Stability of Gorenstein flat categories with respect to a semidualizing module

Abstract

We first introduce in the paper the $\mathcal {W}_F$-Gorenstein modules to establish the following Foxby equivalence: \[\xymatrix @C=80pt{\mathcal {G}(\mathcal {F})\cap \mathcal {A}_C \ar @\lt 0.5ex>[r]^{C\otimes _R-} \amp \,\,\,\,\mathcal {G}(\mathcal {W}_F) \ar @\lt 0.5ex>[l]^{\textrm {Hom}_R(C,-)}} \] where $\mathcal {G}(\mathcal {F})$, $\mathcal {A}_C$ and $\mathcal {G}(\mathcal {W}_F)$ denote the class of Gorenstein flat modules, the Auslander class and the class of $\mathcal {W}_F$-Gorenstein modules, respectively. Then, we investigate two-degree $\mathcal {W}_F$-Gorenstein modules. An $R$-module $M$ is said to be two-degree $\mathcal {W}_F$-Gorenstein if there exists an exact sequence $\mathbb {G}_\bullet =\cdots \rightarrow G_1\rightarrow G_0\rightarrow G^0\rightarrow G^1\rightarrow \cdots $ in $\mathcal {G}(\mathcal {W}_F)$ such that $M \cong $ $\im \,(G_0\rightarrow G^0) $ and $\mathbb {G}_\bullet $ is $\mbox {Hom}_R(\mathcal {G}(\mathcal {W}_F),-)$ and $\mathcal {G}(\mathcal {W}_F)^+\otimes _R-$ exact. We show that two notions of the two-degree $\mathcal {W}_F$-Gorenstein and the $\mathcal {W}_F$-Gorenstein modules coincide when $R$ is a commutative $GF$-closed ring.