$\mathcal{W}$-resolutions and Gorenstein categories with respect to a semidualizing bimodules
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Let $\mathcal{W}$ be an additive full subcategory of the category $R$-Mod of left $R$-modules. We provide a method to construct a proper $\mathcal{W}_C^H$-resolution (resp. coproper $\mathcal{W}_C^T$-coresolution) of one term in a short exact sequence in $R$-Mod from those of the other two terms. By using these constructions, we introduce and study the stability of the Gorenstein categories $\mathcal{G}_C(\mathcal{WW}_C^T)$ and $\mathcal{G}_C(\mathcal{W}_C^H\mathcal{W})$ with respect to a semidualizing bimodule $C$, and investigate the 2-out-of-3 property of these categories of a short exact sequence by using these constructions. Also we prove how they are related to the Gorenstein categories $\mathcal{G}((R\ltimes C)\otimes_R\mathcal{W})_C$ and $\mathcal{G}(\textrm{Hom}_R(R\ltimes C,\mathcal{W}))_C$ over $R\ltimes C$.
Abstract
Let $\mathcal{W}$ be an additive full subcategory of the category $R$-Mod of left $R$-modules. We provide a method to construct a proper $\mathcal{W}_C^H$-resolution (resp. coproper $\mathcal{W}_C^T$-coresolution) of one term in a short exact sequence in $R$-Mod from those of the other two terms. By using these constructions, we introduce and study the stability of the Gorenstein categories $\mathcal{G}_C(\mathcal{WW}_C^T)$ and $\mathcal{G}_C(\mathcal{W}_C^H\mathcal{W})$ with respect to a semidualizing bimodule $C$, and investigate the 2-out-of-3 property of these categories of a short exact sequence by using these constructions. Also we prove how they are related to the Gorenstein categories $\mathcal{G}((R\ltimes C)\otimes_R\mathcal{W})_C$ and $\mathcal{G}(\textrm{Hom}_R(R\ltimes C,\mathcal{W}))_C$ over $R\ltimes C$.
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