N-FI Injective Complexes And N-FI Flat Complexes

In this paper,we mainly discuss n-FI injective complexes andn-FI flat com-plexes,and study their basic properties respectively.Firstly,we introduce then-FI injective complex and prove that the complex C isn-FI injective if and only if each level module C_m is n-FI injective for any integer m and the complex Hom(X,C)is exact for any complex X with FP-id(X)≤9).Then,We discuss the relationship betweenn-FI injective complexes and injective complexes over left coherent ring,and prove that the bounded above complex isn-FI injective if and only if each level module isn-FI injective on left coherent ring.Secondly,we introduce then-FI flat complex and prove that the complex C isn-FI flat if and only if C~+is an-FI in-jective complex.Then,we use this property to investigate the relationship betweenn-FI flat complex and its level module,it is proved that the bounded below com-plex isn-FI flat if and only if each term is an-FI flat module over left coherent ring.Finally,we also discuss the relationship betweenn-FI flat complexes and flat complexes over left coherent ring.