# Transfer results for Frobenius extensions

## TL;DR

This paper investigates Frobenius extensions with free-filtered and free-graded structures, establishing conditions under which the Frobenius property transfers between these structures and their Rees algebras, with applications to new examples.

## Contribution

It proves that under certain conditions, a free-filtered extension is Frobenius if and only if its associated graded extension is Frobenius, linking these properties.

## Key findings

- Frobenius property passes from free-graded to free-filtered extensions.
- Frobenius property passes from free-filtered extensions to Rees algebra extensions.
- Characterization of Frobenius extensions via associated graded structures.

## Abstract

We study Frobenius extensions which are free-filtered by a totally ordered, finitely generated abelian group, and their free-graded counterparts. First we show that the Frobenius property passes up from a free-graded extension to a free-filtered extension, then also from a free-filtered extension to the extension of their Rees algebras. Our main theorem states that, under some natural hypotheses, a free-filtered extension of algebras is Frobenius if and only if the associated graded extension is Frobenius. In the final section we apply this theorem to provide new examples and non-examples of Frobenius extensions.