On the Wakamatsu tilting conjecture

Abstract

Let R be an associative ring with identity. We establish that the generalized Auslander–Reiten conjecture implies the Wakamatsu tilting conjecture. Furthermore, we prove that any Wakamatsu tilting R-module of finite projective dimension that is tensorly faithful is projective. By utilizing this result, we show the validity of the Wakamatsu tilting conjecture for R in two cases: when R is a left Artinian local ring or when it is the group ring of a finite group G over a commutative Artinian ring.