Stable equivalence of Morita type and Frobenius extensions

Abstract

A.S. Dugas and R. Mart\'{i}nez-Villa proved in \cite[Corollary 5.1]{dm} that if there exists a stable equivalence of Morita type between the $k$-algebras $\Lambda$ and $\Gamma$, then it is possible to replace $\Lambda$ by a Morita equivalent $k$-algebra $\Delta$ such that $\Gamma$ is a subring of $\Delta$ and the induction and restriction functors induce inverse stable equivalences. In this note we give an affirmative answer to a question of Alex Dugas about the existence of a $\Gamma$-coring structure on $\Delta$. We do this by showing that $\Delta$ is a Frobenius extension of $\Gamma$.