scholarly journals LATTICE STRUCTURE OF TORSION CLASSES FOR HEREDITARY ARTIN ALGEBRAS

2017 ◽  
Vol 231 ◽  
pp. 89-100 ◽  
Author(s):  
CLAUS MICHAEL RINGEL
Keyword(s):  
Lattice Structure ◽  
Path Algebra ◽  
Artin Algebra ◽  
Simple Modules ◽  
Artin Algebras ◽  
Hereditary Artin Algebra

Let $\unicode[STIX]{x1D6EC}$ be a connected hereditary artin algebra. We show that the set of functorially finite torsion classes of $\unicode[STIX]{x1D6EC}$-modules is a lattice if and only if $\unicode[STIX]{x1D6EC}$ is either representation-finite (thus a Dynkin algebra) or $\unicode[STIX]{x1D6EC}$ has only two simple modules. For the case of $\unicode[STIX]{x1D6EC}$ being the path algebra of a quiver, this result has recently been established by Iyama–Reiten–Thomas–Todorov and our proof follows closely some of their considerations.

scholarly journals On the Auslander–Reiten quiver of a P1-hereditary artin algebra

2000 ◽  
Vol 151 (1) ◽  
pp. 11-29 ◽  
Author(s):  
Lidia Angeleri Hügel ◽  
Flávio U. Coelho
Keyword(s):  
Artin Algebra ◽  
Hereditary Artin Algebra

scholarly journals On the number of terms in the middle of almost split sequences over cycle-finite artin algebras

2014 ◽  
Vol 12 (1) ◽  
Author(s):  
Piotr Malicki ◽  
José Peña ◽  
Andrzej Skowroński
Keyword(s):  
Module Category ◽  
Artin Algebra ◽  
Split Sequence ◽  
Almost Split Sequence ◽  
Artin Algebras ◽  
Almost Split Sequences

AbstractWe prove that the number of terms in the middle of an almost split sequence in the module category of a cycle-finite artin algebra is bounded by 5.

Standardly stratified algebras and cellular algebras

2002 ◽  
Vol 133 (1) ◽  
pp. 37-53 ◽  
Author(s):  
CHANGCHANG XI
Keyword(s):  
Projective Cover ◽  
Complete List ◽  
Injective Envelope ◽  
Finite Chain ◽  
Artin Algebra ◽  
Simple Modules ◽  
Cellular Algebras ◽  
Maximal Submodule ◽  
Standard Module ◽  
Composition Factors

Let A be an Artin algebra. Then there are finitely many non-isomorphic simple A-modules. Suppose S1, S2, …, Sn form a complete list of all non-isomorphic simple A-modules and we fix this ordering of simple modules. Let Pi and Qi be the projective cover and the injective envelope of Si respectively. With this order of simple modules we define for each i the standard module Δ(i) to be the maximal quotient of Pi with composition factors Sj with j [les ] i. Let Δ be the set of all these standard modules Δ(i). We denote by [Fscr ](Δ) the subcategory of A-mod whose objects are the modules M which have a Δ-filtration, namely there is a finite chain0 = M0 ⊂ M1 ⊂ M2 ⊂ … ⊂ Mt = Mof submodules of M such that Mi/Mi−1 is isomorphic to a module in Δ for all i. The modules in [Fscr ](Δ) are called Δ-good modules. Dually, we define the costandard module ∇(i) to be the maximal submodule of Qi with composition factors Sj with j [les ] i and denote by ∇ the collection of all costandard modules. In this way, we have also the subcategory [Fscr ](∇) of A-mod whose objects are these modules which have a ∇-filtration. Of course, we have the notion of ∇-good modules. Note that Δ(n) is always projective and ∇(n) is always injective.

On Algebras Stably Equivalent to an Hereditary Artin Algebra

1978 ◽  
Vol 30 (4) ◽  
pp. 817-829 ◽  
Author(s):  
María Inés Platzeck
Keyword(s):  
Finitely Generated ◽  
Artin Algebra ◽  
Finitely Generated Module ◽  
Artin Ring ◽  
Finitely Generated Modules ◽  
Hereditary Artin Algebra

Let Λ be an artin algebra, that is, an artin ring that is a finitely generated module over its center C which is also an artin ring. We denote by mod Λ the category of finitely generated left Λ-modules. We recall that the category of finitely generated modules modulo projectives is the category given by the following data: the objects are the finitely generated Λ-modules.

Simple reflexive modules over Artin algebras

2019 ◽  
Vol 18 (10) ◽  
pp. 1950193
Author(s):  
René Marczinzik
Keyword(s):  
Simple Module ◽  
Positive Answer ◽  
Finitely Generated ◽  
Artin Algebra ◽  
Large Classes ◽  
Gorenstein Algebras ◽  
Artin Algebras ◽  
Reflexive Modules

Let [Formula: see text] be an Artin algebra. It is well known that [Formula: see text] is selfinjective if and only if every finitely generated [Formula: see text]-module is reflexive. In this paper, we pose and motivate the question whether an algebra [Formula: see text] is selfinjective if and only if every simple module is reflexive. We give a positive answer to this question for large classes of algebras which include for example all Gorenstein algebras and all QF-3 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⊥.

On F-Gorenstein dimensions

2014 ◽  
Vol 13 (06) ◽  
pp. 1450022
Author(s):  
Xi Tang
Keyword(s):  
Equivalent Condition ◽  
Finitistic Dimension ◽  
Artin Algebra ◽  
Gorenstein Algebras ◽  
Artin Algebras ◽  
Dimension Conjecture

Over an artin algebra Λ, for an additive subbifunctor F of [Formula: see text] with enough projectives and injectives, we introduce F-Gorenstein dimensions in this paper. The new relative dimensions are useful to characterize F-Gorenstein algebras and F-self-injective algebras. In addition, with the aid of F-Gorenstein dimensions, we obtain an equivalent condition for the finitistic dimension conjecture to hold, that is, fin.dim Λ < ∞ for all artin algebras Λ if and only if rel.fin.Gdim F Λ < ∞ for all artin algebras Λ.

GORENSTEIN-PROJECTIVE MODULES OVER TRIANGULAR MATRIX ARTIN ALGEBRAS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250066 ◽  
Author(s):  
BAO-LIN XIONG ◽  
PU ZHANG
Keyword(s):  
Triangular Matrix ◽  
Projective Modules ◽  
Artin Algebra ◽  
Gorenstein Projective Modules ◽  
Artin Algebras

Let [Formula: see text] be an Artin algebra. Under suitable conditions, we describe all the modules in ⊥Λ, and obtain criteria for the Gorensteinness of Λ. As applications, we determine explicitly all the Gorenstein-projective Λ-modules if Λ is Gorenstein, and all the Gorenstein-projective Tn(A)-modules if A is Gorenstein.

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