Reduction techniques for the finitistic dimension

TL;DR

This paper introduces new reduction techniques involving arrow and vertex removal on quivers to effectively determine whether a finite dimensional algebra has finite finitistic dimension, supported by numerous examples.

Contribution

It presents novel practical methods using arrow and vertex removal operations to test the finiteness of the finitistic dimension of algebras.

Findings

New reduction techniques for finitistic dimension testing

Arrow removal leads to cleft extensions

Vertex removal results in recollements

Abstract

In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions and the latter to recollements. These two operations provide us new practical methods to detect algebras of finite finitistic dimension. We illustrate our methods with many examples.