On the finitistic dimension conjecture, III: Related to the pair eAe⊆A

Let A be an Artin algebra and e an idempotent element in A. In this paper, we use co-homological conditions on A to control the finitistic dimension of eAe. Such a consideration is of particular interest for understanding the finitistic dimension conjecture. Let us denote the finitistic dimension and global dimension of A by fin.dim(A) and gl.dim(A), respectively. Suppose gl.dim(A)⩽4. Then fin.dim(eAe)<∞ if one of the following conditions holds: (1) A/AeA has representation dimension at most 3; (2) is an A/AeA-module; (3) for all simple A/AeA-modules S. This result can be considered as a first step to the question of whether gl.dim(A)⩽4 implies fin.dim(eAe)<∞. Moreover, we show the following: Let A be an arbitrary Artin algebra and e an idempotent element of A such that the ∗-syzygy dimension or the Gorenstein dimension of the eAe-module Ae is finite. If fin.dim(A)<∞, then fin.dim(eAe)<∞.