Stability of Gorenstein Flat Categories

Abstract

A left R-module M is called two-degree Gorenstein flat if there exists an exact sequence of Gorenstein flat left R-modules ⋅⋅⋅ → G2 → G1 → G0 → G−1 → G−2 → ⋅⋅⋅ such that M ≅ Ker(G0 → G−1) and it remains exact after applying H ⊗R- for any Gorenstein injective right R-module H. In this paper we first give some characterisations of Gorenstein flat objects in the category of complexes of modules and then use them to show that two notions of the two-degree Gorenstein flat and the Gorenstein flat left R-modules coincide when R is right coherent.

Stability of Gorenstein Flat Modules

“…We are informed by the Editor-in-Chief of the Glasgow Mathematical Journal that an independent proof of this result has been obtained by Gang Yang and Zhongkui Liu [10]. The authors would like to thank the referee for his/her careful reading of this paper and his/her valuable comments.…”

Resolutions and stability of $C$-Gorenstein flat modules

“…Since Tor R j≥1 (I C (R), A i ) = 0 for any i ≥ 1, we get an exact sequence 0−→ N 0 −→ G ′ −→ G 2 −→ G 3 −→ • • • in M(R), which is I C (R)⊗ R − exact.Repeating the process, we obtain the desired exact sequence. Thus, A ∈ GF C (R).In the special case C = R, we obtain the main theorem of[2] and[17, Theorem 4.3]. Corollary 3.13.…”

Gorenstein projective complexes with respect to cotorsion pairs

“…For more details, see [1]. The stabiltity of Gorenstein flat R-module has been treated by Bouchiba and Khaloui [2], Xu and Ding [13], Yang and Liu [17], respectively. By using totally different techniques, they showed that over a left GF-closed ring R (a ring R over which the class of the Gorenstein flat R-modules is closed under extensions), an R-module M is Gorenstein flat if and only if there exists an exact sequence of Gorenstein flat (1) M ∈ G(A).…”