The Stacks project

Lemma 20.11.1. Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ be a covering. Let $\mathcal{I}$ be an injective $\mathcal{O}_ X$-module. 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. An injective $\mathcal{O}_ X$-module is also injective as an object in the category $\textit{PMod}(\mathcal{O}_ X)$ (for example since sheafification is an exact left adjoint to the inclusion functor, using Homology, Lemma 12.29.1). Hence we can apply Lemma 20.10.5 (or its proof) to see the result. $\square$


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