Tilting modules and the subcategories CiM

In this article we further study the full subcategories Ci M of the category of finitely generated modules over an Artin algebra introduced in Platzeck and Pratti (2000), consisting of the modules having an addM resolution of length i, which remains exact under the functor Hom A(M, -). In particular, we characterize tilting modules in terms of these categories and determine when the transpose of a tilting module is a tilting module.Fil: Platzeck, Maria Ines. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: Pratti, Nilda Isabel. Universidad Nacional de Mar del Plata; Argentin

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

On the Degree in Categories of Complexes of Fixed Size

This version of the article has been accepted for publication, after peer review and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s10485-019-09557-x.[Abstract]: We consider Λ an artin algebra and n ≥ 2. We study how to compute the left and right degrees of irreducible morphisms between complexes in a generalized standard Auslander-Reiten component of Cn(proj Λ) with length. We give conditions under which the kernel and the cokernel of irreducible morphisms between complexes in Cn(proj Λ) belong to such a category. For a finite dimensional hereditary algebra H over an algebraically closed field, we determine when an irreducible morphism has finite left (or right) degree and we give a characterization, depending on the degrees of certain irreducible morphisms, under which Cn(proj H) is of finite type.The first and second authors thankfully acknowledge partial support from CONICET and EXA558/14 from Universidad Nacional de Mar del Plata, Argentina. The third author thankfully acknowledge support from Ministerio Español de Economía y Competitividad and FEDER (FFI2014-51978-C2-2-R). The first author is a researcher from CONICET.Argentina. Universidad Nacional de Mar del Plata; EXA558/14Cn(proj Λ) H n ≥ 2

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

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

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

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