标题： Gorenstein转置的基变换，k-挠自由模，以及准忠实平坦扩张 显示英文标题

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

摘要： 设$\varphi\colon R \rightarrow A$是一个有限环同态，其中$R$是一个双边诺特环，并设$M$是一个有限生成的左$A$-模。 在关于$A$在$R$上的适当同调条件下，我们建立了在$A$上$M$的经典转置与$R$上$M$的某个挠模的Gorenstein转置之间的密切关系。 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. 我们引入了准忠实平坦扩张的概念，并证明在适当假设下，$k$-无挠模的范畴在$R$上的闭包性与在$A$上的闭包性是等价的。 一个应用是对赵提出的一个关于准Gorenstein性的疑问给出了肯定的回答，在$k$-情形下，当$R$和$A$都是诺特代数时。 最后，当 $\varphi$是一个可分分裂弗罗贝尼乌斯扩张时，证明了 $k$-无挠 $R$-模的范畴具有有限表示类型当且仅当在 $A$上也成立，其应用涉及斜群环。 显示英文摘要

摘要： 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.