The transpose of modules relative to subcategories

Let R be a left noetherian ring and S a right noetherian ring, and let X be a subcategory of finitely generated left R-modules and BSR a finite (R,S)-bimodule. As a generalization of the Auslander transpose, replacing a projective presentation by an X-presentation and the functor HomR(−,R) by the functor HomR(−,B), we introduce the X(B)-transpose of an R-module M admitting an X-presentation. For a suitable subcategory X, some useful properties of X(B)-transposes are obtained. It is shown that the X(B)-transpose shares many nice properties with the Auslander transpose. Some known results are obtained as corollaries.