The Stacks project

Lemma 85.22.4. Let $\mathcal{C}$ be a site with fibre products and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $U$ be a hypercovering of $X$ in $\mathcal{C}$. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ denote the weak Serre subcategory of cartesian $\mathcal{O}$-modules. Then the functor $La^*$ defines an equivalence

\[ D^+(\mathcal{O}_ X) \longrightarrow D_\mathcal {A}^+(\mathcal{O}) \]

with quasi-inverse $Ra_*$.

Proof. This is a special case of Lemma 85.20.4. $\square$

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