Completely 𝒲-Resolved Complexes.

LetRbe a ring and 𝒲 a self-orthogonal class of leftR-modules which is closed under finite direct sums and direct summands. A complexCof leftR-modules is called a 𝒲-complexif it is exact with each cycleZn(C) ∈ 𝒲. The class of such complexes is denoted by 𝒞𝒲. A complexCis calledcompletely𝒲-resolvedif there exists an exact sequence of complexesD·= … → D−1 → D0 → D1 → … with each termDiin 𝒞𝒲such thatC = ker(D0 → D1) andD·is both Hom(𝒞𝒲, −) and Hom(−, 𝒞𝒲) exact. In this article, we show thatC= … → C−1 → C0 → C1 → … is a completely 𝒲-resolved complex if and only ifCnis a completely 𝒲-resolved module for alln∈ ℤ. Some known results are obtained as corollaries. [ABSTRACT FROM AUTHOR]