Localization and Extension of modules Let $R$ be a commutative ring and $S$ be an $R$-algebra. Assume that $S$ is finitely generated as an $R$-module. 
Let $M$ and $N$ be finitely generated $S$-modules and $\mathfrak{m}$ a maximal ideal in $R$.  

Under what condition does the following equality hold? $$
 Ext^{i}_{C}(M,N)_{\mathfrak{m}}=Ext^{i}_{C_{\mathfrak{m}}}(M_{\mathfrak{m}},N_{\mathfrak{m}}) \ \forall i,
$$ where the subscript $_{\mathfrak{m}}$ means the localization with respect to $\mathfrak{m}$.

Thank you for your assistance. 

According to Weibel the following is true (Prop 3.3.10):

Let $A$ be a finitely generated module over a commutative Noetherian ring $R$. Then for every multiplicative set $S$, all modules $B$ and all $n$, $$S^{-1} \text{Ext}_R^n(A,B) \simeq \text{Ext}^n_{S^{-1}R}(S^{-1}A,S^{-1}B)$$

Noetherian-ness is important here, for every finitely generated module over a Noetherian ring is finitely presented. The proof then follows from Lemma 3.3.8:

If $A$ is finitely presented as an $R$-module then for every central multiplicative set $S$ in $R$, there is an isomorphism
  $$S^{-1} \text{Hom}_R(A,B) \simeq \text{Hom}_{S^{-1}R}(S^{-1}A,S^{-1}B)$$

