Cartan-Eilenberg Gorenstein Flat Complexes

Abstract

In this paper, we study Cartan-Eilenberg Gorenstein flat complexes. We show that over coherent rings a Cartan-Eilenberg Gorenstein flat complex can be gotten by a so-called complete Cartan-Eilenberg flat resolution. We argue that over a coherent ring every complex has a Cartan-Eilenberg Gorenstein flat cover.

Cartan-Eilenberg projective, injective and flat complexes

“…Suppose that X is a CE flat complex. Then X + is CE injective by [19], Corollary 2.3, hence CE -id(X + ) = 0, so CE -fd(X ++ ) CE -id(X + ) = 0 by (3). Thus X ++ ∈ CE(F (R)).…”

“…Conversely, if X ++ ∈ CE(F (R)). According to the pure exact sequence 0 → X → X ++ , we can get X is a CE flat complex by [19], Lemma 2.9.…”

“…Furthermore, Enochs [7] considered Cartan-Eilenberg flat complexes which are extensions of Cartan-Eilenberg projective complexes and showed that they are precisely the direct limits of finitely generated Cartan-Eilenberg projective complexes. Recently, Yang and Liang [19] gave some characterizations of Cartan-Eilenberg flat complexes and proved that a ring R is right coherent if and only if every complex of R-modules has a Cartan-Eilenberg flat preenvelope (for the rest of the article, we will use the abbreviation CE for Cartan-Eilenberg).…”

“…We denote these derived functors as Ext i (X, Y ). If C is a complex of right R-modules and D is a complex of left R-modules, by [19], Lemma 2.4 and Theorem 2.6, we can compute left derived functors of -⊗-using the CE flat resolution of either C or D. We denote these derived functors as Tor i (C, D).…”