scholarly journals Composition of irreducible morphisms in quasi-tubes

2017 ◽  
Vol 16 (04) ◽  
pp. 1750071 ◽  
Author(s):  
Claudia Chaio ◽  
Piotr Malicki
Keyword(s):  
Module Category ◽  
Indecomposable Modules ◽  
Artin Algebras ◽  
Irreducible Morphisms

We study the composition of irreducible morphisms between indecomposable modules lying in quasi-tubes of the Auslander–Reiten quivers of artin algebras in relation with the powers of the radical of their module category.

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.

DEFINABLE SUBCATEGORIES OVER PURE SEMISIMPLE RINGS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250099 ◽  
Author(s):  
NGUYEN VIET DUNG ◽  
JOSÉ LUIS GARCÍA
Keyword(s):  
Module Category ◽  
Indecomposable Modules ◽  
Functor Category ◽  
The Family ◽  
Finite Set ◽  
Semisimple Ring

Let R be a right pure semisimple ring, and [Formula: see text] be a family of sources of left almost split morphisms in mod-R. We study the definable subcategory [Formula: see text] in Mod-R determined by the family [Formula: see text], and show that [Formula: see text] has several nice properties similar to those of the category Mod-R. For example, its functor category [Formula: see text] is a module category, and preinjective objects of [Formula: see text] are sources of left almost split morphisms in [Formula: see text] and in mod-R. As an application, it is shown that if R is a right pure semisimple ring with no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in mod-R (in particular, if R is right pure semisimple hereditary), then any definable subcategory of Mod-R determined by a finite set of indecomposable right R-modules contains only finitely many non-isomorphic indecomposable modules.

The Inductive Step of the Second Brauer-Thrall Conjecture

1980 ◽  
Vol 32 (2) ◽  
pp. 342-349 ◽  
Author(s):  
Sverre O. Smalø
Keyword(s):  
Representation Theory ◽  
Finitely Generated ◽  
Inductive Step ◽  
Artin Algebra ◽  
Indecomposable Modules ◽  
Infinite Type ◽  
Categorical Methods ◽  
Artin Algebras ◽  
Almost Split Sequences ◽  
Finitely Generated Modules

In this paper we are going to use a result of H. Harada and Y. Sai concerning composition of nonisomorphisms between indecomposable modules and the theory of almost split sequences introduced in the representation theory of Artin algebras by M. Auslander and I. Reiten to obtain the inductive step in the second Brauer-Thrall conjecture.Section 1 is devoted to giving the necessary background in the theory of almost split sequences.As an application we get the first Brauer-Thrall conjecture for Artin algebras. This conjecture says that there is no bound on the length of the finitely generated indecomposable modules over an Artin algebra of infinite type, i.e., an Artin algebra such that there are infinitely many nonisomorphic indecomposable finitely generated modules. This result was first proved by A. V. Roiter [8] and later in general for Artin rings by M. Auslander [2] using categorical methods.

scholarly journals ORIENTED FLIP GRAPHS, NONCROSSING TREE PARTITIONS, AND REPRESENTATION THEORY OF TILING ALGEBRAS

2019 ◽  
Vol 62 (1) ◽  
pp. 147-182 ◽  
Author(s):  
ALEXANDER GARVER ◽  
THOMAS MCCONVILLE
Keyword(s):  
Representation Theory ◽  
Derived Categories ◽  
Module Category ◽  
Indecomposable Modules ◽  
Quiver Mutation ◽  
Extension Spaces ◽  
Dynkin Quiver ◽  
Torsion Pairs ◽  
Gentle Algebras

AbstractThe purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver.

Minimal representation-infinite artin algebras

1994 ◽  
Vol 116 (2) ◽  
pp. 229-243 ◽  
Author(s):  
Andrzej Skowroński
Keyword(s):  
Jacobson Radical ◽  
Minimal Representation ◽  
Finitely Generated ◽  
Artin Algebra ◽  
Indecomposable Modules ◽  
Artin Ring ◽  
Artin Algebras ◽  
Infinite Power

Let A be an artin algebra over a commutative artin ring R, mod A be the category of finitely generated right A-modules, and rad∞ (modA) be the infinite power of the Jacobson radical rad(modA) of modA. Recall that A is said to be representation-finite if mod A admits only finitely many non-isomorphic indecomposable modules. It is known that A is representation-finite if and only if rad∞ (mod A) = 0. Moreover, from the validity of the First Brauer–Thrall Conjecture [26, 2] we know that A is representation-finite if and only if there is a common bound on the length of indecomposable modules in mod A.

scholarly journals Relative degrees of irreducible morphisms

2015 ◽  
Vol 428 ◽  
pp. 471-489 ◽  
Author(s):  
Flávio U. Coelho ◽  
Danilo D. da Silva
Keyword(s):  
Irreducible Morphisms
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