The use of almost split sequences in the representation theory of artin algebras

1982 ◽  
pp. 29-104 ◽  
Author(s):  
Idun Reiten
Keyword(s):  
Representation Theory ◽  
Artin Algebras ◽  
Almost Split Sequences

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].

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.

Representation Theory of Artin Algebras I

1974 ◽  
Vol 1 (3) ◽  
pp. 177-268 ◽  
Author(s):  
Auslander Maurice
Keyword(s):  
Representation Theory ◽  
Artin Algebras

scholarly journals An equivalence induced by Ext and Tor applied to the finitistic weak dimension of coherent rings

1998 ◽  
Vol 40 (2) ◽  
pp. 167-176 ◽  
Author(s):  
Elwood Wilkins
Keyword(s):  
Representation Theory ◽  
Model Theory ◽  
Arbitrary Ring ◽  
Essential Component ◽  
Functor Categories ◽  
Weak Dimension ◽  
The 1960S ◽  
Almost Split Sequences ◽  
Coherent Rings

Let R be a ring, see below for other notation. The functor categories (mod-R, Ab) and ((R-mod)op, Ab) have received considerable attention since the 1960s. The first of these has achieved prominence in the model theory of modules and most particularly in the investigation of the representation theory of Artinian algebras. Both [11, Chapter 12] and [8] contain accounts of the use (mod-R, Ab) may be put to in the model theoretic setting, and Auslander's review, [1], details the application of (mod-R, Ab) to the study of Artinian algebras. The category ((R-mod)op, Ab) has been less fully exploited. Much work, however, has been devoted to the study of the transpose functor between R-mod and mod-R. Warfield's paper, [13], describes this for semiperfect rings, and this duality is an essential component in the construction of almost split sequences over Artinian algebras, see [4]. In comparison, the general case has been neglected. This paper seeks to remedy this situation, giving a concrete description of the resulting equivalence between (mod-R, Ab) and ((R-mod)op, Ab) for an arbitrary ring R.

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