Fp-Injective Complexes and Fp-Injective Dimension of Complexes

Abstract

In this paper we extend the notion of FP-injective modules to that of complexes and characterize such complexes. We show that some characterizations similar to those for injective complexes exist for FP-injective complexes. We also introduce and study the notion of an FP-injective dimension associated to every complex of left R-modules over an arbitrary ring. We show that there is a close connection between the FP-injective dimension of complexes and flat dimension.2010 Mathematics subject classification: primary 16E05; secondary 16E30.

Cartan–eilenberg Fp-Injective Complexes

“…In the present article, the following result can be obtained. It is well known that a module C is FP-injective if and only if C is pure in every module that contains it [15,18]; a complex C is FP-injective if and only if C is pure in every complex that contains it [24]. Here we get the following result.…”

“…As we know, a complex I is said to be C-E injective if I, Z(I), B(I) and H(I) are complexes consisting of injective modules. In [24], the authors proved that a complex C is FP-injective if and only if C is exact and Z n (C) is FP-injective in R-Mod for each n ∈ Z. In the present article, the following result can be obtained.…”

“…Recall that a complex C is FP-injective if Ext 1 (P, C) = 0 for any finitely presented complex P[24]. Any FP-injective complex is C-E FPinjective; however, the converse is not true (see Example 4.3).…”

Abelian model structures and Ding homological dimensions

“…Proof From [19,Theorem 2.26], we know that the class of chain complexes with finite F P -injective (flat) dimension is the class of exact complexes with cycles of bounded F P -injective (flat) dimension. If R is n-F C, then these classes coincide.…”

Weak Injective and Weak Flat Complexes

“…flat) if and only if C is exact and Z m (C) is injective (resp. flat) as R-modules for any m ∈ Z; In [25,23], Liu et al introduced the notion of FP-injective complexes, they obtained many nice characterizations of them over coherent rings, and they showed that some properties of injective complexes have counterparts for FP-injective complexes. More recently, we introduced and investigated in [16,14] weak injective and weak flat modules, and generalized many results from coherent rings to arbitrary rings.…”