# On the interplay between the Frobenius functor and its dual

## TL;DR

This paper explores the relationship between the Frobenius functor and its dual in commutative Noetherian rings of prime characteristic, extending previous work to better understand their interplay and properties.

## Contribution

It develops and clarifies the interaction between the Frobenius functor and its dual, building on Thomas Marley's and Jürgen Herzog's foundational work.

## Key findings

- Established new properties of the Frobenius functor and its dual.
- Analyzed the interplay between these functors in prime characteristic rings.
- Extended existing theoretical frameworks for Frobenius-related functors.

## Abstract

For a commutative Noetherian ring $R$ of prime characteristic, denote by $^{f}R$ the ring $R$ with the left structure given by the Frobenius map. We develop Thomas Marley's work on the property of the Frobenius functor $\F(-) = - \otimes_R {^f}R$ and show the interplay between $\F$ and its dual $\widetilde{\F}(-) = \Hom_R({}^{f}R, -)$ which is introduced by J\"{u}rgen Herzog.