Lemma 25.5.2 (01GW)—The Stacks project

Lemma 25.5.2. 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

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

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

\begin{eqnarray*} \mathcal{F}(Z) & = & \prod \mathcal{F}(U_ i) \\ & = & \prod \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_{U_ i}, \mathcal{F})\\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(F(Z), \mathcal{F})\\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_{F(Z)}, \mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{Ab}(\mathcal{C})}(\mathbf{Z}_{F(Z)}^\# , \mathcal{F}) \end{eqnarray*}

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

25.5.2.1

\begin{equation} \label{hypercovering-equation-identify-cech} s(\mathcal{F}(K)) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Ab}(\mathcal{C})}(s(\mathbf{Z}_{F(K)}^\# ), \mathcal{F}) \end{equation}

see Definition 25.4.1 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 25.4.5. 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

\[ \check{H}^ i(K, \mathcal{I}) = 0 \]

for $i > 0$. Combined with Lemma 25.5.1 the lemma is proved. $\square$

The Stacks project

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