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A characterization of representation-finite algebras

Andrzej Skowroński, M. Wenderlich (1991)

Fundamenta Mathematicae

Let A be a finite-dimensional, basic, connected algebra over an algebraically closed field. Denote by Γ(A) the Auslander-Reiten quiver of A. We show that A is representation-finite if and only if Γ(A) has at most finitely many vertices lying on oriented cycles and finitely many orbits with respect to the action of the Auslander-Reiten translation.

A duality result for almost split sequences

Lidia Hügel, Helmut Valenta (1999)

Colloquium Mathematicae

Over an artinian hereditary ring R, we discuss how the existence of almost split sequences starting at the indecomposable non-injective preprojective right R-modules is related to the existence of almost split sequences ending at the indecomposable non-projective preinjective left R-modules. This answers a question raised by Simson in [27] in connection with pure semisimple rings.

A family of noetherian rings with their finite length modules under control

Markus Schmidmeier (2002)

Czechoslovak Mathematical Journal

We investigate the category mod Λ of finite length modules over the ring Λ = A k Σ , where Σ is a V-ring, i.e. a ring for which every simple module is injective, k a subfield of its centre and A an elementary k -algebra. Each simple module E j gives rise to a quasiprogenerator P j = A E j . By a result of K. Fuller, P j induces a category equivalence from which we deduce that mod Λ j b a d h b o x P j . As a consequence we can (1) construct for each elementary k -algebra A over a finite field k a nonartinian noetherian ring Λ such that mod A mod Λ , (2) find twisted...

A note on tilting sequences

Clezio Braga, Flávio Coelho (2008)

Open Mathematics

We discuss the existence of tilting modules which are direct limits of finitely generated tilting modules over tilted algebras.

Additive functions for quivers with relations

Helmut Lenzing, Idun Reiten (1999)

Colloquium Mathematicae

Additive functions for quivers with relations extend the classical concept of additive functions for graphs. It is shown that the concept, recently introduced by T. Hübner in a special context, can be defined for different homological levels. The existence of such functions for level 2 resp. ∞ relates to a nonzero radical of the Tits resp. Euler form. We derive the existence of nonnegative additive functions from a family of stable tubes which stay tubes in the derived category, we investigate when...

Additive functions on trees

Piroska Lakatos (2001)

Colloquium Mathematicae

The motivation for considering positive additive functions on trees was a characterization of extended Dynkin graphs (see I. Reiten [R]) and applications of additive functions in representation theory (see H. Lenzing and I. Reiten [LR] and T. Hübner [H]). We consider graphs equipped with integer-valued functions, i.e. valued graphs (see also [DR]). Methods are given for constructing additive functions on valued trees (in particular on Euclidean graphs) and for characterizing...

Almost split sequences and module categories: A complementary view to Auslander-Reiten Theory

Ariel Fernández (1995)

Commentationes Mathematicae Universitatis Carolinae

We take a complementary view to the Auslander-Reiten trend of thought: Instead of specializing a module category to the level where the existence of an almost split sequence is inferred, we explore the structural consequences that result if we assume the existence of a single almost split sequence under the most general conditions. We characterize the structure of the bimodule Δ E x t R ( C , A ) Γ with an underlying ring R solely assuming that there exists an almost split sequence of left R -modules 0 A B C 0 . Δ and Γ are...

Almost split sequences for non-regular modules

S. Liu (1993)

Fundamenta Mathematicae

Let A be an Artin algebra and let 0 X i = 1 r Y i Z 0 be an almost split sequence of A-modules with the Y i indecomposable. Suppose that X has a projective predecessor and Z has an injective successor in the Auslander-Reiten quiver Γ A of A. Then r ≤ 4, and r = 4 implies that one of the Y i is projective-injective. Moreover, if X j = 1 t Y j is a source map with the Y j indecomposable and X on an oriented cycle in Γ A , then t ≤ 4 and at most three of the Y j are not projective. The dual statement for a sink map holds. Finally, if an arrow...

Bipartite coalgebras and a reduction functor for coradical square complete coalgebras

Justyna Kosakowska, Daniel Simson (2008)

Colloquium Mathematicae

Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor C : C - C o m o d H C - C o m o d that restricts to a representation equivalence C : C - c o m o d H C - c o m o d s p , where H C is a coradical square complete hereditary bipartite...

Bound quivers of three-separate stratified posets, their Galois coverings and socle projective representations

Stanisław Kasjan (1993)

Fundamenta Mathematicae

A class of stratified posets I * ϱ is investigated and their incidence algebras K I * ϱ are studied in connection with a class of non-shurian vector space categories. Under some assumptions on I * ϱ we associate with I * ϱ a bound quiver (Q, Ω) in such a way that K I * ϱ K ( Q , Ω ) . We show that the fundamental group of (Q, Ω) is the free group with two free generators if I * ϱ is rib-convex. In this case the universal Galois covering of (Q, Ω) is described. If in addition I ϱ is three-partite a fundamental domain I * + × of this covering is...

Cartan matrices of selfinjective algebras of tubular type

Jerzy Białkowski (2004)

Open Mathematics

The Cartan matrix of a finite dimensional algebra A is an important combinatorial invariant reflecting frequently structural properties of the algebra and its module category. For example, one of the important features of the modular representation theory of finite groups is the nonsingularity of Cartan matrices of the associated group algebras (Brauer’s theorem). Recently, the class of all tame selfinjective algebras having simply connected Galois coverings and the stable Auslander-Reiten quiver...

Classification of discrete derived categories

Grzegorz Bobiński, Christof Geiß, Andrzej Skowroński (2004)

Open Mathematics

The main aim of the paper is to classify the discrete derived categories of bounded complexes of modules over finite dimensional algebras.

Coalgebras, comodules, pseudocompact algebras and tame comodule type

Daniel Simson (2001)

Colloquium Mathematicae

We develop a technique for the study of K-coalgebras and their representation types by applying a quiver technique and topologically pseudocompact modules over pseudocompact K-algebras in the sense of Gabriel [17], [19]. A definition of tame comodule type and wild comodule type for K-coalgebras over an algebraically closed field K is introduced. Tame and wild coalgebras are studied by means of their finite-dimensional subcoalgebras. A weak version of the tame-wild dichotomy theorem of Drozd [13]...

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