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DOI 10.1515/forum-2014-0115 | Forum Math. 2016; 28 (3):425–435 Research Article Henning Krause Morphisms determined by objects and flat covers Abstract:Wedescribe a procedure for constructingmorphisms in additive categories, combining Auslander’s concept of amorphismdeterminedby anobjectwith the existence of flat covers. Also,we showhowflat covers are turned into projective covers and we interpret these constructions in terms of adjoint functors. Keywords: Additive category, morphism determined by objects, flat cover MSC 2010: 18C35, 18E05, 16G70 || Henning

Abstract

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.

Abstract

In the paper, we introduce a wide class of domestic finite dimensional algebras over an algebraically closed field which are obtained from the hereditary algebras of Euclidean type , n≥1, by iterated one-point extensions by two-ray modules. We prove that these algebras are domestic and their Auslander-Reiten quivers admit infinitely many nonperiodic connected components with infinitely many orbits with respect to the action of the Auslander-Reiten translation. Moreover, we exhibit a wide class of almost sincere domestic simply connected algebras of large global dimensions.

Abstract

We consider a class of algebras whose Auslander-Reiten quivers have starting components that are not generalized standard. For these components we introduce a generalization of a slice and show that only in finitely many cases (up to isomorphism) a slice module is a tilting module.

Abstract

Trivial extensions of a certain subclass of minimal 2-fundamental algebras are examined. For such algebras the characterization of components of the Auslander-Reiten quiver which contain indecomposable projective modules is given.

Abstract

We prove that the number of terms in the middle of an almost split sequence in the module category of a cycle-finite artin algebra is bounded by 5.

Abstract.

We generalize Ringel and Schmidmeier's theory on the Auslander–Reiten translation of the submodule category 𝒮2(A) to the monomorphism category 𝒮n(A); the category consists of all chains of (n-1) composable monomorphisms of A-modules. As in the case of n=2, 𝒮n(A) has Auslander–Reiten sequences, and the Auslander–Reiten translation τ𝒮 of 𝒮n(A) can be explicitly formulated via τ of A-mod. Furthermore, if A is a selfinjective algebra, we study the periodicity of τ𝒮 on the objects of 𝒮n(A) and of the Serre functor F𝒮 on the objects of the stable monomorphism category 𝒮n(A)̲. In particular, τ𝒮2m(n+1)XX for X𝒮n(Λ(m,t)), and F𝒮m(n+1)XX for X𝒮n(Λ(m,t))̲, where Λ(m,t), m1, t2, are the selfinjective Nakayama algebras.

Abstract

Inspired by recent work of Carlson, Friedlander and Pevtsova concerning modules for p-elementary abelian groups Er of rank r over a field of characteristic p>0, we introduce the notions of modules with constant d-radical rank and modules with constant d-socle rank for the generalized Kronecker algebra 𝒦r=kΓr with r2 arrows and 1dr-1. We study subcategories given by modules with the equal d-radical property and the equal d-socle property. Utilizing the simplification method due to Ringel, we prove that these subcategories in mod𝒦r are of wild type. Then we use a natural functor 𝔉:mod𝒦rmodkEr to transfer our results to modkEr.

Abstract

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.

Abstract

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 $$\mathcal{L}_A$$ 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 $$\mathcal{R}_A$$ 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 $$\mathcal{L}_A \cup \mathcal{R}_A$$ co-finite in ind A, quasi-tilted algebras and generalized double tilted algebras, have been extensively investigated. The aim of the paper is to show that these two classes of algebras exhaust the class of all artin algebras A for which $$\mathcal{L}_A \cup \mathcal{R}_A$$ is co-finite in ind A, and derive some consequences.