Gorenstein weak injective modules with respect to a semidualizing bimodule
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In this paper, we introduce the notion of $C$-Gorenstein weak injective modules with respect to a semidualizing bimodule $_SC_R$, where $R$ and $S$ are arbitrary associative rings. We show that an iteration of the procedure used to define $G_C$-weak injective modules yields exactly the $G_C$-weak injective modules, and then give the Foxby equivalence in this setting analogous to that of $C$-Gorenstein injective modules over commutative Noetherian rings. Finally, some applications are given, including weak co-Auslander-Buchweitz context, model structure and dual pair induced by $G_C$-weak injective modules.
Abstract
In this paper, we introduce the notion of $C$-Gorenstein weak injective modules with respect to a semidualizing bimodule $_SC_R$, where $R$ and $S$ are arbitrary associative rings. We show that an iteration of the procedure used to define $G_C$-weak injective modules yields exactly the $G_C$-weak injective modules, and then give the Foxby equivalence in this setting analogous to that of $C$-Gorenstein injective modules over commutative Noetherian rings. Finally, some applications are given, including weak co-Auslander-Buchweitz context, model structure and dual pair induced by $G_C$-weak injective modules.
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