Relative torsionfreeness and Frobenius extensions

TL;DR

This paper explores how Frobenius extensions preserve certain module properties, such as relative torsionfreeness and G-dimension, and establishes new connections between Frobenius extensions and Wakamatsu tilting modules.

Contribution

It demonstrates that relative torsionfreeness and G-dimension invariance are maintained under Frobenius extensions, and links these concepts with Wakamatsu tilting modules.

Findings

Relative n-torsionfreeness is preserved under Frobenius extensions.

The natural ring homomorphism is a Frobenius extension.

G-dimension with respect to Wakamatsu modules is invariant under Frobenius extensions.

Abstract

Let $S/R$ be a Frobenius extension with $_RS_R$ centrally projective over $R$. We show that if $_R\omega$ is a Wakamatsu tilting module then so is $_SS\otimes_R\omega$, and the natural ring homomorphism from the endomorphism ring of $_R\omega$ to the endomorphism ring of $_SS\otimes_R\omega$ is a Frobenius extension in addition that pd$(\omega_T)$ is finite, where $T$ is the endomorphism ring of $_R\omega$. We also obtain that the relative $n$-torsionfreeness of modules is preserved under Frobenius extensions. Furthermore, we give an application, which shows that the generalized G-dimension with respect to a Wakamatsu module is invariant under Frobenius extensions.