The Stacks project

Lemma 115.13.2. Let $X$ be a proper scheme over a field $k$ which is regular. Let $K \in \mathop{\mathrm{Ob}}\nolimits (D_\mathit{QCoh}(\mathcal{O}_ X))$. The following are equivalent

  1. $K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X) = D_{perf}(\mathcal{O}_ X)$, and

  2. $\sum _{i \in \mathbf{Z}} \dim _ k \mathop{\mathrm{Ext}}\nolimits ^ i_ X(E, K) < \infty $ for all perfect $E$ in $D(\mathcal{O}_ X)$.

Proof. The equality in (1) holds by Derived Categories of Schemes, Lemma 36.11.8. The implication (1) $\Rightarrow $ (2) follows from Derived Categories of Varieties, Lemma 57.5.3. The implication (2) $\Rightarrow $ (1) follows from More on Morphisms, Lemma 37.69.6. $\square$

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