Duality Pairs Induced by One-Sided Gorenstein Subcategories

Abstract

For a ring $R$ and an additive subcategory $\C$ of the category $\Mod R$ of left $R$-modules, under some conditions we prove that the right Gorenstein subcategory of $\Mod R$ and the left Gorenstein subcategory of $\Mod R^{op}$ relative to $\C$ form a coproduct-closed duality pair. Let $R,S$ be rings and $C$ a semidualizing ($R,S$)-bimodule. As applications of the above result, we get that if $S$ is right coherent and $C$ is faithfully semidualizing, then $({\mathcal{GF}}_C(R),{\mathcal{G}\mathcal{I}}_C(R^{op}))$ is a coproduct-closed duality pair and ${\mathcal{GF}}_C(R)$ is covering in $\Mod R$, where $\mathcal{G}{\mathcal{F}}_C(R)$ is the subcategory of $\Mod R$ consisting of $C$-Gorenstein flat modules and $\mathcal{G}{\mathcal{I}}_C(R^{op})$ is the subcategory of $\Mod R^{op}$ consisting of $C$-Gorenstein injective modules; we also get that if $S$ is right coherent, then $({\mathcal{A}}_C(R^{op}),l\mathcal{G}({\mathcal{F}}_C(R)))$ is a coproduct-closed and product-closed duality pair and ${\mathcal{A}}_C(R^{op})$ is covering and preenveloping in $\Mod R^{op}$, where ${\mathcal{A}}_C(R^{op})$ is the Auslander class in $\Mod R^{op}$ and $l\mathcal{G}({\mathcal{F}}_C(R))$ is the left Gorenstein subcategory of $\Mod R$ relative to $C$-flat modules.

References in corpus (1)

• Homological Aspects of the Dual Auslander Transpose