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

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

Degrees in Auslander-Reiten components with almost split sequences of at most two middle terms

We consider A to be an artin algebra. We study the degrees of irreducible morphisms between modules in Auslander-Reiten components Gamma having only almost split sequences with at most two indecomposable middle terms, that is, alpha(Gamma) leq 2. We prove that if f:X rightarrow Y is an irreducible epimorphism of finite left degree with X or Y indecomposable, then there exists a module Z in Gamma and a morphism varphi in Re^{n}(Z,X) and not in Re^{n+1}(Z,X) for some positive integer n such that f varphi=0. In particular, for such components if A is a finite dimensional algebra over an algebraically closed field and f=(f_1,f_2)^t:X rightarrow Y_1 oplus Y_2 is an irreducible morphism then we show that d_l(f)=d_l(f_1)+ d_l(f_2). We also characterize the artin algebras of finite representation type with alpha(Gamma_A) leq 2 in terms of a finite number of irreducible morphisms with finite degree.Fil: Chaio, Claudia Alicia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; Argentina. Universidad Nacional de Mar del Plata; Argentin

Irreducible morphisms in the bounded derived category via the category of complexes of fixed size

We study which irreducible morphisms in the category of complexes of fixed size Cn(projA) with n≥2 and A an artin algebra, are irreducible in C[0,n](projA) or either in Cn+1(projA). Moreover, we determine which of them are also irreducible in the bounded derived category.Fil: Chaio, Claudia Alicia. Universidad Nacional de Mar del Plata. Facultad de Cs.exactas y Naturales. Centro Marplatense de Investigaciones Matematicas.; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; ArgentinaFil: Gonzalez Chaio, Alfredo. Universidad Nacional de Mar del Plata. Facultad de Cs.exactas y Naturales. Centro Marplatense de Investigaciones Matematicas.; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; ArgentinaFil: Pratti, Isabel. Universidad Nacional de Mar del Plata. Facultad de Cs.exactas y Naturales. Centro Marplatense de Investigaciones Matematicas.; Argentin

Degrees and composition of irreducible morphisms in coils

We characterize the finiteness of degrees of irreducible morphisms between indecomposable modules lying in coils of the Auslander–Reiten quivers of artin algebras. We mainly study non-zero compositions of irreducible morphisms between indecomposable modules lying in coils in relation to powers of the radical of their module category.Fil: Chaio, Claudia Alicia. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; ArgentinaFil: Malicki, Piotr. Uniwersytet Mikołaja Kopernika; Poloni

On the Composition of Three Irreducible Morphisms in the Bounded Homotopy Category

Let A be an artin algebra of finite global dimension. We study when the composition of three irreducible morphisms between indecomposable complexes in {K}^{b}(proj A) is a non-zero morphism in the fourth power of the radical. We apply such results to prove that the composition of three irreducible morphisms between indecomposable complexes in the bounded derived category of a gentle Nakayama algebra, not selfinjective, whose ordinary quiver is an oriented cycle, belongs to the fourth power of the radical if and only if it vanishes.Fil: Chaio, Claudia Alicia. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; ArgentinaFil: González Chaio, Alfredo. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Pratti, Isabel. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentin

On Non-Homogeneous Tubes and Components of Type ℤA∞ in the Bounded Derived Category

By the knitting technique introduced to build the Auslander-Reiten quiver of the category of complexes of fixed size, it is possible to obtain the Auslander-Reiten triangles in the bounded derived category. To show this technique, we consider two different families of finite dimensional algebras over an algebraically closed field. We describe the complexes that belong to the mouth of non-homogeneous tubes in the Auslander-Reiten quiver of their bounded derived category, whenever this algebras are either derived equivalent to hereditary algebras of type A~ n or D~ n. In case the algebras are discrete, we describe the complexes in the mouth of components of type ℤA∞ of the Auslander-Reiten quiver of their bounded derived category.Fil: Chaio, Claudia Alicia. Universidad Nacional de Mar del Plata. Facultad de Cs.exactas y Naturales. Centro Marplatense de Investigaciones Matematicas.; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; ArgentinaFil: Gonzalez Chaio, Alfredo. Universidad Nacional de Mar del Plata. Facultad de Cs.exactas y Naturales. Centro Marplatense de Investigaciones Matematicas.; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; ArgentinaFil: Pratti, Nilda Isabel. Universidad Nacional de Mar del Plata. Facultad de Cs.exactas y Naturales. Centro Marplatense de Investigaciones Matematicas.; Argentin

Morphisms in the infinite radical of the bounded derived category

We study necessary conditions for a morphism in the category of complexes of fixed size Cn(proj A), with n \geq 2 and A an artin algebra, to belong to the infinite radical of K^{-,b}(projA).As an application we consider A a Nakayama gentle algebra, whose ordinary quiver is an oriented cycle and we prove that the irreducible morphisms in Cn(proj A) are irreducible in K^{-,b}(projA) or belong to the infinite radical of K^{-,b}(projA). We characterize the irreducible morphisms between indecomposable complexes in Cn(proj A) that belong to the infinite radical of K^{-,b}(projA) for $A$ a Nakayama gentle algebra not selfinjective. For the case of a Nakayama gentle selfinjective algebra, we characterize the irreducible morphisms between indecomposable complexes in Cn(proj A) that belong to the infinite radical of the bounded homotopic category in terms of their left and right degrees.Fil: Chaio, Claudia Alicia. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; ArgentinaFil: Chaio, Alfredo González. Universidad Nacional de Mar del Plata; ArgentinaFil: Pratti, Nilda Isabel. Universidad Nacional de Mar del Plata; Argentin

On the type of a category of complexes of fixed size and the strong global dimension

We prove that if the strong global dimension η of an algebra A is finite, then Cη+ 1(proj A) is of finite type if and only if for each n ≥ 2, Cn(projA) is of finite type. Moreover, we also prove some implications in order to know if for some positive integer is Cn(projA) of infinite type. We determine the strong global dimension of some piecewise hereditary finite dimensional algebras taking into account their ordinary quivers with relations.Fil: Chaio, Claudia Alicia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; Argentina. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Gonzalez Chaio, Alfredo. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; ArgentinaFil: Pratti, Nilda Isabel. Universidad Nacional de Mar del Plata; Argentin

On the composition of four irreducible morphisms in the fifth power of the radical

Given A an artin algebra we study when the non-zero composition of four irreducible morphisms between indecomposable A-modules belongs to the fifth power of the radical of its module category. In particular, when A is a finite dimensional algebra over an algebraically closed field we prove that if the composition of four irreducible morphisms between indecomposable A-modules belongs to the fifth power of the radical of their module category then any composition of four irreducible morphisms between the same indecomposable A-modules so is.Fil: Chaio, Claudia Alicia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; Argentina. Universidad Nacional de Mar del Plata. Facultad de Cs.exactas y Naturales. Centro Marplatense de Investigaciones Matematicas.; ArgentinaFil: Llodra Schat, Nicolás. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; Argentina. Universidad Nacional de Mar del Plata. Facultad de Cs.exactas y Naturales. Centro Marplatense de Investigaciones Matematicas.; Argentin