Filtering subcategories of modules of an artinian algebra

TL;DR

This paper investigates how certain subcategories of modules over an artinian algebra can be filtered, providing conditions for finiteness of the finitistic dimension and exploring applications to torsion pairs and tilting modules.

Contribution

It characterizes the smallest classes of modules that filter subcategories closed under extensions and kernels or cokernels, linking these to finitistic dimension and torsion theory.

Findings

Identifies conditions for finiteness of the finitistic dimension.

Describes the smallest classes filtering subcategories closed under extensions.

Shows compatibility of filtering classes with Brenner-Butler quasi-equivalences.

Abstract

Let $A$ be an artinian algebra, and let $\mathcal{C}$ be a subcategory of mod$A$ that is closed under extensions. When $\mathcal{C}$ is closed under kernels of epimorphisms (or closed under cokernels of monomorphisms), we describe the smallest class of modules that filters $\mathcal{C}$. As a consequence, we obtain sufficient conditions for the finitistic dimension of an algebra over a field to be finite. We also apply our results to the torsion pairs. In particular, when a torsion pair is induced by a tilting module, we show that the smallest classes of modules that filter the torsion and torsion-free classes are completely compatible with the quasi-equivalences of Brenner and Butler.