Let $\mathcal{E}$ and $\mathcal{F}$ be two coherent sheaves on a polarized projective variety $(X,\mathcal{O}_X(1))$.

Denote by $E=\Gamma_*(\mathcal{E})=\oplus_{k\in\mathbb{Z}}\mathcal{E}(k)$, $F=\Gamma_*(\mathcal{F})$, and $R= \Gamma_\ast(\mathcal{O}_X)$.

Is it true that $$\oplus_{k\in\mathbb{Z}}Ext^i_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}(k))=Ext^i_R(E, F)?$$

If the equality does not hold in general, when and to what extent it will hold?