Lemma 59.18.7. Let $\mathcal{C}$ be a category. If $\mathcal{I}$ is an injective object of $\textit{PAb}(\mathcal{C})$ and $\mathcal{U}$ is a family of morphisms with fixed target in $\mathcal{C}$, then $\check H^ p(\mathcal{U}, \mathcal{I}) = 0$ for all $p > 0$.

Proof. The Čech complex is the result of applying the functor $\mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}(-, \mathcal{I})$ to the complex $\mathbf{Z}^\bullet _\mathcal {U}$, i.e.,

$\check H^ p(\mathcal{U}, \mathcal{I}) = H^ p (\mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})} (\mathbf{Z}^\bullet _\mathcal {U}, \mathcal{I})).$

But we have just seen that $\mathbf{Z}^\bullet _\mathcal {U}$ is exact in negative degrees, and the functor $\mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}(-, \mathcal{I})$ is exact, hence $\mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})} (\mathbf{Z}^\bullet _\mathcal {U}, \mathcal{I})$ is exact in positive degrees. $\square$

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