The Stacks project

Lemma 84.18.3. 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. Then we have a canonical isomorphism

\[ R\Gamma (\mathcal{C}, E) = R\Gamma ((\mathcal{C}/K)_{total}, La^*E) \]

for $E \in D(\mathcal{O}_\mathcal {C})$.

Proof. This follows from Lemma 84.18.2 because $R\Gamma ((\mathcal{C}/K)_{total}, -) = R\Gamma (\mathcal{C}, -) \circ Ra_*$ by Cohomology on Sites, Remark 21.14.4 or by Cohomology on Sites, Lemma 21.20.5. $\square$

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