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.

Almost Split Sequences whose Middle Term has at most Two Indecomposable Summands

1979 ◽  
Vol 31 (5) ◽  
pp. 942-960 ◽  
Author(s):  
M. Auslander ◽  
R. Bautista ◽  
M. I. Platzeck ◽  
I. Reiten ◽  
S. O. Smalø
Keyword(s):  
Exact Sequence ◽  
Representation Theory ◽  
Finitely Generated ◽  
Artin Algebra ◽  
Middle Term ◽  
Split Sequence ◽  
Almost Split Sequence ◽  
Artin Algebras ◽  
Almost Split Sequences

Let Λ be an artin algebra, and denote by mod Λ the category of finitely generated Λ-modules. All modules we consider are finitely generated.We recall from [6] that a nonsplit exact sequence in mod A is said to be almost split if A and C are indecomposable, and given a map h: X → C which is not an isomorphism and with X indecomposable, there is some t: X → B such that gt = h.Almost split sequences have turned out to be useful in the study of representation theory of artin algebras. Given a nonprojective indecomposable Λ-module C (or an indecomposable noninjective Λ-module A), we know thatthere exists a unique almost split sequence [6, Proposition 4.3], [5, Section 3].

LOCALLY FINITELY GENERATED MODULES OVER RINGS WITH ENOUGH IDEMPOTENTS

2009 ◽  
Vol 08 (06) ◽  
pp. 885-901 ◽  
Author(s):  
LIDIA ANGELERI HÜGEL ◽  
JOSÉ ANTONIO DE LA PEÑA
Keyword(s):  
Finite Length ◽  
Associative Ring ◽  
Projective Modules ◽  
Finitely Generated ◽  
Direct Sums ◽  
Indecomposable Modules ◽  
Primitive Idempotents ◽  
Almost Split Sequences ◽  
Finitely Generated Modules ◽  
Infinite Set

Let R = ⊕ Reλ = ⊕ eλ R be an associative ring with enough idempotents indexed over a possibly infinite set Λ. Assume that {eλ : λ ∈ Λ} is a set of pairwise orthogonal primitive idempotents, and that R is locally bounded, that is, the projective modules eλR and Reλ are of finite length for each λ ∈ Λ. We prove the existence of almost split sequences ending at the indecomposable finitely generated non-projective unital R-modules. Moreover, we consider the unital R-modules X that are locally finitely generated, that is, Xeλ is a finitely generated eλ Reλ-module for all λ ∈ Λ. We show that such X accept perfect decompositionsX = ⊕ Xi as direct sums of indecomposable modules.

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 A characterization of right 4-Nakayama artin algebras

2020 ◽  
pp. 2150129
Author(s):  
Alireza Nasr-Isfahani ◽  
Mohsen Shekari
Keyword(s):  
Finitely Generated ◽  
Artin Algebras ◽  
Almost Split Sequences ◽  
Finitely Generated Modules

In this paper, we study the category of finitely generated modules over a class of right [Formula: see text]-Nakayama artin algebras. This class of algebras appear naturally in the study of representation-finite artin algebras. First, we give a characterization of right [Formula: see text]-Nakayama artin algebras. Then, we classify finitely generated indecomposable right modules over right [Formula: see text]-Nakayama artin algebras. We also compute almost split sequences for the class of right [Formula: see text]-Nakayama artin algebras.

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.

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.

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.

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.

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