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On a family of vector space categories

Grzegorz Bobiński, Andrzej Skowroński (2003)

Open Mathematics

In continuation of our earlier work [2] we describe the indecomposable representations and the Auslander-Reiten quivers of a family of vector space categories playing an important role in the study of domestic finite dimensional algebras over an algebraically closed field. The main results of the paper are applied in our paper [3] where we exhibit a wide class of almost sincere domestic simply connected algebras of arbitrary large finite global dimensions and describe their Auslander-Reiten quivers....

On a separation of orbits in the module variety for domestic canonical algebras

Piotr Dowbor, Andrzej Mróz (2008)

Colloquium Mathematicae

Given a pair M,M' of finite-dimensional modules over a domestic canonical algebra Λ, we give a fully verifiable criterion, in terms of a finite set of simple linear algebra invariants, deciding if M and M' lie in the same orbit in the module variety, or equivalently, if M and M' are isomorphic.

On artin algebras with almost all indecomposable modules of projective or injective dimension at most one

Andrzej Skowroński (2003)

Open Mathematics

Let A be an artin algebra over a commutative artin ring R and ind A the category of indecomposable finitely generated right A-modules. Denote A to be the full subcategory of ind A formed by the modules X whose all predecessors in ind A have projective dimension at most one, and by A the full subcategory of ind A formed by the modules X whose all successors in ind A have injective dimension at most one. Recently, two classes of artin algebras A with A A co-finite in ind A, quasi-tilted algebras and...

On Auslander–Reiten components for quasitilted algebras

Flávio Coelho, Andrzej Skowroński (1996)

Fundamenta Mathematicae

An artin algebra A over a commutative artin ring R is called quasitilted if gl.dim A ≤ 2 and for each indecomposable finitely generated A-module M we have pd M ≤ 1 or id M ≤ 1. In [11] several characterizations of quasitilted algebras were proven. We investigate the structure and homological properties of connected components in the Auslander-Reiten quiver Γ A of a quasitilted algebra A.

On Auslander-Reiten translates in functorially finite subcategories and applications

K. Erdmann, D. Madsen, V. Miemietz (2010)

Colloquium Mathematicae

We consider functorially finite subcategories in module categories over Artin algebras. One main result provides a method, in the setup of bounded derived categories, to compute approximations and the end terms of relative Auslander-Reiten sequences. We also prove an Auslander-Reiten formula for the setting of functorially finite subcategories. Furthermore, we study the category of modules filtered by standard modules for certain quasi-hereditary algebras and we classify precisely when this category...

On cyclic vertices in valued translation quivers

Piotr Malicki (2006)

Colloquium Mathematicae

Let x and y be two vertices lying on an oriented cycle in a connected valued translation quiver (Γ,τ,δ). We prove that, under certain conditions, x and y belong to the same cyclic component of (Γ,τ,δ) if and only if there is an oriented cycle in (Γ,τ,δ) passing through x and y.

On domestic algebras of semiregular type

Alicja Jaworska-Pastuszak, Andrzej Skowroński (2013)

Colloquium Mathematicae

We describe the structure of finite-dimensional algebras of domestic representation type over an algebraically closed field whose Auslander-Reiten quiver consists of generalized standard and semiregular components. Moreover, we prove that this class of algebras contains all special biserial algebras whose Auslander-Reiten quiver consists of semiregular components.

On hereditary rings and the pure semisimplicity conjecture II: Sporadic potential counterexamples

José L. García (2015)

Colloquium Mathematicae

It was shown in [Colloq. Math. 135 (2014), 227-262] that the pure semisimplicity conjecture (briefly, pssC) can be split into two parts: first, a weak pssC that can be seen as a purely linear algebra condition, related to an embedding of division rings and properties of matrices over those rings; the second part is the assertion that the class of left pure semisimple sporadic rings (ibid.) is empty. In the present article, we characterize the class of left pure semisimple sporadic rings having finitely...

On Hom-spaces of tame algebras

Raymundo Bautista, Yuriy Drozd, Xiangyong Zeng, Yingbo Zhang (2007)

Open Mathematics

Let Λ be a finite dimensional algebra over an algebraically closed field k and Λ has tame representation type. In this paper, the structure of Hom-spaces of all pairs of indecomposable Λ-modules having dimension smaller than or equal to a fixed natural number is described, and their dimensions are calculated in terms of a finite number of finitely generated Λ-modules and generic Λ-modules. In particular, such spaces are essentially controlled by those of the corresponding generic modules.

On minimal non-tilted algebras

Flávio U. Coelho, José A. de la Peña, Sonia Trepode (2008)

Colloquium Mathematicae

A minimal non-tilted triangular algebra such that any proper semiconvex subcategory is tilted is called a tilt-semicritical algebra. We study the tilt-semicritical algebras which are quasitilted or one-point extensions of tilted algebras of tame hereditary type. We establish inductive procedures to decide whether or not a given strongly simply connected algebra is tilted.

On nonstandard tame selfinjective algebras having only periodic modules

Jerzy Białkowski, Thorsten Holm, Andrzej Skowroński (2003)

Colloquium Mathematicae

We investigate degenerations and derived equivalences of tame selfinjective algebras having no simply connected Galois coverings but the stable Auslander-Reiten quiver consisting only of tubes, discovered recently in [4].

On quasitilted algebras which are one-point extensions of hereditary algebras

Dieter Happel, Inger Slungård (1999)

Colloquium Mathematicae

Quasitilted algebras have been introduced as a proper generalization of tilted algebras. In an earlier article we determined necessary conditions for one-point extensions of decomposable finite-dimensional hereditary algebras to be quasitilted and not tilted. In this article we study algebras satisfying these necessary conditions in order to investigate to what extent the conditions are sufficient.

On self-injective algebras of finite representation type

Marta Błaszkiewicz, Andrzej Skowroński (2012)

Colloquium Mathematicae

We describe the structure of finite-dimensional self-injective algebras of finite representation type over a field whose stable Auslander-Reiten quiver has a sectional module not lying on a short chain.

On selfinjective algebras of tilted type

Andrzej Skowroński, Kunio Yamagata (2015)

Colloquium Mathematicae

We provide a characterization of all finite-dimensional selfinjective algebras over a field K which are socle equivalent to a prominent class of selfinjective algebras of tilted type.

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