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(∆, A) for a fixed diagram category ∆. 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 ∈ ∆. 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.
In this note, we prove that if Λ is an Artin algebra with a simple module S of finite projective dimension, then the finiteness of the finitistic dimension of Λ implies that of (1−e)Λ(1−e) where e is the primitive idempotent supporting S. We derive some consequences of this. In particular, we recover a result of Green-Solberg-Psaroudakis: if Λ is the quotient of a path algebra by an admissible ideal I whose defining relations do not involve a certain arrow α, then the finitistic dimension of Λ is finite if and only if the finitistic dimension of Λ/ΛαΛ is finite.2010 Mathematics Subject Classification. 16E10, 16G20.
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