The Stacks project

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

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

as desired. $\square$