Lemma 20.10.1. The functor given by Equation (20.10.0.1) is an exact functor (see Homology, Lemma 12.7.2).

**Proof.**
For any open $W \subset U$ the functor $\mathcal{F} \mapsto \mathcal{F}(W)$ is an additive exact functor from $\textit{PMod}(\mathcal{O}_ X)$ to $\text{Mod}_{\mathcal{O}_ X(U)}$. The terms $\check{\mathcal{C}}^ p(\mathcal{U}, \mathcal{F})$ of the complex are products of these exact functors and hence exact. Moreover a sequence of complexes is exact if and only if the sequence of terms in a given degree is exact. Hence the lemma follows.
$\square$

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