Lemma 57.5.3 (0FYC)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 57: Derived Categories of Varieties
Section 57.5: Characterizing coherent modules
Lemma 57.5.3 (cite)

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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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