Lemma 85.6.4 (0D75)—The Stacks project

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Lemma 85.6.4. In Situation 85.3.3. 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 85.6.3. $\square$

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