Foxby equivalence relative to \(C\)-\(fp_n\)-injective and \(C\)-\(fp_{n}\)-flat modules

Let \(R\) and \(S\) be rings, \(C= {}_SC_R\) a (faithfully) semidualizing bimodule, and \(n\) a positive integer or \(n=\infty\). In this paper, we introduce the concepts of \(C\)-\(fp_n\)-injective \(R\)-modules and \(C\)-\(fp_n\)-flat \(S\)-modules as a common generalization of some known modules such as \(C\)-\(FP_{n}\)-injective (resp. \(C\)-weak injective) \(R\)-modules and \(C\)-\(FP_{n}\)-flat (resp. \(C\)-weak flat) \(S\)-modules. Then we investigate \(C\)-\(fp_{n}\)-injective and \(C\)-\(fp_{n}\)-flat dimensions of modules, where the classes of these modules, namely \(Cfp_nI(R)_{\leq k}\) and \(Cfp_nF(S)_{\leq k}\), respectively. We study Foxby equivalence relative to these classes, and also the existence of \(Cfp_nI(R)_{\leq k}\) and \(Cfp_nF(S)_{\leq k}\) preenvelopes and covers. Finally, we study the exchange properties of these classes, as well as preenvelopes (resp. precovers) and Foxby equivalence, under almost excellent extensions of rings.