The Stacks project

Lemma 84.6.4. In Situation 84.3.3. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$ such that $f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ is flat for all $\varphi : [n] \to [m]$. If $\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{O})$, then $\mathcal{I}_ n$ is injective in $\textit{Mod}(\mathcal{O}_ n)$.

Proof. This follows from Homology, Lemma 12.29.1 and Lemma 84.6.3. $\square$

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