$\omega$-left approximation dimensions under Stable equivalence

TL;DR

This paper explores how $$-left approximation dimensions behave under stable equivalences of Artin algebras, establishing correspondences and invariance of certain tilting modules and conjectures.

Contribution

It demonstrates the transfer properties of $$-left approximation dimensions under stable equivalences and confirms the invariance of the Wakamatsu tilting conjecture.

Findings

Established a one-to-one correspondence between basic tilting modules.

Proved the Wakamatsu tilting conjecture is preserved under stable equivalences.

Analyzed transfer properties of approximation dimensions in specific algebra classes.

Abstract

In this paper, we investigate some transfer properties of $\omega$-left approximation dimensions of modules of stably equivalent Artin algebras having neither nodes nor semisimple direct summands. As applications, we give a one-to-one correspondence between basic (Wakamatsu) tilting modules, and prove that the Wakamatsu tilting conjecture is preserved under those equivalences.