= Localization of Ext over a Noetherian ring
{title2=$S^{-1}\operatorname{Ext}_R^i(N,M)\cong\operatorname{Ext}_{S^{-1}R}^i(S^{-1}N,S^{-1}M)$}
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 $i\geq0$ and arbitrary $M$. The finiteness condition is on the first argument; the second need not be finitely generated.
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