Lemma 93.9.4. In Lemma 93.9.3 if $X$ is proper over $k$, then $\text{Inf}(\mathcal{D}\! \mathit{ef}_ X)$ and $T\mathcal{D}\! \mathit{ef}_ X$ are finite dimensional.
Proof. By the lemma we have to show $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/k}, \mathcal{O}_ X)$ and $\mathop{\mathrm{Ext}}\nolimits ^0_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/k}, \mathcal{O}_ X)$ are finite dimensional. By More on Morphisms, Lemma 37.13.4 and the fact that $X$ is Noetherian, we see that $\mathop{N\! L}\nolimits _{X/k}$ has coherent cohomology sheaves zero except in degrees $0$ and $-1$. By Derived Categories of Schemes, Lemma 36.11.7 the displayed $\mathop{\mathrm{Ext}}\nolimits $-groups are finite $k$-vector spaces and the proof is complete. $\square$
Post a comment
Your email address will not be published.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)