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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 behaviour of the Igusa–Todorov functions for radical square zero algebras. We show that the left and the right ϕ -dimensions coincide, in this case. Some general results are given, but we concentrate more in the radical square zero algebras. Our study is based on two notions of hearth and member of a quiver Q . We give some bounds for the ϕ and the ψ -dimensions and we describe the algebras for which the bound of ψ is obtained. We also exhibit modules for which the ϕ -dimension is realised. [ABSTRACT FROM AUTHOR]

Abstract: Let R be a finite dimensional k-algebra over an algebraically closed field k and modR be the category of all finitely generated left R-modules. For a given full subcategory of modR, we denote by the projective finitistic dimension of . That is, . It was conjectured by H. Bass in the 60''s that the projective finitistic dimension has to be finite. Since then, much work has been done toward the proof of this conjecture. Recently, K. Igusa and G. Todorov defined in [K. Igusa, G. Todorov, On the finitistic global dimension conjecture for artin algebras, in: Representations of Algebras and Related Topics, in: Fields Inst. Commun., vol. 45, Amer. Math. Soc., Providence, RI, 2005, pp. 201–204] a function , which turned out to be useful to prove that is finite for some classes of algebras. In order to have a different approach to the finitistic dimension conjecture, we propose to consider a class of full subcategories of modR instead of a class of algebras. That is, we suggest to take the class of categories , of θ-filtered R-modules, for all stratifying systems in modR. We prove that the Finitistic Dimension Conjecture holds for the categories of filtered modules for stratifying systems with one or two (and some cases of three) modules of infinite projective dimension. [Copyright &y& Elsevier]