Approximations and the Little Finitistic Dimension of Artinian Rings

✍ Scribed by Jan Trlifaj

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 141 KB
• Volume: 246
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2001.8951

✦ Synopsis

dedicated to professor rüdiger göbel on his 60th birthday Let R be a ring and let simp-R be a representative set of all simple (right R-) modules. Denote by <ω the class of all modules which are finitely generated and have finite projective dimension. The little finitistic dimension of R is defined by fdim R = sup proj dim M M ∈ <ω . Let be the complete cotorsion theory cogenerated by <ω . For each S ∈ simp-R, let f S X S → S be a special -precover of S. We prove that fdim R = max proj dim X S S ∈ simp-R provided that R is right artinian. As a corollary, we extend to right artinian rings the well-known Auslander-Reiten sufficient condition for finiteness of the little finitistic dimension.