Abstract
A complex C is said to be FR-injective (resp., FR-flat) if Ext1(D,C) = 0 (resp., $\overline{\mathrm{Tor}}_1$ (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 Zm(C) is FR-injective (resp., FR-flat) in R-Mod for all m $in$ 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.
Citation
Bo Lu. Zhongkui Liu. "Relative injectivity and flatness of complexes." Kodai Math. J. 36 (2) 343 - 362, June 2013. https://doi.org/10.2996/kmj/1372337523
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