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

Let φ R 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 φ 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.