The Stacks project

Lemma 21.43.12. Let $\epsilon : (\mathcal{C}_\tau , \mathcal{O}_\tau ) \to (\mathcal{C}_{\tau '}, \mathcal{O}_{\tau '})$ be as in Section 21.27. Assume

  1. $\tau '$ is the chaotic topology on the category $\mathcal{C}$,

  2. for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and all K-flat complexes of $\mathcal{O}(U)$-modules $M^\bullet $ the map

    \[ M^\bullet \longrightarrow R\Gamma ((\mathcal{C}/U)_\tau , (M^\bullet \otimes _{\mathcal{O}(U)} \mathcal{O}_ U)^\# ) \]

    is a quasi-isomorphism (see proof for an explanation).

Then $\epsilon ^*$ and $R\epsilon _*$ define mutually quasi-inverse equivalences between $\mathit{QC}(\mathcal{O})$ and the full subcategory of $D(\mathcal{C}_\tau , \mathcal{O}_\tau )$ consisting of objects $K$ such that $R\epsilon _*K$ is in $\mathit{QC}(\mathcal{O})$1.

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

\[ R\Gamma (U, R\epsilon _*(\epsilon ^*M)) = R\Gamma ((\mathcal{C}/U)_\tau , (\mathcal{M}^\bullet )^\# |_{\mathcal{C}/U}) = R\Gamma ((\mathcal{C}/U)_\tau , (\mathcal{M}^\bullet |_{\mathcal{C}/U})^\# ) \]

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

\[ \mathcal{M}^\bullet (U) \otimes _{\mathcal{O}(U)} \mathcal{O}_ U \longrightarrow \mathcal{M}^\bullet |_{\mathcal{C}/U} \]

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

\[ R\Gamma (U, R\epsilon _*(\epsilon ^*M)) = R\Gamma ((\mathcal{C}/U)_\tau , (M^\bullet \otimes _{\mathcal{O}(U)} \mathcal{O}_ U)^\# ) \]

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

[1] This means that $R\Gamma (V, K) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U) \to R\Gamma (U, K)$ is an isomorphism for all $U \to V$ in $\mathcal{C}$.

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