Biseparable extensions are not necessarily Frobenius

TL;DR

This paper characterizes when Ore extensions over finite dimensional algebras are Frobenius extensions over polynomial rings, providing a negative answer to a previously posed problem about biseparable extensions and Frobenius properties.

Contribution

It offers necessary and sufficient conditions for Ore extensions to be Frobenius over polynomial rings, addressing a question about the relationship between biseparable and Frobenius extensions.

Findings

Ore extensions can be Frobenius under specific conditions.

Negative answer to the problem by Caenepeel and Kadison.

Provides criteria linking algebraic structures to Frobenius properties.

Abstract

We give necessary and sufficient conditions on an Ore extension $A[x;\sigma,\delta]$, where $A$ is a finite dimensional algebra over a field $\mathbb{F}$, for being a Frobenius extension over the ring of commutative polynomials $\mathbb{F}[x]$. As a consequence, as the title of this paper highlights, we provide a negative answer to a problem stated by Caenepeel and Kadison.