\(D_C\)-projective dimensions, Foxby equivalence and \(\mathcal{SD}_C\)-projective modules (Q2803564)

Let \(R\) be a commutative ring with unity and let \(C\) be a semidualizing module. Associate to \(C\) there are several important classes: \(\mathcal F_C(R)\) (resp. \(\mathcal P_C(R)\), \(\mathcal I_C(R))=\{C\otimes F, F\) is \(R\)-flat (resp. projective, injective). The authors introduce the notion of \(D_C\)-projective as those modules wich are the cokernel of \((P_1\to P_0)\) in a exact complex \(\cdots \to P_1\to P_0\to C\otimes P^0\to C\otimes P^1\to \cdots\) with the property that it remains exact under \(\mathrm{Hom}_R(-,X)\) for each \(X\) in \(\mathcal F_C(R)\). These modules generalize the \(G_C\)-projectives modules introduced by \textit{H. Holm} and \textit{P. Jørgensen} [J. Pure Appl. Algebra 205, No. 2, 423--445 (2006; Zbl 1094.13021)]. Next, the notion of dimension relative to these classes of modules is studied. Analougously the concepts of \(D_C\)-injective modules and dimension relative to them are defined. Using these modules and Ding projective and injective modules [\textit{N. Ding} et al., J. Aust. Math. Soc. 86, No. 3, 323--338 (2009; Zbl 1200.16010)] a new version of the Foxby's equivalence is obtained. In the last section, new characterizations, by the preceding classes of modules, of perfect rings and QF rings are presented.