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We prove that the representation dimension of a selfinjective algebra of wild tilted type is equal to three, and give an explicit construction of an Auslander generator of its module category. We also show that if a connected selfinjective algebra admits an acyclic generalised standard Auslander–Reiten component then its representation dimension is equal to three. [ABSTRACT FROM AUTHOR]

In this paper, we give a sufficient (which is also necessary under a compatibility hypothesis) condition on a set of arrows in the quiver of an algebra A so that A is a split extension of A/M, where M is the ideal of A generated by the classes of these arrows. We also compare the notion of split extension with that of semiconvex extension of algebras. [ABSTRACT FROM AUTHOR]

Abstract: Let be a hereditary abelian -category with tilting object and denote the bounded derived category of . This paper is devoted to a study of suspended subcategories of by means of their Ext-projectives. [Copyright &y& Elsevier]

Let k be an algebraically closed field, B be a finite dimensional k-algebra and A be the one-point extension of B by the projective B-module P0. We compare the posets TA and TB of tilting A-modules and B-modules, respectively. We prove that the restriction and the extension functors define morphisms of posets r: TA → TB and e: TB → TA such that re = id. Moreover, e induces a full embedding of the quiver of TB into that of TA, whose image is closed under successors, and mapping distinct connected components of the first into distinct connected components of the second. [ABSTRACT FROM AUTHOR]

In this note, we show that, if A ≅ kQA/IA is a schurian strongly simply connected algebra given by its normed presentation, and Σ is the unique poset whose Hasse quiver coincides with QA, then A ≅ kΣ if and only if IA has a generating set consisting of exactly χ(QA) elements, where χ(QA) is the Euler characteristic of QA. We also prove that a quotient of an incidence algebra A = kΣ/J is strongly simply connected if and only if A is simply connected and kΣ is strongly simply connected. [ABSTRACT FROM AUTHOR]

In this paper, we study, for an artin algebra, the class (and ) which is a full subcategory of the category modA of finitely generated A-modules, and which consists of all indecomposable A-modules whose predecessors (and successors) have projective dimension (and injective dimension, respectively) at most one. We consider quotient algebras of A, which contain the information on these classes, then define and characterize those algebras for which the class is is contravariantly finite (and is covariantly finite, respectively). [Copyright &y& Elsevier]

We prove that a tame weakly shod algebra A which is not quasi-tilted is simply connected if and only if the orbit graph of its pip-bounded component is a tree. or if and only if its first Hochschild cohomology group H²(A) with coefficients in AAA vanishes. We also show that it is strongly simply connected if and only if the orbit graph of each of its directed components is a tree, or if and only if H¹ (A) = 0 and it contains no full convex subcategory which is hereditary of type A. or if and only lilt is separated and contains no full convex subcategory which is hereditary of type A. [ABSTRACT FROM AUTHOR]