Stable equivalence of Morita type and Frobenius extensions

A.S. Dugas and R. Martínez-Villa proved in \cite[Corollary 5.1]{dm} that if there exists a stable equivalence of Morita type between the $k$-algebras $Λ$ and $Γ$, then it is possible to replace $Λ$ by a Morita equivalent $k$-algebra $Δ$ such that $Γ$ is a subring of $Δ$ 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 $Γ$-coring structure on $Δ$. We do this by showing that $Δ$ is a Frobenius extension of $Γ$.

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