Lemma 57.5.3. Let $X$ be a scheme proper over a field $k$. Let $K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ and let $E$ in $D(\mathcal{O}_ X)$ be perfect. Then $\sum _{i \in \mathbf{Z}} \dim _ k \mathop{\mathrm{Ext}}\nolimits ^ i_ X(E, K) < \infty $.

**Proof.**
This follows for example by combining Derived Categories of Schemes, Lemmas 36.11.7 and 36.13.5. Alternative proof: combine Derived Categories of Schemes, Lemmas 36.11.6 and 36.11.3.
$\square$

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