The Stacks project

Proof. We will use the observations made in Section 21.27 without further mention. Since $R\epsilon _*$ is fully faithful and $\epsilon ^* \circ R\epsilon _* = \text{id}$, to prove the lemma it suffices to show that for $M$ in $\mathit{QC}(\mathcal{O})$ we have $R\epsilon _*(\epsilon ^*M) = M$. Condition (2) is exactly the condition needed to see this. Namely, we choose a K-flat complex $\mathcal{M}^\bullet$ of $\mathcal{O}$-modules with flat terms representing $M$. Then we see that $\epsilon ^*M$ is represented by the $\tau$-sheafification $(\mathcal{M}^\bullet )^\#$ of $\mathcal{M}^\bullet$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. By Leray we get

The last equality since sheafification commutes with restriction to $\mathcal{C}/U$. As usual, denote $\mathcal{O}_ U$ the restriction of $\mathcal{O}$ to $\mathcal{C}/U$. Consider the map

of complexes of $\mathcal{O}_ U$-modules (in $\tau '$-topology). By our choice of $\mathcal{M}^\bullet$ the complex $\mathcal{M}^\bullet (U)$ is a K-flat complex of $\mathcal{O}(U)$-modules; see Lemma 21.18.1 and use that the inclusion of $U$ into $\mathcal{C}$ defines a morphism of ringed topoi $(\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}(U)) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{\tau '}), \mathcal{O})$. Since $M$ is in $\mathit{QC}(\mathcal{O})$ we conclude that the displayed arrow is a quasi-isomorphism. Since sheafification is exact, we see that the same remains true after sheafification. Hence

and assumption (2) tells us this is equal to $R\Gamma (U, M) = \mathcal{M}^\bullet (U)$ as desired. $\square$