The Stacks project

Lemma 21.37.1. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the corresponding morphism of topoi. Let $\mathcal{O}_\mathcal {D}$ be a sheaf of rings and let $\mathcal{I}$ be an injective $\mathcal{O}_\mathcal {D}$-module. Then $H^ p(U, g^{-1}\mathcal{I}) = 0$ for all $p > 0$ and $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

Proof. The vanishing of the lemma follows from Lemma 21.10.9 if we can prove vanishing of all higher Čech cohomology groups $\check H^ p(\mathcal{U}, g^{-1}\mathcal{I})$ for any covering $\mathcal{U} = \{ U_ i \to U\} $ of $\mathcal{C}$. Since $u$ is continuous, $u(\mathcal{U}) = \{ u(U_ i) \to u(U)\} $ is a covering of $\mathcal{D}$, and $u(U_{i_0} \times _ U \ldots \times _ U U_{i_ n}) = u(U_{i_0}) \times _{u(U)} \ldots \times _{u(U)} u(U_{i_ n})$. Thus we have

\[ \check H^ p(\mathcal{U}, g^{-1}\mathcal{I}) = \check H^ p(u(\mathcal{U}), \mathcal{I}) \]

because $g^{-1} = u^ p$ by Sites, Lemma 7.21.5. Since $\mathcal{I}$ is an injective $\mathcal{O}_\mathcal {D}$-module these Čech cohomology groups vanish, see Lemma 21.12.3. $\square$

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