Degrees of irreducible morphisms in standard components

We study the left degree of an irreducible morphism with X and Y_i indecomposable modules in a standard component of the Auslander-Reiten quiver, for 1≤i≤r. Two criteria to determine whether the left degree of these irreducible morphisms is finite or infinite are given, for standard algebras. We also study which of them has left degree two.     

On the Composite of Three Irreducible Morphisms in the Fourth Power of the Radical

We study here the nonzero composite of three irreducible morphisms between indecomposable modules lying in the fourth power of the radical.     

A Note on the Composite of Two Irreducible Morphisms

In this note, we show that a composite of two irreducible morphisms between indecomposable modules cannot lie in ?³\?⁵.     

Degrees of irreducible morphisms in generalized standard coherent almost cyclic components

We study the degrees of irreducible morphisms between indecomposable modules lying in generalized standard coherent almost cyclic components of Auslander-Reiten quivers of artin algebras.     

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.     

Coactions on Spaces of Morphisms

We study certain comodule structures on spaces of linear morphisms between H-comodules, where H is a Hopf algebra over the field k. We apply the results to show that H has non-zero integrals if and only if there exists a non-zero finite dimensional injective right H-comodule. Using this approach, we prove an extension of a result of Sullivan, by showing that if H is involutory and has non-zero integrals, and there exists an injective indecomposable right comodule whose dimension is not a multiple of char(k), then H is cosemisimple. Also we prove without using character theory that if H is cosemisimple and M is an absolutely irreducible right H-comodule, then char(k) does not divide dim(M).     

Irreducible morphisms in the bounded derived category

We study irreducible morphisms in the bounded derived category of finitely generated modules over an Artin algebra Λ, denoted , by means of the underlying category of complexes showing that, in fact, we can restrict to the study of certain subcategories of finite complexes. We prove that as in the case of modules there are no irreducible morphisms from X to X if X is an indecomposable complex. In case Λ is a selfinjective Artin algebra we show that for every irreducible morphism f in either f^j is split monomorphism for all j∈Z or split epimorphism, for all j∈Z. Moreover, we prove that all the non-trivial components of the Auslander-Reiten quiver of are of the form ZA_∞.     

Composite of irreducible morphisms in the bounded derived category

We study necessary and sufficient conditions for the existence of n irreducible morphisms in the bounded derived category of an Artin algebra, with non-zero composite in the n+1-power of the radical. In the case of , the bounded derived category of an Ext-finite hereditary k-category with tilting object, such irreducible morphisms exist if and only if H is derived equivalent to a wild hereditary algebra or to a wild canonical algebra. We also characterize the cluster tilted algebras having such irreducible morphisms.     

On the composite of irreducible morphisms in almost sectional paths

We study here when the composite of n irreducible morphisms in almost sectional paths is non-zero and lies in ℜⁿ⁺¹.     

Descent Theory and Morita Theory for Ultrametric Banach Modules

In this paper we consider ultrametric Banach modules over commutative ultrametric Banach algebras with unit. We study the descent problem along a morphism f: R S of such algebras and show that descent morphisms coincide with weak retracts. We give further conditions for having an effective descent morphism or for having a Morita equivalence between the corresponding categories of ultrametric Banach modules.     

Preinjective modules over pure semisimple rings

For a left pure semisimple ring R, it is shown that the local duality establishes a bijection between the preinjective left R-modules and the preprojective right R-modules, and any preinjective left R-module is the source of a left almost split morphism. Moreover, if there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R-mod, the direct sum of all non-preinjective indecomposable direct summands of products of preinjective left R-modules is a finitely generated product-complete module. This generalizes a recent theorem of Angeleri Hügel [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007) 361-376] for hereditary rings.     

Irreducible Morphisms Between Modules over a Repetitive Algebras

We describe the irreducible morphisms in the category of modules over a repetitive algebra. We find three special canonical forms: The first canonical form happens when all the component morphisms are split monomorphisms, the second when all the component morphisms are split epimorphisms and the third when there is exactly one irreducible component map. Also, we obtain the same result for the irreducible homomorphisms in the stable category of modules over a repetitive algebra.     

Morphisms between complete Riemannian pseudogroups

We introduce the concept of morphism of pseudogroups generalizing the étalé morphisms of Haefliger. With our definition, any continuous foliated map induces a morphism between the corresponding holonomy pseudogroups. The main theorem states that any morphism between complete Riemannian pseudogroups is complete, has a closure and its maps are C^∞ along the orbit closures. Here, completeness and closure are versions for morphisms of concepts introduced by Haefliger for pseudogroups. This result is applied to approximate foliated maps by smooth ones in the case of transversely complete Riemannian foliations, yielding the foliated homotopy invariance of their spectral sequence. This generalizes the topological invariance of their basic cohomology, shown by El Kacimi-Alaoui-Nicolau. A different proof of the spectral sequence invariance was also given by the second author.     

Decomposition and simulation of sequential dynamical systems

Sequential dynamical systems have been developed as a basis for a theory of computer simulation. This paper contains a generalization of this concept. The notion of morphism of sequential dynamical systems is introduced, formalizing the concept of simulating one system by another. Several examples of morphisms are given. Using the morphism concept, it is shown that every sequential dynamical system decomposes uniquely into a product of indecomposable systems.     

Integrable representations for the twisted full toroidal Lie algebras

The purpose of this work is to classify irreducible integrable modules of the twisted full toroidal Lie algebra τ, with finite-dimensional weight spaces and non-zero central charges. There are three families of such modules, two of which are classified.     

Descent for Discrete (Co)fibrations

We characterize the (effective) E-descent morphisms in the category Cat of small categories, when E is the class of discrete fibrations or the one of discrete cofibrations, and prove that every effective global-descent morphism is an effective E-descent morphism while its converse fails.     

Toroidalization of locally toroidal morphisms from N-folds to surfaces

The toroidalization conjecture of D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk asks whether any given morphism of nonsingular varieties over an algebraically closed field of characteristic zero can be modified into a toroidal morphism. Following a suggestion by Dale Cutkosky, we define the notion of locally toroidal morphisms and ask whether any locally toroidal morphism can be modified into a toroidal morphism. In this paper, we answer the question in the affirmative when the morphism is between any arbitrary variety and a surface.     

Cyclically Presented Modules Over Rings of Finite Type

A ring is of finite type if it has only finitely many maximal right ideals, all two-sided. In this article, we give a complete set of invariants for finite direct sums of cyclically presented modules over a ring R of finite type. More generally, our results apply to finite direct sums of direct summands of cyclically presented right R-modules (DCP modules). Using a duality, we obtain as an application a similar set of invariants for kernels of morphisms between finite direct sums of pair-wise non-isomorphic indecomposable injective modules over an arbitrary ring. This application motivates the study of DCP modules.