Let $(R,m)$ be a Noetherian *local ring and suppose that $m$ is maximal in the ordinary sense. Then why is it true that $\operatorname{Ext}^i_R(R/m^j,M) \cong \operatorname{Ext}^i_{R_m}(R_m/m^jR_m,M_m)$ for every graded $R$-module $M$?
Reference: Bruns and Herzog, Cohen-Macaulay Rings, Remark 3.6.18