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
$K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X) = D_{perf}(\mathcal{O}_ X)$, 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 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. $\square$
Your email address will not be published.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)