[Lemma 7.46, Rouquier-dimensions] and implicit in [Theorem A.1, BvdB]

Lemma 57.5.4. Let $X$ be a projective scheme over a field $k$. Let $K \in \mathop{\mathrm{Ob}}\nolimits (D_\mathit{QCoh}(\mathcal{O}_ X))$. The following are equivalent

1. $K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$, 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 implication (1) $\Rightarrow$ (2) follows from Lemma 57.5.3.

Assume (2). Choose a closed immersion $i : X \to \mathbf{P}^ n_ k$. It suffices to show that $Ri_*K$ is in $D^ b_{\textit{Coh}}(\mathbf{P}^ n_ k)$ since a quasi-coherent module $\mathcal{F}$ on $X$ is coherent, resp. zero if and only if $i_*\mathcal{F}$ is coherent, resp. zero. For a perfect object $E$ of $D(\mathcal{O}_{\mathbf{P}^ n_ k})$, $Li^*E$ is a perfect object of $D(\mathcal{O}_ X)$ and

$\mathop{\mathrm{Ext}}\nolimits ^ q_{\mathbf{P}^ n_ k}(E, Ri_*K) = \mathop{\mathrm{Ext}}\nolimits ^ q_ X(Li^*E, K)$

Hence by our assumption we see that $\sum _{q \in \mathbf{Z}} \dim _ k \mathop{\mathrm{Ext}}\nolimits ^ q_{\mathbf{P}^ n_ k}(E, Ri_*K) < \infty$. We conclude by Lemma 57.5.2. $\square$

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