Degrees of irreducible morphisms in generalized standard coherent almost cyclic components

We study the degrees of irreducible morphisms between indecomposablemodules lying in generalized standard coherent almost cyclic componentsof Auslander-Reiten quivers of artin algebras.Fil: Chaio, Claudia Alicia. Universidad Nacional de Mar del Plata; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; ArgentinaFil: Malicki, Piotr. Nicolaus Copernicus University. Faculty of Mathematics and Computer Science; Poloni

Degrees of irreducible morphisms and finite-representation type

We study the degree of irreducible morphisms in any Auslander-Reiten component of a finite dimensional algebra over an algebraically closed field. We give a characterization for an irreducible morphism to have finite left (or right) degree. This is used to prove our main theorem: An algebra is of finite representation type if and only if for every indecomposable projective the inclusion of the radical in the projective has finite right degree, which is equivalent to require that for every indecomposable injective the epimorphism from the injective to its quotient by its socle has finite left degree. We also apply the techniques that we develop: We study when the non-zero composite of a path of $n$ irreducible morphisms between indecomposable modules lies in the $n+1$-th power of the radical; and we study the same problem for sums of such paths when they are sectional, thus proving a generalisation of a pioneer result of Igusa and Todorov on the composite of a sectional path.Comment: 20 page

Composition of irreducible morphisms in coils

We study the non-zero composition of n irreducible morphisms between modules lying in coils in relation with the powers of the radical of their module category

Covering techniques in Auslander-Reiten theory

Given a finite dimensional algebra over a perfect field the text introduces covering functors over the mesh category of any modulated Auslander-Reiten component of the algebra. This is applied to study the composition of irreducible morphisms between indecomposable modules in relation with the powers of the radical of the module category.Comment: Minor modifications. Final version to appear in the Journal of Pure and Applied Algebr

A generalization of the nilpotency index of the radical of the module category of an algebra

Let $A$ be a finite dimensional representation-finite algebra over an algebraically closed field. The aim of this work is to generalize the results proven in CGS. Precisely, we determine which vertices of $Q_A$ are sufficient to be considered in order to compute the nilpotency index of the radical of the module category of a monomial algebra and a toupie algebra $A$, when the Auslander-Reiten quiver is not necessarily a component with length.Comment: arXiv admin note: text overlap with arXiv:2003.0418

The Auslander-Reiten quiver of the category of m-periodic complexes

Let $\mathcal{A}$ be an additive $k-$category and $\mathbf{C}_{\equiv m}(\mathcal{A})$ be the category of $m-$periodic objects. For any integer $m>1$, we study conditions under which the compression functor ${\mathcal F}_m :\mathbf{C}^{b}(\mathcal{A}) \rightarrow \mathbf{C}_{\equiv m}(\mathcal{A})$ preserves or reflects irreducible morphisms. Moreover, we find sufficient conditions for the functor ${\mathcal F}_m$ to be a Galois $G$-covering in the sense of \cite{BL}. If in addition $\mathcal{A}$ is a dualizing category and \mbox{mod}\, \mathcal{A} has finite global dimension then $\mathbf{C}_{\equiv m}(\mathcal{A})$ has almost split sequences. In particular, for a finite dimensional algebra $A$ with finite strong global dimension we determine how to build the Auslander-Reiten quiver of the category \mathbf{C}_{\equiv m}(\mbox{proj}\, A). Furthermore, we study the behavior of sectional paths in \mathbf{C}_{\equiv m}(\mbox{proj}\, A), whenever $A$ is any finite dimensional $k-$algebra over a field $k$.Comment: 24 page

COMPOSITION OF IRREDUCIBLE MORPHISMS IN QUASI-TUBES

We study the composition of irreducible morphisms between indecomposable modules lying in quasi-tubes of the Auslander-Reiten quivers of artin algebras in relation with the powers of the radical of their module category

On sectional paths in a category of complexes of fixed size

We show how to build the Auslander-Reiten quiver of the category Cn(proj A)of complexes of size n ≥ 2, for any artin algebra A. We also give conditions over the complexes in Cn(proj A) under which the composition of irreducible morphisms in sectional paths vanishes.Fil: Chaio, Claudia Alicia. Universidad Nacional de Mar del Plata; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; ArgentinaFil: Pratti, Nilda Isabel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; Argentina. Universidad Nacional de Mar del Plata; ArgentinaFil: Souto-Salorio, M. José. Universidad de Coruña; Españ