The Stacks project

Lemma 21.14.2. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. If $f$ is flat, then $f_*\mathcal{I}$ is an injective $\mathcal{O}_\mathcal {D}$-module for any injective $\mathcal{O}_\mathcal {C}$-module $\mathcal{I}$.

Proof. In this case the functor $f^*$ is exact, see Modules on Sites, Lemma 18.31.2. Hence the result follows from Homology, Lemma 12.29.1. $\square$

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