Remarks on modules approximated by G-projective modules

Let $R$ be a commutative Noetherian Henselian local ring. Denote by $\mathrm{m}\mathrm{o}\mathrm{d}R$ the category of finitely generated $R$-modules, and by $\mathcal{G}$ the full subcategory of $\mathrm{m}\mathrm{o}\mathrm{d}R$ 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 $\mathrm{r}\mathrm{a}\mathrm{p}\mathcal{G}$ of $\mathrm{m}\mathrm{o}\mathrm{d}R$ consisting of all $R$-modules that admit right $\mathcal{G}$-approximations. We investigate the structure of $\mathrm{r}\mathrm{a}\mathrm{p}\mathcal{G}$ by observing $\mathcal{G}$, ${\mathcal G}^{\bot}$and$$\mathrm{l}\mathrm{a}\mathrm{p}${\mathcal G}$,where$$\mathrm{l}\mathrm{a}\mathrm{p}${\mathcal G}$denotesthefullsubcategoryof$$\mathrm{m}\mathrm{o}\mathrm{d}$ R$consistingofall$R$-modules that admit left $\mathcal{G}$-approximations. On the other hand, we also characterize $\mathrm{r}\mathrm{a}\mathrm{p}\mathcal{G}$ in terms of Tate cohomologies. We give several sufficient conditions for $\mathcal{G}$ to be contravariantly finite in $\mathrm{m}\mathrm{o}\mathrm{d}R$.