Lemma 36.6.7. With $X$, $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$, and $\mathcal{F}$ as in Remark 36.6.4. If $\mathcal{F}$ is quasi-coherent, then the complex (36.6.4.1) represents $i_* R\mathcal{H}_ Z(\mathcal{F})$ in $D_ Z(\mathcal{O}_ X)$.
Proof. Let us denote $\mathcal{F}^\bullet $ the complex (36.6.4.1). The statement of the lemma means that the map $\mathcal{F}^\bullet \to i_*R\mathcal{H}_ Z(\mathcal{F})$ of Remark 36.6.6 is an isomorphism. Since $\mathcal{F}^\bullet $ is in $D_ Z(\mathcal{O}_ X)$ (see remark cited), we see that $i_*R\mathcal{H}_ Z(\mathcal{F}^\bullet ) = \mathcal{F}^\bullet $ by Cohomology, Lemma 20.34.2. The morphism $U_{i_0 \ldots i_ p} \to X$ is affine as it is given over affine opens of $X$ by inverting the function $f_{i_0} \ldots f_{i_ p}$. Thus we see that
\[ \mathcal{F}_{i_0 \ldots i_ p} = (U_{i_0 \ldots i_ p} \to X)_*\mathcal{F}|_{U_{i_0 \ldots i_ p}} = R(U_{i_0 \ldots i_ p} \to X)_*\mathcal{F}|_{U_{i_0 \ldots i_ p}} \]
by Cohomology of Schemes, Lemma 30.2.3 and the assumption that $\mathcal{F}$ is quasi-coherent. We conclude that $R\mathcal{H}_ Z(\mathcal{F}_{i_0 \ldots i_ p}) = 0$ by Cohomology, Lemma 20.34.7. Thus $i_*R\mathcal{H}_ Z(\mathcal{F}^ p) = 0$ for $p > 0$. Putting everything together we obtain
\[ \mathcal{F}^\bullet = i_*R\mathcal{H}_ Z(\mathcal{F}^\bullet ) = i_*R\mathcal{H}_ Z(\mathcal{F}) \]
as desired. $\square$
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Your email address will not be published.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)
There are also: