Homological dimensions and strongly idempotent ideals

TL;DR

This paper investigates the relationships between homological dimensions of an Artin algebra and its subalgebras, establishing conditions under which the finitistic dimension of the algebra is finite based on properties of its ideals.

Contribution

It generalizes previous results by relating the finitistic dimensions of an algebra and its subalgebras via strongly idempotent ideals with finite projective dimension.

Findings

Finitistic projective dimension of A is finite if those of eAe and A/AeA are finite.

Finitistic injective dimension of A is finite under similar conditions.

Provides a broader understanding of homological dimensions in Artin algebras.

Abstract

Let A be an Artin algebra and e an idempotent in A. It is an interesting topic to compare the homological dimension of the algebras A,A/AeA and eAe. For example, in [2], the relation among the global dimension of these algebras is discussed under the condition that AeA is a strongly idempotent ideal. Motivated by this, we try to compare the finitistic dimension of these algebras under certain homological conditions on AeA. In particular, under the condition that AeA is a strongly idempotent ideal with finite projective dimension, we prove that if the finitistic projective (or injective) dimension of eAe and A/AeA are finite, then the finitistic projective (or injective) dimension of A is finite. This is a generalized version of the main result in [1].