The balance of relative Ext groups defined by semi dualizing modules

Let $R$ be a commutative Noetherian ring with identity and $C$ a semidualizing module for $R$. Let ${\mathcal{P}}_C(R)$ and ${\mathcal{I}}_C(R)$ denote, respectively, the classes of $C$-projective and $C$-injective $R$-modules. We show that their induced Ext bifunctors ${\text{Ext}}_{{\mathcal{P}}_C}^i(-,\sim )$ and ${\text{Ext}}_{{\mathcal{I}}_C}^i(-,\sim )$ coincide for all $i\ge 0$ if and only if $C$ is projective. Also, we provide some other criteria for $C$ to be projective by using some special cotorsion theories.