Lemma 10.71.9. Let $R$ be a Noetherian ring. Let $M$, $N$ be finite $R$-modules. Then $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N)$ is a finite $R$-module for all $i$.

**Proof.**
This holds because $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N)$ is computed as the cohomology groups of a complex $\mathop{\mathrm{Hom}}\nolimits _ R(F_\bullet , N)$ with each $F_ n$ a finite free $R$-module, see Lemma 10.71.1.
$\square$

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