Lemma 21.20.10 (093Y)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.20: Some properties of K-injective complexes
Lemma 21.20.10 (cite)

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Lemma 21.20.10. 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 flat morphism of ringed topoi. If $\mathcal{I}^\bullet$ is a K-injective complex of $\mathcal{O}_\mathcal {C}$ -modules, then $f_*\mathcal{I}^\bullet$ is K-injective as a complex of $\mathcal{O}_\mathcal {D}$ -modules.

Proof. This is true because

$\mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_\mathcal {D})}(\mathcal{F}^\bullet , f_*\mathcal{I}^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_\mathcal {C})}(f^*\mathcal{F}^\bullet , \mathcal{I}^\bullet )$

by Modules on Sites, Lemma 18.13.2 and the fact that $f^*$ is exact as $f$ is assumed to be flat. $\square$

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