Lemma 21.12.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a covering of $\mathcal{C}$. Let $\mathcal{I}$ be an injective $\mathcal{O}$-module, i.e., an injective object of $\textit{Mod}(\mathcal{O})$. Then
\[ \check{H}^ p(\mathcal{U}, \mathcal{I}) = \left\{ \begin{matrix} \mathcal{I}(U)
& \text{if}
& p = 0
\\ 0
& \text{if}
& p > 0
\end{matrix} \right. \]
Proof.
Lemma 21.9.3 gives the first equality in the following sequence of equalities
\begin{align*} \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}) & = \mathop{\mathrm{Mor}}\nolimits _{\textit{PAb}(\mathcal{C})}( \mathbf{Z}_{\mathcal{U}, \bullet }, \mathcal{I}) \\ & = \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathbf{Z})}( \mathbf{Z}_{\mathcal{U}, \bullet }, \mathcal{I}) \\ & = \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O})}( \mathbf{Z}_{\mathcal{U}, \bullet } \otimes _{p, \mathbf{Z}} \mathcal{O}, \mathcal{I}) \end{align*}
The third equality by Modules on Sites, Lemma 18.9.2. By Lemma 21.12.1 we see that $\mathcal{I}$ is an injective object in $\textit{PMod}(\mathcal{O})$. Hence $\mathop{\mathrm{Hom}}\nolimits _{\textit{PMod}(\mathcal{O})}(-, \mathcal{I})$ is an exact functor. By Lemma 21.9.5 we see the vanishing of higher Čech cohomology groups. For the zeroth see Lemma 21.8.2.
$\square$
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