The Stacks project

Lemma 21.14.1. 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. Then for any injective object $\mathcal{I}$ in $\textit{Mod}(\mathcal{O}_\mathcal {C})$ the pushforward $f_*\mathcal{I}$ is totally acyclic.

Proof. Let $K$ be a sheaf of sets on $\mathcal{D}$. By Modules on Sites, Lemma 18.7.2 we may replace $\mathcal{C}$, $\mathcal{D}$ by “larger” sites such that $f$ comes from a morphism of ringed sites induced by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ such that $K = h_ V$ for some object $V$ of $\mathcal{D}$.

Thus we have to show that $H^ q(V, f_*\mathcal{I})$ is zero for $q > 0$ and all objects $V$ of $\mathcal{D}$ when $f$ is given by a morphism of ringed sites. Let $\mathcal{V} = \{ V_ j \to V\} $ be any covering of $\mathcal{D}$. Since $u$ is continuous we see that $\mathcal{U} = \{ u(V_ j) \to u(V)\} $ is a covering of $\mathcal{C}$. Then we have an equality of Čech complexes

\[ \check{\mathcal{C}}^\bullet (\mathcal{V}, f_*\mathcal{I}) = \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}) \]

by the definition of $f_*$. By Lemma 21.12.3 we see that the cohomology of this complex is zero in positive degrees. We win by Lemma 21.10.9. $\square$

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