The Stacks project

Lemma 25.6.3. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{G}$ be a presheaf on $\mathcal{C}$. Let $K$ be a hypercovering of $\mathcal{G}$. Let $\mathcal{I}$ be an injective sheaf of abelian groups on $\mathcal{C}$. Then

\[ \check{H}^ p(K, \mathcal{I}) = \left\{ \begin{matrix} H^0(\mathcal{G}, \mathcal{I}) & \text{if} & p = 0 \\ 0 & \text{if} & p > 0 \end{matrix} \right. \]

Proof. By (25.5.2.1) we have

\[ s(\mathcal{F}(K)) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Ab}(\mathcal{C})}(s(\mathbf{Z}_{F(K)}^\# ), \mathcal{F}) \]

The complex $s(\mathbf{Z}_{F(K)}^\# )$ is quasi-isomorphic to $\mathbf{Z}_\mathcal {G}^\# $, see Lemma 25.4.4. We conclude that if $\mathcal{I}$ is an injective abelian sheaf, then the complex $s(\mathcal{I}(K))$ is acyclic except possibly in degree $0$. In other words, we have $\check{H}^ i(K, \mathcal{I}) = 0$ for $i > 0$. Combined with Lemma 25.6.2 the lemma is proved. $\square$

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