Relative injectivity and flatness of complexes

Abstract

A complex C is said to be FR-injective (resp., FR-flat) if Ext¹(D,C) = 0 (resp., $\overline{Tor}₁ (C,D) = 0) for any finitely represented complex D. We prove that a complex C is FR-injective (resp., FR-flat) if and only if C is exact and Z_m(C) is FR-injective (resp., FR-flat) in R-Mod for all m ∈ Z. We show that the class of FR-injective complexes is closed under direct limits and the class of FR-flat complexes is closed under direct products over any ring R. We use this result to prove that every complex have FR-flat preenvelopes and FR-injective covers.