Wakamatsu's equivalence revisited

TL;DR

This paper revisits Wakamatsu's equivalence, proving that the constructed functor between stable module categories of trivial extension algebras is a triangle equivalence, strengthening the understanding of these algebraic structures.

Contribution

It demonstrates that Wakamatsu's functor is a triangle functor, establishing a triangle equivalence between the stable categories of trivial extension algebras.

Findings

Wakamatsu's functor is a triangle functor

The equivalence between stable categories is a triangle equivalence

Strengthens the theoretical understanding of algebraic equivalences

Abstract

For a certain Wakamatsu-tilting bimodule over two artin algebras $A$ and $B$, Wakamatsu constructed an explicit equivalence between the stable module categories over the trivial extension algebra of $A$ and that of $B$. We prove that Wakamatsu's functor is a triangle functor, thus a triangle equivalence.