$k$-torsionfree modules and Frobenius extensions

TL;DR

This paper explores the relationship between $k$-torsionfree modules over Frobenius extensions and demonstrates how certain Gorenstein properties transfer from a base ring to its Frobenius extension.

Contribution

It establishes an equivalence of $k$-torsionfree modules between Frobenius extensions and shows the transfer of quasi $k$-Gorenstein properties, with a counterexample for the converse.

Findings

$k$-torsionfree modules are preserved under Frobenius extensions

Quasi $k$-Gorenstein property transfers from $S$ to $R$

The converse of property transfer does not always hold

Abstract

Let $R/S$ be a Frobenius extension and $k$ be a positive integer. We prove that an $R$-module is $k$-torsionfree if and only if so is its underlying $S$-module. As an application, we obtain that if $S$ is a quasi $k$-Gorenstein ring then so is $R$, but the converse does not hold in general.