Remarks on modules approximated by G-projective modules

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