The Stacks project

Lemma 20.11.11. Let $f : X \to Y$ be a morphism of ringed spaces. Assume $f$ is flat. Then $f_*\mathcal{I}$ is an injective $\mathcal{O}_ Y$-module for any injective $\mathcal{O}_ X$-module $\mathcal{I}$.

Proof. In this case the functor $f^*$ transforms injections into injections (Modules, Lemma 17.20.2). Hence the result follows from Homology, Lemma 12.29.1. $\square$

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