Completely 𝒲-Resolved Complexes

Abstract

Let R be a ring and 𝒲 a self-orthogonal class of left R-modules which is closed under finite direct sums and direct summands. A complex C of left R-modules is called a 𝒲-complex if it is exact with each cycle Z ⁿ (C) ∈ 𝒲. The class of such complexes is denoted by 𝒞_𝒲. A complex C is called completely 𝒲-resolved if there exists an exact sequence of complexes D _· = … → D ₋₁ → D ₀ → D ₁ → … with each term D _i in 𝒞_𝒲 such that C = ker(D ₀ → D ₁) and D _· is both Hom(𝒞_𝒲, −) and Hom(−, 𝒞_𝒲) exact. In this article, we show that C = … → C ⁻¹ → C ⁰ → C ¹ → … is a completely 𝒲-resolved complex if and only if C ⁿ is a completely 𝒲-resolved module for all n ∈ ℤ. Some known results are obtained as corollaries.

Key Words:

• Completely 𝒲-resolved complex
• Completely 𝒲-resolved module
• Precover
• Preenvelope

2010 Mathematics Subject Classification:

• 16E05
• 16E30

ACKNOWLEDGMENTS

This research is supported by the National Natural Science Foundation of China (10971024, 11201063), the Specialized Research Fund for the Doctoral Program of Higher Education (200802860024), and the Natural Science Foundation of Jiangsu Province (BK2010393). The authors thank the referee for several helpful suggestions.

Notes

Communicated by I. Swanson.