Complexes of Gorenstein Flat Modules and Gorenstein Cotorsion Modules

Abstract

A complex (C, δ) is called strongly Gorenstein flat if C is exact and Ker δ ⁿ is Gorenstein flat in R-Mod for all n ∈ ℤ. Let 𝒮𝒢 stand for the class of strongly Gorenstein flat complexes. We show that a complex C of left R-modules over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ℤ and Hom^.(G, C) is exact for any strongly Gorenstein flat complex G. Furthermore, a bounded below complex C over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ℤ. Finally, strongly Gorenstein flat covers and 𝒮𝒢^⊥-envelopes of complexes are considered. For a right coherent ring R, we show that every bounded below complex has a 𝒮𝒢^⊥-envelope.

Key Words:

• Cover
• Envelope
• Gorenstein cotorsion module
• Right orthogonal class
• Strongly Gorenstein flat complex

2000 Mathematics Subject Classification:

• 16E05
• 16E10
• 16E30

ACKNOWLEDGMENTS

The authors would like to thank the referee for helpful suggestions and corrections. This work is partially supported by the National Natural Science Foundation of China (No. 10961021).

Notes

Communicated by I. Swanson.