On stable equivalences of Morita type with twisted diagonal vertices

TL;DR

This paper provides a simplified proof of Puig's result on stable equivalences of Morita type with twisted diagonal vertices, extending it to arbitrary fields.

Contribution

It offers a new, simplified proof of Puig's theorem and generalizes the result to broader field conditions.

Findings

Bimodule with twisted diagonal vertex induces endopermutation source.

Simplified proof using clearer notation and terminology.

Extension of Puig's result to arbitrary fields.

Abstract

We give a new proof, by using simplified terminology and notation, to a result of Puig stating that if a bimodule of two block algebras of finite groups over an algebraically closed field induces a stable equivalence of Morita type and has a twisted diagonal vertex, then it has an endopermutation module as a source. We also extend this result to arbitrary fields under a mild assumption.