Lemma 21.13.3. 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

**Proof.**
Lemma 21.10.3 gives the first equality in the following sequence of equalities

The third equality by Modules on Sites, Lemma 18.9.2. By Lemma 21.13.1 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 21.10.5 we see the vanishing of higher Čech cohomology groups. For the zeroth see Lemma 21.9.2. $\square$

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