On the Wakamatsu tilting conjecture

TL;DR

This paper explores the relationship between the generalized Auslander-Reiten conjecture and the Wakamatsu tilting conjecture, proving specific cases where the latter holds, especially for certain Artinian rings and group rings.

Contribution

It establishes that the generalized Auslander-Reiten conjecture implies the Wakamatsu tilting conjecture and proves the conjecture for specific classes of rings.

Findings

Generalized Auslander-Reiten conjecture implies Wakamatsu tilting conjecture.

Wakamatsu tilting modules of finite projective dimension that are tensorly faithful are projective.

Wakamatsu tilting conjecture holds for left Artinian local rings and group rings of finite groups over Artinian rings.

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.