical square zero Nakayama algebras

2020 ◽  
pp. 2250041
Author(s):  
Xiaojin Zhang
Keyword(s):  
Direct Summand ◽  
Tilting Modules ◽  
Simple Modules ◽  
Nakayama Algebras ◽  
Nakayama Algebra ◽  
Auslander Algebra

Let [Formula: see text] be a radical square zero Nakayama algebra with [Formula: see text] simple modules and let [Formula: see text] be the Auslander algebra of [Formula: see text]. Then every indecomposable direct summand of a tilting [Formula: see text]-module is either simple or projective. Moreover, if [Formula: see text] is self-injective, then the number of tilting [Formula: see text]-modules is [Formula: see text]; otherwise, the number of tilting [Formula: see text]-modules is [Formula: see text].

Gorenstein global dimension and Hopf algebroid actions

2016 ◽  
Vol 15 (05) ◽  
pp. 1650090
Author(s):  
Yong Wang ◽  
Fang Li
Keyword(s):  
Triangular Matrix ◽  
Global Dimension ◽  
Smash Product ◽  
Artin Algebra ◽  
Representation Dimension ◽  
Hopf Algebroid ◽  
Hopf Algebroids ◽  
The Stability ◽  
Gorenstein Injective ◽  
The Relationship

Let [Formula: see text] be a Hopf algebroid and [Formula: see text] a left [Formula: see text]-module algebra. In this paper, we mainly present the duality theorem for the smash product [Formula: see text], and making use of integral theory for Hopf algebroids, we investigate the stability of Gorenstein injective pre-envelopes and Gorenstein projective precovers between the category of [Formula: see text]-modules and the category of [Formula: see text]-modules. Moreover, we establish the relationship between Gorenstein global dimension of [Formula: see text] and that of [Formula: see text], and prove that [Formula: see text] has finite representation type, resp. is selfinjective, resp. is CM-finite [Formula: see text]-Gorenstein, if and only if [Formula: see text] has the same property under suitable conditions. As an application, we investigate the representation dimension of the lower triangular matrix Artin algebra [Formula: see text].

scholarly journals The finiteness of the Gorenstein dimension for Artin algebras

2019 ◽  
Vol 18 (06) ◽  
pp. 1950112
Author(s):  
René Marczinzik
Keyword(s):  
Projective Dimension ◽  
Indecomposable Module ◽  
Global Dimension ◽  
Injective Dimension ◽  
Artin Algebra ◽  
Artinian Rings ◽  
Gorenstein Dimension ◽  
Finite Global Dimension ◽  
Artin Algebras

In [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478], the authors proved that an Artin algebra [Formula: see text] with infinite global dimension has an indecomposable module with infinite projective and infinite injective dimension, giving a new characterization of algebras with finite global dimension. We prove in this paper that an Artin algebra [Formula: see text] that is not Gorenstein has an indecomposable [Formula: see text]-module with infinite Gorenstein projective dimension and infinite Gorenstein injective dimension, which gives a new characterization of algebras with finite Gorenstein dimension. We show that this gives a proper generalization of the result in [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478] for Artin algebras.

scholarly journals A new characterization of Auslander algebras

2017 ◽  
Vol 16 (11) ◽  
pp. 1750219
Author(s):  
Shen Li ◽  
Shunhua Zhang
Keyword(s):  
Projective Dimension ◽  
Global Dimension ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional ◽  
Auslander Algebra

Let [Formula: see text] be a finite dimensional Auslander algebra. For a [Formula: see text]-module [Formula: see text], we prove that the projective dimension of [Formula: see text] is at most one if and only if the projective dimension of its socle soc[Formula: see text][Formula: see text] is at most one. As an application, we give a new characterization of Auslander algebras [Formula: see text] and prove that a finite dimensional algebra [Formula: see text] is an Auslander algebra provided its global dimension gl.d[Formula: see text][Formula: see text] and an injective [Formula: see text]-module is projective if and only if the projective dimension of its socle is at most one.

Tilting modules over Auslander algebras of Nakayama algebras with radical cube zero

2021 ◽  
pp. 1-22
Author(s):  
Zongzhen Xie ◽  
Hanpeng Gao ◽  
Zhaoyong Huang
Keyword(s):  
Tilting Modules ◽  
Simple Modules ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Nakayama Algebras ◽  
Nakayama Algebra ◽  
Auslander Algebra

Let [Formula: see text] be the Auslander algebra of a finite-dimensional basic connected Nakayama algebra [Formula: see text] with radical cube zero and [Formula: see text] simple modules. Then the cardinality [Formula: see text] of the set consisting of isomorphism classes of basic tilting [Formula: see text]-modules is [Formula: see text]

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