Lemma 24.5.2. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $\mathcal{I}$ be an injective sheaf of abelian groups on $\mathcal{C}$. Then

**Proof.**
Observe that for any object $Z = \{ U_ i \to X\} $ of $\text{SR}(\mathcal{C}, X)$ and any abelian sheaf $\mathcal{F}$ on $\mathcal{C}$ we have

Thus we see, for any simplicial object $K$ of $\text{SR}(\mathcal{C}, X)$ that we have

see Definition 24.4.1 for notation. The complex of sheaves $s(\mathbf{Z}_{F(K)}^\# )$ is quasi-isomorphic to $\mathbf{Z}_ X^\# $ if $K$ is a hypercovering, see Lemma 24.4.5. We conclude that if $\mathcal{I}$ is an injective abelian sheaf, and $K$ a hypercovering, then the complex $s(\mathcal{I}(K))$ is acyclic except possibly in degree $0$. In other words, we have

for $i > 0$. Combined with Lemma 24.5.1 the lemma is proved. $\square$

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