The Stacks project

Lemma 85.18.4. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $K$ be a hypercovering. 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

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

with quasi-inverse $Ra_*$.

Proof. Observe that $\mathcal{A}$ is a weak Serre subcategory by Lemma 85.12.6 (the required hypotheses hold by the discussion in Remark 85.16.5). The equivalence is a formal consequence of the results obtained so far. Use Lemmas 85.18.1 and 85.18.2 and Cohomology on Sites, Lemma 21.28.5. $\square$

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