Remarks on modules approximated by G-projective modules

Abstract

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 $\mathcal{G}$, ${\mathcal{G}}^{\perp }$ 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 $\mathrm{mod}R$.

28 pages, to appear in J. Algebra