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 G 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 G-approximation. For this, we study the full subcategory rapG of modR consisting of all R-modules that admit right G-approximations. We investigate the structure of rapG by observing G, G⊥ and lapG, where lapG denotes the full subcategory of modR consisting of all R-modules that admit left G-approximations. On the other hand, we also characterize rapG in terms of Tate cohomologies. We give several sufficient conditions for G to be contravariantly finite in modR.