Related papers: On the degree in categories of complexes of fixed …

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.…

Let $\mathbf{k}$ be an algebraically closed field, let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra, and let $\widehat{\Lambda}$ be the repetitive algebra of $\Lambda$. For the stable category of finitely generated left…

Let $\Lambda$ be a $\mathbb{Z}$-graded artin algebra. Two classical results of Gordon and Green state that if $\Lambda$ has only finitely many indecomposable gradable modules, up to isomorphism, then $\Lambda$ has finite representation…

Let $\Lambda$ be a finite dimensional algebra of type ${\mathbb A}_n$ over an algebraically closed field $K$ with the quiver $Q$ and let $|\Det(\Lambda)|$ be the number of the minimal right determiners of all irreducible morphisms between…

The module category of any artin algebra is filtered by the powers of its radical, thus defining an associated graded category. As an extension of the degree of irreducible morphisms, this text introduces the degree of morphisms in the…

Let $\Lambda$ be an artin algebra. We give an upper bound for the dimension of the bounded derived category of the category $\mod \Lambda$ of finitely generated right $\Lambda$-modules in terms of the projective and injective dimensions of…

Let $\Lambda$ be an artin algebra. We obtain that $\Lambda$ is syzygy-finite when the radical layer length of $\Lambda$ is at most two; as two consequences, we give a new upper bound for the dimension of the bounded derived category of the…

Let $\mathbb{Z}$ be the integer numbers, $\mathbb{k}$ an algebraically closed field, $\Lambda$ a finite dimensional $\mathbb{k}$-algebra, mod$\Lambda$ the category of finitely generated right modules, proj$\Lambda$ the full subcategory of…

Almost split sequences lie in the heart of Auslander-Reiten theory. This paper deals with the structure of almost split sequences with certain ending terms in the morphism category of an Artin algebra $\Lambda$. Firstly we try to interpret…

Let $\mathbb{Z}$ be the integer numbers, $\mathbb{K}$ an algebraically closed field, $\Lambda$ a finite dimensional $\mathbb{K}$-algebra, mod$\Lambda$ the category of finitely generated right modules, proj$\Lambda$ the full subcategory of…

For an abelian category $\mathcal{A}$, we establish the relation between its derived and extension dimensions. Then for an artin algebra $\Lambda$, we give the upper bounds of the extension dimension of $\Lambda$ in terms of the radical…

Let $A$ be an artin algebra. The aim of this work is to describe the enlargements of an indecomposable complex in $\mathbf{C}_{n}(\mbox{proj} \,A)$, and to study the irreducible morphisms between them. Precisely, we prove that any…

Our main theorem classifies the Auslander-Reiten triangles according to properties of the morphisms involved. As a consequence, we are able to compute the mapping cone of an irreducible morphism. We finish by showing a technique for…

We investigate the structure of certain almost split sequences in $\mathcal{P}(\Lambda)$, i.e., the category of morphisms between projective modules over an Artin algebra $\Lambda$. The category $\mathcal{P}(\Lambda)$ has very nice…

Let $\Lambda$ be a finite dimensional string algebra over a field with the quiver $Q$ such that the underlying graph of $Q$ is a tree, and let $|\Det(\Lambda)|$ be the number of the minimal right determiners of all irreducible morphisms…

Let $\Lambda$ be an $n$-Auslander algebra with global dimension $n+1$. In this paper, we prove that $\Lambda$ is representation-finite if and only if the number of non-isomorphic indecomposable $\Lambda$-modules with projective dimension…

We deduce a necessary condition for Auslander-Reiten components of the bounded derived category of a finite dimensional algebra to have Euclidean tree class by classifying certain types of irreducible maps in the category of complexes. This…

In this article, we give a family of examples of algebras, showing that for every $n \geq 2$ and $m \geq 0$, there is an algebra displaying a path of n irreducible morphisms between indecomposable modules whose composite lies in the…

Let $\Lambda$ be an Artin algebra. A GR segment of $\Lambda$ is a sequence of GR measures which is closed under direct successors and direct predecessors. The number of the GR segments was conjectured to relate to the representation type of…

In this paper, we classify the finite dimensional irreducible modules for affine BMW algebra over an algebraically closed field with arbitrary characteristic.