In the projective case this is [Lemma 7.46, Rouquier-dimensions] and implicit in [Theorem A.1, BvdB]
Lemma 57.5.4. Let $X$ be a proper scheme over a field $k$. Let $K \in \mathop{\mathrm{Ob}}\nolimits (D_\mathit{QCoh}(\mathcal{O}_ X))$. The following are equivalent
$K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$, and
$\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. The implication (2) $\Rightarrow $ (1) follows from More on Morphisms, Lemma 37.69.6 (see Derived Categories of Schemes, Example 36.35.2 for the meaning of a relatively perfect object over a field); the easier proof in the projective case is in the next paragraph.
Assume (2) and $X$ projective over $k$. 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
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. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)