Search

For an abelian category A , we define the category PEx(A) of pullback diagrams of short exact sequences in A , as a subcategory of the functor category Fun(Δ , A) for a fixed diagram category Δ. For any object M in PEx (A) , we prove the existence of a short exact sequence 0 → K → P → M → 0 of functors, where the objects are in PEx(A) and P (i) ∈ Proj (A) for any i ∈ Δ. As an application, we prove that if (C , D , E) is a triple of syzygy finite classes of objects in mod Λ satisfying some special conditions, then Λ is an Igusa–Todorov algebra. Finally, we study lower triangular matrix Artin algebras and determine in terms of their components, under reasonable hypothesis, when these algebras are syzygy finite or Igusa–Todorov. [ABSTRACT FROM AUTHOR]

In this paper we study the behavior of the Igusa–Todorov functions for Artin algebras A with finite injective dimension, and Gorenstein algebras as a particular case. We show that the ϕ -dimension and ψ -dimension are finite in both cases. Also we prove that monomial, gentle and cluster tilted algebras have finite ϕ -dimension and finite ψ -dimension. [ABSTRACT FROM AUTHOR]

Abstract: We show that the main results of Happel–Rickard–Schofield (1988) and Happel–Reiten–Smalø (1996) on piecewise hereditary algebras are coherent with the notion of group action on an algebra. Then, we take advantage of this compatibility and show that if is a finite group acting on a piecewise hereditary algebra over an algebraically closed field whose characteristic does not divide the order of , then the resulting skew group algebra is also piecewise hereditary. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce and study a class of algebras which we call ada algebras. An artin algebra is ada if every indecomposable projective and every indecomposable injective module lies in the union of the left and the right parts of the module category. We describe the Auslander–Reiten components of an ada algebra which is not quasi-tilted, showing in particular that its representation theory is entirely contained in that of its left and right supports, which are both tilted algebras. Also, we prove that an ada algebra over an algebraically closed field is simply connected if and only if its first Hochschild cohomology group vanishes. [Copyright &y& Elsevier]