The Stacks project

Lemma 84.17.2. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. The Čech complex of Lemma 84.9.2 associated to $a^{-1}\mathcal{F}$

\[ a_{0, *}a_0^{-1}\mathcal{F} \to a_{1, *}a_1^{-1}\mathcal{F} \to a_{2, *}a_2^{-1}\mathcal{F} \to \ldots \]

is equal to the complex $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (s(\mathbf{Z}_{F(K)}^\# ), \mathcal{F})$. Here $s(\mathbf{Z}_{F(K)}^\# )$ is as in Hypercoverings, Definition 25.4.1.

Proof. By Lemma 84.15.2 we have

\[ a_{n, *}a_ n^{-1}\mathcal{F} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits '(F(K_ n)^\# , \mathcal{F}) \]

where $\mathop{\mathcal{H}\! \mathit{om}}\nolimits '$ is as in Sites, Section 7.26. The boundary maps in the complex of Lemma 84.9.2 come from the simplicial structure. Thus the equality of complexes comes from the canonical identifications $\mathop{\mathcal{H}\! \mathit{om}}\nolimits '(\mathcal{G}, \mathcal{F}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathbf{Z}_\mathcal {G}, \mathcal{F})$ for $\mathcal{G}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. $\square$

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