Weak AB-context for FP-injective modules with respect to semidualizing modules (Q2853975)

\textit{H. B. Foxby} [Math. Scand. 31, 267--284 (1973; Zbl 0272.13009)], \textit{W. V. Vasconcelos} [Divisor theory in module categories. Amsterdam-Oxford: North-Holland Publishing Company (1974; Zbl 0296.13005)] and \textit{E. S. Golod} [Tr. Mat. Inst. Steklova 165, 62--66 (1984; Zbl 0577.13008)] independently initiated the study of semidualizing modules over a commutative Noetherian local ring. Recently, many authors studied (semi)dualizing modules. The authors introduce \(\mathcal{C}\)-FP-injective and \(\mathcal{C}\)-FP-projective \(R\)-modules (denoted by \(\mathcal{FI}_C^{fp}(R)\)) and \(\mathcal{G}_C\)-FP-injective \(R\)-modules (denoted by \(\mathcal{GFI}_C(R)\)) and prove that the triple (\(\mathcal{GFI}_C(R)\), cores \(\widehat{\mathcal{FI}_C^{fp}(R)}\), \(\mathcal{FI}_C^{fp}(R)\)) satisfies the axioms that are dual to the ones for a weak AB-context defined by \textit{M. Hashimoto} [Auslander-Buchweitz approximations of equivariant modules. Cambridge: Cambridge University Press (2000; Zbl 0993.13007)]. As an application the authors prove the following result.NEWLINENEWLINETheorem. Assume that \(R\) is left coherent. If (\(\mathcal{GFI}_C(R)\), cores \(\widehat{\mathcal{FI}_C^{fp}(R)}\), \(\mathcal{FI}_C^{fp}(R)\)) is an AB-context, then there is a model structure of \(\operatorname {Mod}R\) in which the cofibrant objects are the \(\mathcal{G}\)-FP-injective \(R\)-modules and trivial objects are the modules with finite \(\mathcal{FI}(R)\)-injective dimension.