Definition 69.6.2 (0724)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 69: Cohomology of Algebraic Spaces
Section 69.6: The alternating Čech complex
Definition 69.6.2 (cite)

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Definition 69.6.2. 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 complex¹ $\check{\mathcal{C}}^\bullet _{alt}(f, \mathcal{F})$ associated to $\mathcal{F}$ and $f$ is the complex

$\mathop{\mathrm{Hom}}\nolimits (K^0, \mathcal{F}) \to \mathop{\mathrm{Hom}}\nolimits (K^1, \mathcal{F}) \to \mathop{\mathrm{Hom}}\nolimits (K^2, \mathcal{F}) \to \ldots$

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

[1] This may be nonstandard notation

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