Lemma 36.11.7. Let $A$ be a Noetherian ring. Let $X$ be a proper scheme over $A$. For $L$ in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ and $K$ in $D^-_{\textit{Coh}}(\mathcal{O}_ X)$, the $A$-modules $\mathop{\mathrm{Ext}}\nolimits _{\mathcal{O}_ X}^ n(K, L)$ are finite.

Proof. Recall that

$\mathop{\mathrm{Ext}}\nolimits _{\mathcal{O}_ X}^ n(K, L) = H^ n(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, L)) = H^ n(\mathop{\mathrm{Spec}}(A), Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, L))$

see Cohomology, Lemma 20.40.1 and Cohomology, Section 20.13. Thus the result follows from Lemmas 36.11.5 and 36.11.4. $\square$

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