The Stacks project

Lemma 85.6.2. In Situation 85.3.3. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. If $\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{O})$, then $\mathcal{I}_ n$ is a totally acyclic sheaf on $\mathcal{C}_ n$.

Proof. This follows from Cohomology on Sites, Lemma 21.37.4 applied to the inclusion functor $\mathcal{C}_ n \to \mathcal{C}_{total}$ and its properties proven in Lemma 85.3.5. $\square$

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