Pullback diagrams, syzygy finite classes and Igusa-Todorov algebras

TL;DR

This paper introduces a new categorical framework for pullback diagrams in abelian categories, applies it to syzygy finite classes, and characterizes when certain matrix algebras are Igusa-Todorov, advancing understanding in algebra representation theory.

Contribution

It defines the category PEx(𝒜) of pullback diagrams, proves existence of certain short exact sequences within it, and characterizes when matrix algebras are syzygy finite or Igusa-Todorov.

Findings

Established conditions for matrix algebras to be syzygy finite.

Proved that certain classes imply the algebra is Igusa-Todorov.

Developed a categorical approach to pullback diagrams in abelian categories.

Abstract

For an abelian category $\mathcal{A}$, we define the category PEx($\mathcal{A}$) of pullback diagrams of short exact sequences in $\mathcal{A}$, as a subcategory of the functor category Fun($\Delta, \mathcal{A}$) for a fixed diagram category $\Delta$. For any object $M$ in ${\rm PEx}(\mathcal{A}),$ we prove the existence of a short exact sequence $0 {\to} K {\to} P {\to} M {\to} 0$ of functors, where the objects are in PEx($\mathcal{A}$) and $P(i) \in {\rm Proj(\mathcal{A})}$ for any $i \in \Delta$. As an application, we prove that if $(\mathcal{C}, \mathcal{D}, \mathcal{E})$ is a triple of syzygy finite classes of objects in $\mathrm{mod}\,\Lambda$ satisfying some special conditions, then $\Lambda$ 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.