The Stacks project

Lemma 85.17.5. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Let $\mathcal{A} \subset \textit{Ab}((\mathcal{C}/K)_{total})$ denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor $a^{-1}$ defines an equivalence

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

with quasi-inverse $Ra_*$.

Proof. Observe that $\mathcal{A}$ is a weak Serre subcategory by Lemma 85.12.6. The equivalence is a formal consequence of the results obtained so far. Use Lemmas 85.17.1 and 85.17.3 and Cohomology on Sites, Lemma 21.28.5 $\square$

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