Base change of (Gorenstein) transpose, k-torsionfree modules, and quasi-faithfully flat extensions

TL;DR

This paper explores the relationship between transpose modules and torsionfree modules over ring extensions, introducing quasi-faithfully flat extensions and applying results to Gorenstein properties and skew group rings.

Contribution

It establishes new links between classical and Gorenstein transpose modules under ring extensions and introduces the concept of quasi-faithfully flat extensions.

Findings

Characterizes when modules are k-torsionfree over extensions and base rings.

Shows equivalence of module categories under separable split Frobenius extensions.

Provides conditions for finite representation type over skew group rings.

Abstract

Let $\varphi\colon R \rightarrow A$ be a finite ring homomorphism, where $R$ is a two-sided Noetherian ring, and let $M$ be a finitely generated left $A$-module. Under suitable homological conditions on $A$ over $R$, we establish a close relationship between the classical transpose of $M$ over $A$ and the Gorenstein transpose of a certain syzygy module of $M$ over $R$. As an application, for each integer $k>0$, we provide a sufficient condition under which $M$ is $k$-torsionfree over $A$ if and only if a certain syzygy of $M$ over $R$ is $k$-torsionfree over $R$, extending a result of Zhao. We introduce the notion of quasi-faithfully flat extensions and show that, under suitable assumptions, the extension closedness of the category of $k$-torsionfree modules over $R$ is equivalent to that over $A$. An application is an affirmative answer to a question posed by Zhao concerning quasi $k$-Gorensteiness, in the case where both $R$ and $A$ are Noetherian algebras. Finally, when $\varphi$ is a separable split Frobenius extension, it is proved that the category of $k$-torsionfree $R$-modules has finite representation type if and only if the same holds over $A$, with applications to skew group rings.