The Stacks project

Lemma 20.32.9. Let $f : X \to Y$ be a flat morphism of ringed spaces. If $\mathcal{I}^\bullet $ is a K-injective complex of $\mathcal{O}_ X$-modules, then $f_*\mathcal{I}^\bullet $ is K-injective as a complex of $\mathcal{O}_ Y$-modules.

Proof. This is true because

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ Y)}(\mathcal{F}^\bullet , f_*\mathcal{I}^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ X)}(f^*\mathcal{F}^\bullet , \mathcal{I}^\bullet ) \]

by Sheaves, Lemma 6.26.2 and the fact that $f^*$ is exact as $f$ is assumed to be flat. $\square$

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