Irreducible Morphisms Research Articles

The composite of irreducible morphisms in standard components

In this work, we consider standard components of the Auslander–Reiten quiver with trivial valuation. We give a characterization of when there are n irreducible morphisms between modules in such a component with non-zero composite belonging to the n + 1 -th power of the radical. We prove that a necessary condition for their existence is that it has to be a non-zero cycle or a non-zero bypass in the component. For directed algebras, we prove that the composite of n irreducible morphisms between indecomposable modules belongs to a greater power of the radical, greater than n, if and only if it is zero.

Degrees of irreducible morphisms in standard components

We study the left degree of an irreducible morphism f : X → ⨁ i = 1 r Y i 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.

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 ℜ3\\ℜ5.

Irreducible morphisms of categories of complexes

We study irreducible morphisms of complexes. In particular, we show that the irreducible morphisms having one (finite) irreducible submorphism fall into three canonical forms and we give necessary and sufficient conditions for a given morphism of that type to be irreducible. Our characterization of the above mentioned type of irreducible morphisms of complexes characterizes also some class of irreducible morphisms of the derived category D − ( Λ ) for Λ a finite dimensional k-algebra, where k is a field.

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 ℜ n + 1 .

On the composite of two irreducible morphisms in radical cube

We study here when there are irreducible morphisms h : X → Y and h ′ : Y → Z between indecomposable modules such that the composite h ′ h is a non-zero morphism in R 3 ( X , Z ) . In particular, we characterize the representation-finite string algebras and the tilted algebras having such irreducible morphisms.

Irreducible morphisms, the Gabriel-valued quiver and colocalizations for coalgebras

Given a basicK-coalgebraC, we study the left Gabriel-valued quiver(QC,dC)ofCby means of irreducible morphisms between indecomposable injective leftC-comodules and by means of the powersradmof the radicalradof the categoryC-injof the socle-finite injective leftC-comodules. Connections between the valued quiver(QC,dC)ofCand the valued quiver(QC¯,dC¯)of a colocalization coalgebra quotientfE:C→C¯ofCare established.

On the degree of irreducible morphisms

We study the degree of irreducible morphisms in generalized standard convex components of the Auslander–Reiten quiver of an artin algebra with the property that paths with the same origin and end vertices have equal length. We call the components with this last property components with length. In particular, we give two criteria to determine wether the degree of such an irreducible morphism f is finite or infinite. One of them is given in terms of the compositions of f with non-zero maps between modules in the component. The other states that the left degree of an irreducible map f is finite if and only if Ker f belongs to the component. We apply our results to irreducible morphisms over artin algebras of finite representation type and over tame hereditary algebras.

Projective indecomposable modules over the small quantum osp (1|2)

In this paper, we construct the finite dimensional Hopf superalgebra uq(osp(1|2)) arising from Uq(osp(1|2)) when q is a root of unity and describe the projective objects and the irreducible morphisms in a category of Z-graded uq(osp(l|2))-modules.

Graded artin algebras, rational series, and bounds for homological dimensions

is a minimal injective resolution of A, then all the I,, are projective. Then n is selfinjecrive. It is of interest to note that the Nakayama conjecture follows in a trivial way from the finitistic dimension conjecture. This one says the following: Let n be an artin algebra. Then sup{proj dim M( proj dimM<oc$ is finite where M ranges over the finitely generated Amodules. The finitistic projective dimension has its origins in the early 1960s and was formulated from results of Rosenberg and Zelinsky. In this paper we study the behavior or certain rational power series associated to every graded module over a graded finite dimensional algebra, series which are slightly different from the usual Poincare-Betti series. It is interesting to note that rationality is enough to enable us to prove that over a graded finite dimensional algebra, the linitistic projective dimension is finite over the class of finitely generated graded modules whose nonprojective syzygies have no projective summands. This implies the Nakayama conjecture for graded algebras. (See also Wilson [3].) Also using rationality we find bounds for global dimension in terms of graded length and the number of nonisomorphic simple modules. Finally let A be an algebra of finite representation type having n nonisomorphic indecomposable modules. We prove that every composition of more than 2n 2 nonisomorphisms of Amodules is zero. This is done by using the result of Auslander-Reiten which states that every nonzero homomorphism which is not an isomorphism is a sum of compositions of irreducible morphisms.

Representation Theory of Algebras Stably Equivalent to an Hereditary Artin Algebra

Two artin algebras are stably equivalent if their categories of finitely generated modules modulo projectives are equivalent. The author studies the representation theory of algebras stably equivalent to hereditary algebras, using the notions of almost split sequences and irreducible morphisms. This gives a new unified approach to the theories developed for hereditary and radical square zero algebras by Gabriel, Gelfand, Bernstein, Ponomarev, Dlab, Ringel and Müller, as well as other algebras not covered previously. The techniques are purely module theoretical and do not depend on representations of diagrams. They are similar to those used by M. Auslander and the author to study hereditary algebras.

Representation theory of algebras stably equivalent to an hereditary Artin algebra

Two artin algebras are stably equivalent if their categories of finitely generated modules modulo projectives are equivalent. The author studies the representation theory of algebras stably equivalent to hereditary algebras, using the notions of almost split sequences and irreducible morphisms. This gives a new unified approach to the theories developed for hereditary and radical square zero algebras by Gabriel, Gelfand, Bernstein, Ponomarev, Dlab, Ringel and Müller, as well as other algebras not covered previously. The techniques are purely module theoretical and do not depend on representations of diagrams. They are similar to those used by M. Auslander and the author to study hereditary algebras.

Representation theory of artin algebras v: methods for computing aimost split sequences and irreducible morphisms

(1977). Representation theory of artin algebras v: methods for computing aimost split sequences and irreducible morphisms. Communications in Algebra: Vol. 5, No. 5, pp. 519-554.