The Stacks project

Lemma 85.21.4. Let $\mathcal{C}$ be a site with fibre product and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above. Let $\mathcal{A} \subset \textit{Ab}((\mathcal{C}/U)_{total})$ denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor $a^{-1}$ defines an equivalence

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

with quasi-inverse $Ra_*$.

Proof. This is a special case of Lemma 85.19.4 $\square$

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