Gorenstein Modules Induced by Foxby Equivalence

Abstract

Let $R$, $S$ be arbitrary associative rings and $C$ a semidualizing $(R,S)$-bimodule. For a subcategory $\mathcal{H}$ (resp. $\mathcal{T}$) of the category of left $R$-modules (resp. left $S$-modules), we introduce ${\mathcal{H}}_C$-Gorenstein projective and flat modules (resp. ${\mathcal{T}}_C$-Gorenstein injective modules). Under certain conditions, we prove that the ${\mathcal{H}}_C$-Gorenstein projective dimension of any left $R$-module is at most $n$ if and only if the projective dimension of any $C$-injective left $S$-module and the injective dimension of any module in $\mathcal{H}$ are at most $n$. The dual result about the ${\mathcal{T}}_C$-Gorenstein injective dimension of modules also holds true. As a consequence, we get that the supremum of the $C$-Gorenstein projective dimensions of all left $R$-modules and that of the $C$-Gorenstein injective dimensions of all left $S$-modules are identical; and the maximum of the common value of the quantities and its symmetric common value is at least the supremum of the $C$-Gorenstein flat dimensions of all left $R$-modules. Moreover, we obtain some equivalent characterizations for the finiteness of the left and right injective dimensions of ${}_R{C}_S$ in terms of the properties of the projective and injective dimensions of modules relative to various classes of $C$-Gorenstein modules. As an application, we provide some support for the Wakamatsu tilting conjecture.