Resolutions and stability of \(C\)-Gorenstein flat modules (Q2374297)

Gorenstein projective, injective, and flat modules were introduced in [\textit{E. E. Enochs} and \textit{O. M. G. Jenda}, Math. Z. 220, No. 4, 611--633 (1995; Zbl 0845.16005)] and [\textit{E. E. Enochs} et al., J. Nanjing Univ., Math. Biq. 10, No. 1, 1--9 (1993; Zbl 0794.16001)]. A module is Gorenstein flat if it is a syzygy in a complex of flat modules which stays exact after tensoring with any injective module. Denote the class of all Gorenstein flat \(R\)-modules by \(\mathcal{G}\mathcal{F}(R)\). One question is if an iteration of this definition leads to the same result, in other words, does the equality \(\mathcal{G}\mathcal{F}(\mathcal{G}\mathcal{F}(R))=\mathcal{G}\mathcal{F}(R)\) hold? Questions of this kind were first studied in [\textit{S. Sather-Wagstaff} et al., J. Lond. Math. Soc., II. Ser. 77, No. 2, 481--502 (2008; Zbl 1140.18010)]. The case of Gorenstein flats has been studied in at least two papers [\textit{G. Yang} and \textit{Z. Liu}, Glasg. Math. J. 54, No. 1, 177--191 (2012; Zbl 1248.16007)] and [\textit{S. Bouchiba} and \textit{M. Khaloui}, Glasg. Math. J. 54, No. 1, 169--175 (2012; Zbl 1235.16009)]. In [\textit{H. Holm} and \textit{P. Jørgensen}, J. Pure Appl. Algebra 205, No. 2, 423--445 (2006; Zbl 1094.13021)] the authors define a relative version of Gorenstein projective, injective and flat modules with respect to a semidualizing module \(C\). Denote the class of all Gorenstein flat \(R\)-modules with respect to \(C\) by \(\mathcal{G}\mathcal{F}_{\mathcal{C}}(R)\). The authors in their main result [Theorem 3.12] prove that if \(R\) is a coherent ring then \(\mathcal{G}\mathcal{F}_{C}(\mathcal{G}\mathcal{F}_{C}(R))=\mathcal{G}\mathcal{F}_{C}(R)\).