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.

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].

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 Irreducible homomorphisms for lattices over orders

1977 ◽  
Vol 17 (1) ◽  
pp. 109-124
Author(s):  
Joachim W. Schmidt
Keyword(s):  
Finite Group ◽  
Direct Summand ◽  
Group Ring ◽  
Middle Term ◽  
Split Sequence ◽  
Almost Split Sequence ◽  
Almost Split Sequences ◽  
A Finite Group

Let Λ be a complete R-order in the semi-simple K-algebra A. Then it has been shown that for each indecomposable Λ-lattice M which is not projective, there exists a unique almost split sequence 0 → N → E → M → 0. Here we study the middle term E and characterize those almost split sequences where E has a projective direct summand. In the case where Λ is the group-ring RG for a finite group G, we get information about the almost split sequences for the syzygies and apply our results in an example.

scholarly journals EXISTENCE OF ALMOST SPLIT SEQUENCES VIA REGULAR SEQUENCES

2013 ◽  
Vol 88 (2) ◽  
pp. 218-231 ◽  
Author(s):  
HOSSEIN ESHRAGHI
Keyword(s):  
Local Ring ◽  
Endomorphism Ring ◽  
Krull Dimension ◽  
Inductive Argument ◽  
Split Sequence ◽  
Regular Sequences ◽  
Complete Local Ring ◽  
Almost Split Sequence ◽  
Almost Split Sequences

AbstractLet $(R, \mathfrak{m})$ be a Cohen–Macaulay complete local ring. We will apply an inductive argument to show that for every nonprojective locally projective maximal Cohen–Macaulay object $ \mathcal{X} $ of the morphism category of $R$ with local endomorphism ring, there exists an almost split sequence ending in $ \mathcal{X} $. Regular sequences are exploited to reduce the Krull dimension of $R$ on which the inductive argument is established. Moreover, the Auslander–Reiten translate of certain objects is described.

Right ADA algebras

2017 ◽  
Vol 16 (11) ◽  
pp. 1750210
Author(s):  
Edson Ribeiro Alvares ◽  
Ibrahim Assem ◽  
Diane Castonguay ◽  
Rosana R. S. Vargas
Keyword(s):  
Projective Module ◽  
Module Category ◽  
Artin Algebra ◽  
Indecomposable Projective Module ◽  
The Right

We introduce and study the class of right ADA algebras. An artin algebra is right ADA if every indecomposable projective module lies in the left or in the right part of its module category. We study the Auslander–Reiten components of a right ADA algebra which is not quasi-tilted and prove that they are of three types: components of the left and of the right support, and transitional components each containing a right section.

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.

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.

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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