Monomial arrow removal and the finitistic dimension conjecture

TL;DR

This paper introduces a new reduction technique called monomial arrow removal for bound quiver algebras, aiding in determining the finitistic dimension's finiteness using abelian category theory and non-commutative Gröbner bases.

Contribution

It develops a novel method based on monomial arrow removal and abelian category cleft extensions to analyze the finitistic dimension conjecture.

Findings

The monomial arrow removal operation effectively reduces the problem.

The method is validated through concrete examples.

Connections with non-commutative Gröbner bases are established.

Abstract

In this paper, we introduce the monomial arrow removal operation for bound quiver algebras, and show that it is a novel reduction technique for determining the finiteness of the finitistic dimension. Our approach first develops a general method within the theory of abelian category cleft extensions. We then demonstrate that the specific conditions of this method are satisfied by the cleft extensions arising from strict monomial arrow removals. This crucial connection is established through the application of non-commutative Gr\"{o}bner bases in the sense of Green. The theory is illustrated with various concrete examples.