Relative FP-gr-injective and gr-flat modules

Abstract

Let [Formula: see text] be an integer. We introduce the notions of [Formula: see text]-FP-gr-injective and [Formula: see text]-gr-flat modules. Then we investigate the properties of these modules by using the properties of special finitely presented graded modules and obtain some equivalent characterizations of [Formula: see text]-gr-coherent rings in terms of [Formula: see text]-FP-gr-injective and [Formula: see text]-gr-flat modules. Moreover, we prove that the pairs (gr-[Formula: see text], gr-[Formula: see text]) and (gr-[Formula: see text], gr-[Formula: see text]) are duality pairs over left [Formula: see text]-coherent rings, where gr-[Formula: see text] and gr-[Formula: see text] denote the subcategories of [Formula: see text]-FP-gr-injective left [Formula: see text]-modules and [Formula: see text]-gr-flat right [Formula: see text]-modules respectively. As applications, we obtain that any graded left (respectively, right) [Formula: see text]-module admits an [Formula: see text]-FP-gr-injective (respectively, [Formula: see text]-gr-flat) cover and preenvelope.

Category of $n$-FCP-gr-projective modules with respect to special copresented graded modules

“…Then by (1), K * n−1 is special gr-copresented in gr-R. So if M is n-FCP-grprojective right R-module, then similar to the proof (1), 0 [20,Definition 3.1]. Therefore, M * is an n-FP-gr-injective left R-module, and then we conclude that (gr-FCP n ) * ⊆ gr-FI n .…”

“…Let n ≥ 0 be an integer. Then, a graded left R-module F is called n-presented [20] if there exists an exact sequence…”

“…For a graded ring R, let X be a class of graded right R-modules and M a graded left R-module. Following [3,20], we say that a graded morphism f : X → M is an X -precover of M if X ∈ X and HOM R (X , X) → HOM R (X , M ) → 0 is exact for all X ∈ X . Moreover, if whenever a graded morphism g : X → X such that f g = f is an automorphism of X, then f : X → M is called an X -cover of M .…”

“…For example, Asensio et al in [4] introduced the notions of FP-gr-injective modules, then Yang and Liu in [18] investigated homological behavior of the FPgr-injective modules on gr-coherent rings. Recently in 2018, Zhao, Gao and Huang [20] gave a definition of n-presented graded modules and n-gr-coherent rings and also, by using of n-presented graded modules, they introduced the concept of n-FPgr-injective and n-gr-flat modules, and then examined the homological behavior of these modules over n-gr-coherent rings. In case n = 1, see [13,18].…”

“…So by Proposition 3.8, M N is n-FCP-gr-projective and hence by Theorem 4.4, N is n-FCP-gr-projective. (⇐=) It is clear.Before the next results, we first introduce the following symbols and definitions given in[9,20].For every class Y of graded right R-modules, denote the classesY ⊥ = {X ∈ gr-R : Ext 1 gr−R (Y, X) = 0 f or all Y ∈ Y} and ⊥ Y = {X ∈ gr-R : Ext gr−R (X, Y ) = 0 f or all Y ∈ Y}.Given two classes of graded right R-modules F and C, then we say that (F, C)is a cotorsion theory in gr-R if F ⊥ = C and F = ⊥ C. A cotorsion theory (F, C) is called hereditary if whenever 0 → F → F → F → 0 is exact in gr-R with F, F ∈ F then F is also in F.A duality pair over a graded ring R is a pair (F, C), where F is a class of graded right (resp. left) R-modules and C is a class of graded left (resp.…”