Remarks on modules approximated by G-projective modules

Let R be a commutative Noetherian Henselian local ring. Denote by modR the category of finitely generated R-modules, and by the full subcategory of modR consisting of all G-projective R-modules. In this paper, we consider when a given R-module has a right -approximation. For this, we study the full subcategory of modR consisting of all R-modules that admit right -approximations. We investigate the structure of by observing , and , where denotes the full subcategory of modR consisting of all R-modules that admit left -approximations. On the other hand, we also characterize in terms of Tate cohomologies. We give several sufficient conditions for to be contravariantly finite in modR.