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. $\square$
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