Homological dimensions and strongly idempotent ideals

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/AeAA,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/AeAA/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].