Stable equivalence of Morita type and Frobenius extensions

TL;DR

This paper proves that under stable Morita equivalence, the algebra can be replaced by a Morita equivalent algebra that forms a Frobenius extension of the original, answering a question about coring structures.

Contribution

It establishes that the algebra replacing the original in a stable Morita equivalence can be structured as a Frobenius extension, confirming a conjecture about coring structures.

Findings

$ ext{Delta}$ is a Frobenius extension of $ extGamma$.

Stable equivalences can be realized via Frobenius extensions.

Answer to Dugas's question about $ extGamma$-coring structure.

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 $\Gamma$.