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2015 Stability of Gorenstein flat categories with respect to a semidualizing module
Zhenxing Di, Zhongkui Liu, Jianlong Chen
Rocky Mountain J. Math. 45(6): 1839-1859 (2015). DOI: 10.1216/RMJ-2015-45-6-1839

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.

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Zhenxing Di. Zhongkui Liu. Jianlong Chen. "Stability of Gorenstein flat categories with respect to a semidualizing module." Rocky Mountain J. Math. 45 (6) 1839 - 1859, 2015. https://doi.org/10.1216/RMJ-2015-45-6-1839

Information

Published: 2015
First available in Project Euclid: 14 March 2016

zbMATH: 1362.13012
MathSciNet: MR3473157
Digital Object Identifier: 10.1216/RMJ-2015-45-6-1839

Subjects:
Primary: 16E05, 16E10, 55U15

Rights: Copyright © 2015 Rocky Mountain Mathematics Consortium

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Vol.45 • No. 6 • 2015
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