Localization of Ext over a Noetherian ring ($S^{-1}{\mathrm{Ext}}_R^i(N,M)\cong {\mathrm{Ext}}_{S^{-1}R}^i(S^{-1}N,S^{-1}M)$)

Over a Noetherian ring, a finitely generated module $N$ has a free resolution with every term finitely generated. Hom from a finite free module commutes with localization, and exactness of localization commutes with cohomology. Thus the displayed isomorphism holds for every $i\ge 0$ and arbitrary $M$. The finiteness condition is on the first argument; the second need not be finitely generated.