Stability of Gorenstein flat categories with respect to a semidualizing module

Abstract

In this paper, we first introduce ${\mathcal{W}}_F$-Gorenstein modules to establish the following Foxby equivalence: $\xymatrix@C=80pt{\mathcal {G}(\mathcal {F})\cap \mathcal {A}_C(R) \ar@<0.5ex>[r]^{C\otimes_R-} & \mathcal {G}(\mathcal {W}_F) \ar@<0.5ex>[l]^{\textrm{Hom}_R(C,-)}} $ where $\mathcal{G}(\mathcal{F})$, ${\mathcal{A}}_C(R)$ 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 $\mathbb{G}_\bullet=\indent ...\longrightarrow G_1\longrightarrow G_0\longrightarrow G^0\longrightarrow G^1\longrightarrow...$ in $\mathcal{G}({\mathcal{W}}_F)$ such that $M\cong$ $\im(G_0\rightarrow G^0) $ and that ${\mathbb{G}}_{\bullet }$ is 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.

18 pages