The Stacks project

Lemma 84.18.2. 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. For $E \in D(\mathcal{O}_\mathcal {C})$ the map

\[ E \longrightarrow Ra_*La^*E \]

is an isomorphism.

Proof. Since $a^{-1}\mathcal{O}_\mathcal {C} = \mathcal{O}$ we have $La^* = a^* = a^{-1}$. Moreover $Ra_*$ agrees with $Ra_*$ on abelian sheaves, see Cohomology on Sites, Lemma 21.20.7. Hence the lemma follows immediately from Lemma 84.17.3. $\square$

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