Definition 68.6.2. Let $S$ be a scheme. Let $f : U \to X$ be a surjective étale morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be an object of $\textit{Ab}(X_{\acute{e}tale})$. The *alternating Čech complex*^{1} $\check{\mathcal{C}}^\bullet _{alt}(f, \mathcal{F})$ associated to $\mathcal{F}$ and $f$ is the complex

with Hom groups computed in $\textit{Ab}(X_{\acute{e}tale})$.

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