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

• Published in: Journal of algebra Vol. 319; no. 9; pp. 3666 - 3688
• Main Author: Xi, Changchang
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.05.2008
• Subjects: Finitistic dimension; Global dimension; Gorenstein dimension; Idempotent element; Syzygy module; Finitistic dimension; Syzygy module; Gorenstein dimension; Global dimension; Idempotent element
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1016/j.jalgebra.2008.01.021

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 ( e A e ) < ∞ if one of the following conditions holds: (1) A / A e A has representation dimension at most 3; (2) Ω A −3 ( A ) is an A / A e A -module; (3) proj.dim ( S A ) ⩽ 3 for all simple A / A e A -modules S. This result can be considered as a first step to the question of whether gl.dim ( A ) ⩽ 4 implies fin.dim ( e A e ) < ∞ . 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 ( e A e ) < ∞ .