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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.

2 Answers 2