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

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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: