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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...

Algebras stably equivalent to trivial extensions of hereditary algebras of type à n

Zygmunt Pogorzały (1993)

Colloquium Mathematicae

The study of stable equivalences of finite-dimensional algebras over an algebraically closed field seems to be far from satisfactory results. The importance of problems concerning stable equivalences grew up when derived categories appeared in representation theory of finite-dimensional algebras [8]. The Tachikawa-Wakamatsu result [17] also reveals the importance of these problems in the study of tilting equivalent algebras (compare with [1]). In fact, the result says that if A and B are tilting...

Algebras standardly stratified in all orders

Fidel Hernández Advíncula, Eduardo do Nascimento Marcos (2007)

Colloquium Mathematicae

The aim of this work is to characterize the algebras which are standardly stratified with respect to any order of the simple modules. We show that such algebras are exactly the algebras with all idempotent ideals projective. We also deduce as a corollary a characterization of hereditary algebras, originally due to Dlab and Ringel.

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...

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