Pullback diagrams, syzygy finite classes and Igusa–Todorov algebras.

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]