The finitistic dimension of algebras with a directed stratification]

TL;DR

This paper introduces directed stratifications for finite-dimensional algebras, characterizes projective resolutions, and provides new insights into the finitistic dimension using techniques from EI-category algebra theory.

Contribution

It defines directed stratifications, characterizes projective resolutions, and offers a new inductive approach to understanding the finitistic dimension of such algebras.

Findings

Characterization of projective resolutions for modules

An inductive formula for finitistic dimension

Identification of algebras without directed stratification

Abstract

We introduce the notion of a directed stratification for a finite-dimensional algebra. For algebras that admit such a stratification we characterise the projective resolutions of finitely generated modules and obtain a result for the finitistic dimension, which is an inductive version of a result of Fossum, Griffith and Reiten. With the developed techniques, which are adopted from the theory of EI-category algebras, we gain deeper insight in the combinatorial nature of this result. A characterisation of algebras which do not admit a directed stratification is given in terms of the Ext-quiver.